The Pennsylvania State UniversityThe Graduate School

Eberly College of Science

ON TWO PROBLEMS IN ANALYSIS ON MANIFOLDS

A Dissertation inMathematics

byJinpeng Lu

© 2019 Jinpeng Lu

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Doctor of Philosophy

May 2019

The dissertation of Jinpeng Lu was reviewed and approved∗ by the following:

Dmitri BuragoDistinguished Professor of MathematicsDissertation Advisor, Committee Chair

Mark LeviProfessor of MathematicsDepartment Head

Alexei NovikovProfessor of MathematicsAssociate Head for Graduate Studies

Mark StrikmanDisinguished Professor of Physics

∗Signatures are on file in the Graduate School.

ii

AbstractThis dissertation contains meaningful results on two problems.1. I prove that the spectrum of the Laplace-Beltrami operator with the Neumannboundary condition on a compact Riemannian manifold with boundary admits afast approximation by the spectra of suitable graph Laplacians on the manifold,and more generally similar graph approximation works for metric-measure spaceswhich are glued out of compact Riemannian manifolds of the same dimension.2. Given a diffeomorphism which is homotopic to the identity from the 2-torus toitself, we construct an isotopy whose norm is controlled by that of the diffeomor-phism in question.

iii

Table of Contents

List of Symbols v

Acknowledgments vii

Chapter 1Introduction 11.1 Graph approximations of the Laplacian spectra . . . . . . . . . . . 11.2 An isotopy between diffeomorphisms (of T2) . . . . . . . . . . . . . 9

Chapter 2Graph approximations for manifolds with boundary 132.1 Geometry of manifolds with boundary . . . . . . . . . . . . . . . . 132.2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 3Graph approximations for metric-measure spaces glued out of

manifolds 273.1 The Laplacian eigenvalue problem . . . . . . . . . . . . . . . . . . . 273.2 Proof of Theorem 2 and 3 . . . . . . . . . . . . . . . . . . . . . . . 31

Chapter 4An isotopy between diffeomorphisms of T2 394.1 The curve shortening flow on T2 . . . . . . . . . . . . . . . . . . . . 394.2 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

References 48

iv

List of Symbols

M A compact Riemannian manifold of dimension n without boundary orwith smooth boundary ∂M , or a metric-measure space glued out ofcompact Riemannian manifolds of the same dimension, as specified atthe beginning of each chapter.

νn The volume of the unit ball in Rn.

Br(x) The standard open geodesic ball of radius r of a manifold centered at apoint x.

i(M) The injectivity radius of a manifold M as specified in Section 2.1.

X Short for Xε: a ε-net xiNi=1 on M for ε ≪ 1. N is the number of thepoints of the net.

Vi A partition of M satisfying Vi ⊂ Bε(xi).

µi The Riemannian volume of Vi assigned as the weight of the vertex xi,which defines a discrete measure µ on the manifold.

Γ Short for the weighted graph Γε,ρ = Γ(Xε, µ, ρ) with ε≪ ρ≪ 1 definedin the third paragraph of Chapter 1.

||δu||2 The energy of a function u on X defined at the beginning of Section2.1.

λk(Γ) The k-th eigenvalue of a graph Laplacian −∆Γ defined as (1.1.2) or(1.1.4).

L(γ) The length of a curve γ.

d A distance-like function for a Riemannian distance function d definedin (1.1.1).

v

Br(x) The reflected cap y : d(x, y) < r.

Br(x) The disjoint union of Br(x) and Br(x).

Er(f, V ) Defined in (2.2.1) for an L2 function f of M and a subset V ⊂ M . Inparticular, denote Er(f,M) simply by Er(f).

d A distance-like function defined by y : d(x, y) < r = Br(x).

Br(0) The open ball of radius r centered at the origin in the tangent spaceover a point on a manifold.

Wr(x) The set of tangent vectors of lengths small than r in the tangent spaceover x, whose images via the exponential map have multiple collisionswith the boundary of the manifold, as specified in Section 2.1.

P The discretization operator defined at the beginning of Section 2.2.

P ∗ The operator which maps a discrete function on a manifold to a piece-wise constant function, defined at the beginning of Section 2.2.

I The interpolation (smoothing) operator defined in the latter half ofSection 2.2.

S The whole gluing locus ∪i<j(Mi ∩Mj), in the case of M = ∪Mi being ametric-measure space glued out of compact Riemannian manifolds Mi.

di The Riemannian distance function of the manifold Mi.

Ωijr Defined by x ∈Mi : di(x,Mi ∩Mj) < r.

Ωr The union of Ωijr for all i = j.

Φ The generic notation of the assumed homeomorphisms Φij betweensmall neighborhoods of the gluing locus in the Assumption.

xj The mirror image of x in Mj defined immediately following the Assump-tion.

F A Cm,α homotopically trivial diffeomorphism of T2 with m > 3 andα ∈ (0, 1).

vi

Acknowledgments

First of all, I would like to express my deepest gratitude to my advisor ProfessorDmitri Burago for your patient guidance and countless fruitful discussions overthe years. Your valuable critiques of my work always keep me on track. You arethere every step of the way to help me succeed in academia, from searching formy favourite project to wholeheartedly supporting my postdoc applications, andfrom painstakingly scrutinizing my academic writing word by word to finding meopportunities for collaborations on exciting new projects.

I am particularly grateful to Professor Sigurd Angenent, Professor Mark Leviand Professor Alexei Novikov for their interest in my work and many helpful dis-cussions and remarks, and Professor Mark Strikman for serving on my doctoralcommittee. I would also like to thank Professor Anton Petrunin for several valu-able discussions on a problem which does not appear in this dissertation. Thankall the faculty and staff in our department for their effort to make graduate studieshere a memorable experience.

I cannot put into words my gratitude to my parents, my grandparents, my fam-ily and friends for their emotional support and everything. I wish my grandfatherwere here to see.

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Chapter 1 |Introduction

1.1 Graph approximations of the Laplacian spectraThe problem is inspired by D. Burago, S. Ivanov and Y. Kurylev’s work [7] forclosed manifolds. They showed that the spectrum of the Laplace-Beltrami opera-tor on a closed Riemannian manifold admits a fast approximation by the spectrumof a weighted graph Laplacian (a finite dimensional matrix) on the manifold aslong as the graph is dense enough. The advantage of this method is that it avoidsthe triangulation of a manifold and does not require data points to be distributedin any regular way. From this construction, one can use rough local data to recoverglobal information such as the spectrum of the classical Laplacian. Graph Lapla-cians and their continuous (as opposed to discrete) analogues, called ρ-Laplaciansin [8], can be defined not only on manifold structures but also for general metric-measure spaces. It was proved in [8] that for a large class of metric-measure spaces,the convergence of the spectra of graph Laplacians is equivalent to the convergenceof those of ρ-Laplacians. In the first part of this dissertation, I introduce a fast ap-proximation of the Laplacian spectra by suitable graph Laplacians (and hence byρ-Laplacians), for compact Riemannian manifolds with boundary and more gener-ally metric-measure spaces which are glued out of compact Riemannian manifoldsof the same dimension [20]. The result is stated as Theorem 1-3.

My result implies that the closeness between the spectrum of the classical Lapla-cian and the spectra of graph Laplacians extends beyond manifolds. To considera potential counterexample where no restriction on volume growth is assumed,we are trying to find out if the convergence holds for spaces which are glued outof manifolds of different dimensions (e.g. a ball with a segment attached, where

1

the segment is equipped with 1-dimensional Lebesgue measure) and if the limit isclosely related to the classical Laplacian. Studying such space could lead to a newunderstanding of graph Laplacians and ρ-Laplacians, considering that the doublingcondition of volume growth fails there. So far we are aiming at proving the conver-gence of the spectra of graph Laplacians and ρ-Laplacians holds for a large classof metric-measure spaces, presumably only with volume growth assumptions. Fur-thermore, under what conditions on a metric-measure space can one guarantee thatgraph Laplacians and ρ-Laplacians converge to some operator in a proper sense?On the other hand, we are considering similar approximations of other Laplacianssuch as the Riemannian connection Laplacian [9] and the Hodge Laplacian. In thecase of 1-forms, the discretized Hodge Laplacian can be presented through tripletsof points since the discretization of 1-forms is simply functions of two points, andthe rest is to obtain necessary estimates. However for n-forms, the formulation ofthe discretized Hodge Laplacian is not yet clear. Once completed, this work couldgive a way to study topological invariants such as Betti numbers via operators onfunctions on graph structures.

Suppose M is an n-dimensional compact Riemannian manifold (without bound-ary or with smooth boundary ∂M), and we present a weighted graph structure foran arbitrary net of the manifold, which was introduced in [7] and is used through-out this dissertation. First we fix two positive parameters ε, ρ with ε ≪ ρ ≪ 1.Suppose Xε = xiNi=1 (for short X) is a finite ε-net of the manifold; it forms thevertices of our graph. Two vertices xi, xj are connected by an edge, denoted byxi ∼ xj, if and only if their Riemannian distance is smaller than ρ. We take anypartition of the manifold into measurable subsets Vi satisfying that Vi ⊂ Bε(xi),where Bε(xi) is the standard geodesic ball of the manifold around xi with theradius ε. Then we assign the Riemannian volume of Vi as the weight µi to eachvertex xi. Denote this weighted graph by Γε,ρ = Γ(Xϵ, µ, ρ) (for short Γ) withε ≪ ρ ≪ 1. The space of functions on X is equivalent to RN . In the base case ofclosed manifolds, for any function u on X, the weighted graph Laplacian is definedby

∆Γu(xi) =2(n+ 2)

νnρn+2

∑j:xj∼xi

µj(u(xj)− u(xi)),

where νn is the volume of the unit ball in Rn. This is a nonpositive self-adjoint op-

2

erator with respect to the weighted discrete L2(X) norm. Its continuous analogue,called the ρ-Laplacian, is defined in [8] with a single parameter ρ. It was provedin [7] that the spectra of the weighted graph Laplacians converge to the spectrumof the Laplace-Beltrami operator as ρ+ ε/ρ→ 0, which, in the light of [8], impliesthe convergence of the spectra of ρ-Laplacians as ρ→ 0.

In the case of manifolds with boundary, the graph Laplacian needs to be definedin order to gather sufficient information from the boundary, resulting in a slightlydifferent form. Let d be the Riemannian distance function on M , and define d asthe infimum over the lengths of all curves which collide (not tangentially) with theboundary ∂M . Namely

d(x, y) = infγL(γ(t)) | γ(0) = x, γ(1) = y, γ(t0) ∈ ∂M for some isolated

t0 ∈ [0, 1] satisfying γ′(t±0 ) /∈ Tγ(t0)∂M, (1.1.1)

and we call the minimizers reflected geodesics. Note that d is not a distance.If d is sufficiently small, it is realized by a unique reflected geodesic (with onlyone collision) which is characterized by reversing the normal component of thetangent vector at the collision point with respect to the boundary and preservingthe tangential component (Lemma 2.1.1). One can proceed to define the injectivityradius i(M) for a manifold M with boundary, as specified in Section 2.1. We nowdefine the graph Laplacian operator ∆Γ by

∆Γu(xi) =2(n+ 2)

νnρn+2[

∑j:d(xi,xj)<ρ

µj(u(xj)− u(xi)) +∑

j:d(xi,xj)<ρ

µj(u(xj)− u(xi))],

(1.1.2)where νn is the volume of the unit ball in Rn. Note that ∆Γ is a finite dimensionallinear operator−a matrix of dimension N , where N is the total number of vertices.Denote the k-th eigenvalue of the graph Laplacian −∆Γ (1.1.2) by λk(Γ), and the k-th eigenvalue of the Laplace-Beltrami operator −∆M with the Neumann boundarycondition by λk(M). I prove that for every nonnegative integer k 6 N(Γ) − 1,where N(Γ) is the number of vertices of the graph Γ, λk(Γ) converges to λk(M) asρ+ ε

ρ→ 0. More precisely,

Theorem 1. Suppose M is an n-dimensional compact Riemannian manifold withsmooth boundary ∂M . Assume all sectional curvatures of both M and ∂M are

3

bounded by K1 and |∇R| 6 K2, where ∇R is the covariant derivative of the cur-vature tensor R of M . Then for every nonnegative integer k 6 N(Γε,ρ)− 1, thereexists ρ0 = ρ0(n,K1, K2, λk(M), i(M)) and C = C(n,K1, K2), such that for anyρ < ρ0, one has

|λk(Γε,ρ)− λk(M)| 6 C(ρ+ε

ρ)λk(M) + Cρλk(M)

32 .

Consequently, the eigenfunctions (see Remark 1) of graph Laplacians converge to arespective eigenfunction of the Laplace-Beltrami operator with the Neumann bound-ary condition in L2(M).

Remark 1. We need a precise statement for the convergence of eigenfunctions ofgraph Laplacians, since they are not actually L2(M) functions. Given a functionu on the vertices X = xiNi=1, we can define a piecewise constant function P ∗u onM by P ∗u =

∑Ni=1 u(xi)1Vi

. In Theorem 1 and the two theorems that follow, theconvergence of eigenfuctions of graph Laplacians means exactly the convergence ofthe piecewise constant functions defined via the operator P ∗.

Now suppose M is a more general metric-measure space which is isometricallyglued out of n-dimensional compact Riemannian manifolds Mi without boundaryor with smooth boundary, and every connected component of each gluing locusMi ∩Mj(i = j) is a submanifold of both Mi and Mj of codimension at least 1.Denote the whole gluing locus by S = ∪i<j(Mi ∩Mj). Without loss of generality,we can assume each Mi ∩ Mj(i = j) to be connected. Denote the Riemanniandistance of Mi by di, and the r-neighborhood of the gluing locus within Mi byΩij

r = x ∈ Mi : di(x,Mi ∩ Mj) < r and Ωr = ∪i,jΩijr , whenever i = j and

Mi ∩Mj = ∅. Define the reflected r-neighborhood by

Ωijr = Ωij

r ⊔ x ∈Mi : di(x,Mi ∩Mj) < r, (1.1.3)

where the union is a disjoint union. We impose the following the assumption onthe space M . The motivation behind this assumption is explained at the end ofthis section.

Assumption. There exists 0 < r0 < 1, such that for any i = j and Mi ∩Mj = ∅,Ωij

r0is homeomorphic to Ωji

r0, and the homeomorphisms Φij : Ω

ijr0→ Ωji

r0satisfy:

4

(1) Φij|Mi∩Mj= Id|Mi∩Mj

;(2) Φij are diffeomorphisms outside of S where their differential may be multi-fold;(3) For any ϵ > 0, there exists r1 < r0, such that for any r < r1 and any vector vin tangent spaces over Ωij

r ,

| |(∇Φij)(v)||v|

− 1| < ϵ.

Remark 2. Denote [∇Φ]r = infv|(∇Φ)v|

|v| and ||∇Φ||r = supv|(∇Φ)v|

|v| , where v rangesover all vectors in tangent spaces over Ωij

r . The third condition can be stated as:both [∇Φ]r and ||∇Φ||r converge to 1 as r → 0.Suppose for one moment that all manifolds Mi are identical, the condition that eachnonempty gluing locus is a submanifold can be removed, since the homeomorphismsrequired in the Assumption automatically exist, no matter how wild the gluing locusis.

In other words, the r-neighborhood of the gluing locus within one manifoldpart can be mapped to r-neighborhoods within other manifold parts via almostisometries up to an infinitesimal error as r → 0. This assumption gives us thedesired symmetry: we can map a ball in tangent spaces via the homeomorphismsto almost another ball with little distortion. One can immediately see why weuse the reflected neighborhood. Suppose the gluing locus is the boundary of onemanifold and does not intersect with the boundary of the other manifold. Theregular neighborhood of the gluing locus within the first manifold is collar whiletubular within the other manifold. The homeomorphism satisfying the Assump-tion clearly does not exist. This is why we need the reflected neighborhood tomatch the neighborhoods in the first place. It is also worth pointing out that ther0−neighborhoods described in the Assumption do not have to be strict neighbor-hoods within the distance r0 from the gluing locus. In fact, the existence of anysmall open neighborhoods of the gluing locus and their mapping homeomorphismssatisfying the conditions will suffice.

If each nonempty gluing locus Mi ∩Mj(i = j) is a codimension 1 submanifoldwithout boundary, the Assumption is satisfied thanks to the dimension homogene-ity condition. The homeomorphisms can be constructed via the gradient flow withunit velocity associated to h(x) = d(x, S), which is essentially generated by thegeodesics with unit normal vector. Starting from the gluing locus, the points at

5

equal small time along the flow within each manifold are mapped to each other.Due to the estimates of the Jacobi field, we have ||(∇Φ)v| − |v|| 6 C|v|3 for allv ∈ Br(0) in tangent spaces over Ωr for r ≪ 1.

By virtue of the Assumption, we are able to recover the symmetry near the glu-ing locus as follows. We use the same construction and notations for the weightedgraph Γε,ρ = Γ(Xε, µ, ρ) on the metric-measure space in question as in the manifoldcase. The only requirement in this case is on the partition that every Vi is con-tained in one single manifold. That is to say: we do not choose vertices xi from thegluing locus. For any ρ < 1

10r

430 ≪ 1 and a point x ∈ Mi ⊂ M within ρ

34 distance

(the Riemannian distance of Mi) from a nonempty gluing locus Mi ∩Mj(i = j),we define mirror images xj ∈ Mj of x, by xj = Φij(x). The power 3

4is chosen to

reconcile relevant inequalities to produce the convergence, as we will see later inthe proof. Actually the power can be chosen to be any number between 2

3and 1 to

guarantee the convergence. And for x, y ∈ M , define d(x, y) = minl dl(xl, yl) and

the reflected case d(x, y) = minl dl(x

l, yl), wherever the mirror images are defined.For simplicity, we just write them as d and d. Note that in the case of M beingone single manifold, these notations of "distances" collapse to the notations in theprevious discussed manifold case. The graph Laplacian in this case is

∆Γu(xi) =2(n+ 2)

νnρn+2[

∑j:d(xi,xj)<ρ

µj(u(xj)− u(xi)) +∑

j:d(xi,xj)<ρ

µj(u(xj)− u(xi))],

(1.1.4)where νn is the volume of the unit ball in Rn. Keep in mind that the "distances" dand d used here involve all the mirror images. On the other hand, the Laplacianeigenvalue problem with the Neumann boundary condition can be defined weaklyon the metric-measure space in question as shown in Section 3.1, and denote the k-th eigenvalue by λk(M). For a codimension 1 gluing locus, a natural Kirchhoff-typecondition is imposed: the sum of all normal derivatives at the gluing locus vanishes.For higher codimension there is no restriction at the gluing locus, meaning that thespaces can be regarded as disjoint for the purpose of spectra. The k-th eigenvalueof the graph Laplacian (1.1.4) is denoted by −λk(Γ). I prove the following result.

Theorem 2. Suppose M is a metric-measure space which is isometrically gluedout of n-dimensional compact Riemannian manifolds Mi without boundary or withsmooth boundary, and every connected component of each nonempty gluing locus

6

Mi ∩ Mj(i = j) is a submanifold of both Mi and Mj of codimension at least 1,and satisfies the Assumption. Then for every nonnegative integer k 6 N(Γε,ρ)− 1,λk(Γε,ρ) converges to λk(M) as ρ+ ε

ρ→ 0.

Consequently the eigenfunctions of graph Laplacians converge to a respective eigen-function of the Laplacian eigenvalue problem with the Neumann boundary conditionin L2(M).

Thanks to the dimension homogeneity condition, the Assumption is satisfiedby a large class of the metric-measure spaces in question. In particular, if eachnonempty gluing locus Mi ∩Mj(i = j) is a codimension 1 submanifold withoutboundary, the Assumption is satisfied, which leads to an explicit convergence rate.

Theorem 3. Suppose M is a metric-measure space which is isometrically gluedout of m number of n-dimensional compact Riemannian manifolds Mi with smoothboundary or without boundary, and every connected component of each nonemptygluing locus Mi ∩Mj(i = j) is a codimension 1 submanifold of both Mi and Mj

without boundary. Assume all sectional curvatures of both Mi and ∂Mi are boundedby K1 and |∇Ri| 6 K2 for all i, where ∇Ri is the covariant derivative of thecurvature tensor Ri of Mi. Then for every nonnegative integer k 6 N(Γε,ρ) −1, there exists ρ0 = ρ0(n,m,K1, K2, r0, λk(M),minl6m i(Ml),maxj6k ||∇fj||∞) andC = C(n,m,K1, K2, voln−1(S),maxj6k ||∇fj||∞), such that for any ρ < ρ0, onehas

|λk(Γε,ρ)− λk(M)| 6 C(ρ14 +

ε

ρ)λk(M) + Cρ

14λk(M)

32 ,

where where the voln−1(S) is the n − 1 dimensional volume of the gluing locus Sand fj is the j-th normalized eigenfunction of the Laplacian eigenvalue problemwith the Neumann boundary condition on M with respect to the eigenvalue λj.

Theorem 3 can be generalized to gluing loci of higher codimensions. The con-dition that each gluing locus has no boundary can also be relaxed, as long as theAssumption is satisfied. Generally if the gluing locus has boundary, the conver-gence rate heavily depends on the geometry of the gluing locus. However, thereis one clear case: if each nonempty gluing locus, possibly with smooth boundary,does not intersect with the boundary of any manifold part, the convergence holdsand the convergence rate stays the same.

7

Ideas of the proof

The blueprint of the proof follows from the original proof for manifolds withoutboundary in [7], which is constructing a discretization operator and a smoothingoperator to connect functions on graphs and smooth functions via min-max for-mulas. The main obstruction of proving the convergence in our cases is that thekey estimates are sharp for closed manifolds and heavily disrupted by any possiblesingularity (manifold boundary is viewed as a type of singularity), due to majorchanges in the symmetry and the volumes of small balls. The discretization opera-tor is still there, but the same thing cannot be said about the smoothing operator,which needs to be constructed to satisfy the following two conditions:(1) The smoothing operator should send every discrete function to a Lipschitz func-tion. This ensures a proper connection via min-max formulas between discrete sideand continuous side. For metric-measure spaces which are glued out of manifoldparts, we use a proper cut-off function to ensure continuity.(2) The smoothing operator should send constant discrete functions to constantsmooth functions within a proper error. Suppose the averaging radius of thesmoothing operator is r ≪ 1 and it turns out that the error of pointwise derivativeshas to be better than r−1, which cannot be achieved without the help of some sym-metry. For manifolds with boundary, this can be resolved simply by doubling themanifold. More precisely, geodesics can be extended through the classical reflec-tion upon hitting the boundary, and due to the fact that the geodesic flow along areflected geodesic preserves the Liouville measure on the unit tangent bundle, keyestimates can be obtained in the same way as in the base case. Since constant dis-crete functions on graphs are clearly eigenfunctions of the graph Laplacians withrespect to zero eigenvalue, it is no surprise that eventually the spectra of graphLaplacians approach the spectrum of the Laplace-Beltrami operator with the Neu-mann boundary condition.For metric-measure spaces glued out of manifolds, the singularities are much moreserious. For instance, a standard metric ball near a gluing locus can intersect withmultiple manifold parts and have an arbitrary shape and volume. This asymme-try of balls caused by the gluing worsens estimates and consequently destroys theconvergence. It is crucial to find a uniform and consistent way of averaging, whichintuitively works as follows. Consider the simple space M of three planar rectan-

8

gles M1,M2,M3 glued along an edge (e.g. book pages) and a point x ∈ M1 nearthe edge. We take the two points corresponding to x in the other two rectanglesin the most natural way, since a small neighborhood of the gluing locus withinM1 is isometric to some small neighborhoods within M2,M3. Instead of averagingin a standard metric ball of M around x, we take three seperate balls: the stan-dard geodesic ball of M1 around x, and the two standard geodesic balls of M2,M3

around the corresponding points of x in the respective rectangle. In this way, thesymmetry of balls is restored. Such construction can be done for the spaces inquestion as long as the Assumption is satisfied.

1.2 An isotopy between diffeomorphisms (of T2)Given a diffeomorphism which is homotopic to the identity from a closed surface toitself, it is known that one can improve the homotopy to an isotopy in a topologicalway [12] and further improve the differentiability of the isotopy [5], but the knownprocedures are subtle, causing the norms of the isotopy being difficult to control.In the second part of this dissertation, we introduce a procedure to construct a newisotopy on the 2-torus generated by a time-dependent vector field, and the normsof the isotopy are controlled in terms of the norms of the initial diffeomorphism.This is a joint work with D. Burago and T. Ozuch.

This problem of finding a canonically defined isotopy between diffeomorphismswith controlled norms, seemingly naive, is non-trivial even for a 2-torus. Given asequence of diffeomorphisms of, say a 2-torus, they may be all conjugate to eachother, or their conjugates converge (meaning they are the same or almost the samein some coordinate systems). This is one of the main motivations for the abovementioned problem. One method of detecting this is to find a conjugation-invariantnorm which grows to infinity on the diffeomorphims in question. Another approachis finding a canonical (normal) form, which is what we are trying in our paper.

Given a horizontal geodesic on the standard torus S1×S1, the initial diffeomor-phism can send it to a complicated embedded closed curve if the diffeomorphismhas large norms. One of the most efficient ways to straighten this complicatedcurve is to apply the curve shortening flow along which every point moves withthe velocity that is the geodesic curvature. This geometric flow behaves like a heat

9

equation keeping all the evolving curves embedded, and singularity does not occurunless the flow shrinks to a point (see e.g. [14] [15] [16]). The latter case doesnot happen, since our diffeomorphism is homotopic to the identity and hence thecurve in question is not contractible. The flow at first straightens the curve rapidly,though possibly introducing points with very large but finite curvature, and slowsdown once the curvature becomes small. When the curvature is sufficiently smallthe curve becomes simply a graph (as a function of the second coordinate in theproduct S1 × S1), in which case straightening the curve is a simple task.

Consider the whole family of horizontal geodesics on the standard torus S1×S1.Then apply the curve shortening flow to their images1 via the diffeomorphism F

all at once up to an explicit time T0 = (||F ||C1 + 1)2. By a compactness argument(Proposition 4.1.2), the curvatures are uniformly bounded for all such flows up totime T0. Denote the uniform curvature bound by K.

Theorem 4. Suppose F is a Cm,α (m > 3, α ∈ (0, 1)) diffeomorphism of the2-torus and it is homotopic to the identity. Then there is a C [m−1

2],α isotopy (gen-

erated by a time-dependent vector field) between F and the identity, and the isotopyrestricted to each fixed time is a Cm−2,α diffeomorphism. Furthermore the norms ofthe isotopy are explicitly controlled depending only on ||F ||Cm,α , ||F−1||C1 , K,m, α,where K is the uniform curvature bound depending only on ||F ||C3,α , ||F−1||C1.

Remark 3. It turns out to be difficult to explicitly estimate the uniform curvaturebound K only in terms of the initial diffeomorphism. What we do know is thatthe bound depends only on certain norms of the initial diffeomorphism. Considera closed curve which winds many times within a small region. In a short timepart of the curve can shrink rapidly causing large curvature to appear. Howeveronce the curvature becomes too large, it cannot keep growing much more otherwisea singularity will form. The curve shortening flow is one of the fastest way todecrease large curvatures, but it appears to be too fast as far as we are concernedwith controlling the curvature of the curves and the derivatives of the isotopy.

We believe that the right characteristic which can be explicitly controlled alonga curve shortening procedure is the largest radius r(γ) (known as the reach of a

1By image we mean the curve with parametrization coming from the composition of thediffeomorphism and the natural parametrization of the horizontal geodesic.

10

curve) such that one can touch a curve γ by a ball of radius r at every point andeither side so that the interior of the ball does not intersect the curve. We believe wecan construct a “ tame” curve shortening procedure such that the following propertyholds: on the space of homotopically non-trivial curves γ with r(γ) > r0 on a 2-torus, the length functional has no critical points other than closed geodesics. (Inthe plane the critical points are circles of radius r0.) This enables us to explicitlycontrol the curvature k 6 1/r(γ0), where γ0 is the initial curve. The problem isthat this procedure is only at most C1,1, and its smoothening remains to be thedifficulty. This part is a work in progress.

The problem came from a discussion between D. Burago and L. Polterovich.We apply the curve shortening flow to all the embedded closed curves which arethe images of the horizontal geodesics via the diffeomorphism F at once, and theybecome a family of horizontal geodesics as time goes to infinity by Grayson’s Theo-rem (Theorem 4.1.1). We stop the curve shortening flow at some time after whichall the curves become graphs. Then we can simply move them back to the originalfamily of horizontal geodesics via the “ height function”, up to a reparametrization.Thus the time-dependent vector field defined by the curve shortening flow gener-ates an isotopy. Since the isotopy is defined through a family of curves, the naturalobstruction is the regularity of the isotopy in the transversal direction, which is es-sentially the smooth dependence of the curve shortening flow on initial conditions.It is settled by means of the smooth dependence of the solution of a fully nonlinearparabolic equation on parameters, based on the method in [3]. The details can befound in Section 4.2.

Theorem 4 provides a control on the isotopy between two homotopically triv-ial diffeomorphisms on the standard 2-torus S1 × S1. Without loss of generality,we assume that one of the diffeomorphisms is the identity. The theorem also ap-plies to homotopically non-trivial orientation-preserving diffeomorphisms becauseof the classical fundamental theorem of Dehn that the group of isotopy classes oforientation-preserving diffeomorphisms of S1×S1 is isometric to SL(2,Z) which isgenerated by two Dehn twists (see e.g. [12]), in which case similar control can beobtained in the same way. We prove Theorem 4 for the standard 2-torus S1 × S1.For a general 2-torus, there exists a diffeomorphism from it to the standard 2-torus. So all the distortions to the isotopy caused by the non-standard metric arecontrolled by the norms of that diffeomorphism. Our isotopy exists if the initial

11

diffeomorphism has lower regularity and one can expect a Lipschitz estimate onthe isotopy.

We think our isotopy may play an important role in defining an “ intrinsic” dis-tance between diffeomorphisms by finding the “ minimal” isotopy in a proper sense,considering how efficient this curve shortening procedure is. We do not know ifsimilar control on the isotopy can be obtained for S2. Motivated by works of A.Nabutovsky and S. Weinberger (see [22] [23] [24]), we have serious doubts if it ispossible for n-tori for n > 5. The main result of [22] implies that diffeomorphismsof Sn (n > 5) embedded in Rn+1 cannot be extended to the ambient space withalgorithmly controllable norms, and we think our work is closely related to thepossible extendability for the base dimension.

Organization of the dissertation

In Chapter 2, we prove Theorem 1 of graph approximations for manifolds withboundary. Then we turn to the case of M being a metric-measure space isomet-rically glued out of Riemannian manifolds of the same dimension in Chapter 3.Section 3.1 is devoted to studying the Laplacian eigenvalue problem and the prop-erties of the eigenfunctions on the metric-measure spaces in question. We thenprove Theorem 2 and consequently Theorem 3 in Section 3.2. Finally in Chapter4 we prove Theorem 4.

12

Chapter 2 |Graph approximations for manifoldswith boundary

2.1 Geometry of manifolds with boundaryIn this chapter, suppose M is an n-dimensional compact Riemannian manifoldwith smooth boundary ∂M . Assume all sectional curvatures of both M and ∂M

are bounded by K1 and |∇R| 6 K2, where ∇R is the covariant derivative ofthe curvature tensor R of M . In this section, we explain the geometry that weneed for manifolds with boundary. Let Xε = xiNi=1 be a finite ε-net on M andΓε,ρ = Γ(Xε, µ, ρ) with ε ≪ ρ ≪ 1 be the weighted graph defined in the previouschapter. For simplicity, we write X and Γ in short for Xε and Γε,ρ. Define

L2(X) = u : X → R, ||u||2 =N∑i=1

µi|u(xi)|2 <∞.

In L2(X), we have inner product < u, v >=∑N

i=1 µiu(xi)v(xi), for u, v ∈ L2(X).By straightforward calculations, the graph Laplacian ∆Γ (1.1.2) is self-adjoint andnegative semi-definite under this L2(X) inner product, and the discrete energy isgiven by

||δu||2 =< −∆Γu, u >=n+ 2

νnρn+2

∑i

(∑

j:d(xi,xj)<ρ

+∑

j:d(xi,xj)<ρ

)µiµj|u(xj)− u(xi)|2.

13

Denote by λk(Γ) the k-th eigenvalue of the graph Laplacian −∆Γ, and set λ0(Γ) = 0.The min-max principle applies:

λk(Γ) = minLk+1

maxu∈Lk+1−0

||δu||2

||u||2,

where Lk+1 ranges over all (k + 1)-dimensional subspaces of L2(X).On the other hand, ∆M is the Laplace-Beltrami operator on M and λk(M) is thek-th eigenvalue of −∆M with the Neumann boundary condition. Set λ0(M) = 0,and we also have the following min-max formula,

λk(M) = infQk+1

supf∈Qk+1−0

||∇f ||2L2(M)

||f ||2L2(M)

,

where Qk+1 ranges over all (k + 1)-dimensional subspaces of the Sobolev spaceW 1,2(M).

Now we study the property of the function d defined in (1.1.1). Consider astandard Riemannian geodesic (a locally shortest path) which collides with theboundary, we need a way to extend this geodesic further past the collision point.We define the reflected geodesic (with only one collision) by reversing the normalcomponent of the tangent vector of the standard geodesic at the collision point withrespect to the boundary and preserving the tangential component. We choose ourparameters to be smaller than the radius of the collar neighborhood of ∂M , toavoid the situations where a reflected geodesic could collide with far parts of theboundary. Due to the first variation formula,

Lemma 2.1.1. There exists r2 > 0 such that if d(x, y) < r2, d(x, y) is realized bya unique minimizing reflected geodesic.

The uniqueness is due to Theorem 4 in [1]. The choice of r2 depends on thetubular radius of M which is a single positive number reflecting all the curvaturebounds and the lower bounds of the injectivity radius of both ∂M and the interiorof M (see [1]). If a standard Riemannian geodesic touches the boundary some-where tangentially, it can extend both in the interior of M and in the boundary.By Theorem 3 in [2], there is exactly one geodesic which first unrolls from theboundary and extends in the interior of M , called the primary minimizer. This isthe one geodesic we choose in the tangential case, with the endpoint of this primary

14

minimizer being the image of the exponential map. In this way, the exponentialmap can be extended to the whole tangent space. However multiple collisions canstill happen, in which case collision points are nearby. We denote by Wr(x) ⊂ TxM

the set of tangent vectors of lengths smaller than r whose images via the extendedexponential map have multiple collision points. We argue that the set Wr(x) hassmall measure.

Lemma 2.1.2.m(Wr(x)) 6 C(n,K1)r

n+1, for r ≪ 1.

Proof. From a point on the boundary, a geodesic of length r can only collide withthe boundary at another point if the initial angle is small than K1r with respect tothe tangent space over the initial point. Therefore on the tangent space, Wr(x) issimply the complement of two antipodal hyperspherical caps with the polar angleπ/2−K1r. Its volume is controlled by Crn+1.

Denote by Br(x) the open geodesic ball of radius r centered at x ∈ M , andby Br(x) the open reflected geodesic ball of radius r defined as the disjoint unionof the standard geodesic ball Br(x) = y : d(x, y) < r and the reflected capBr(x) = y : d(x, y) < r. Note that Br(x) ⊂ Br(x), and Br(x) = ∅ if x is farfrom the boundary. By Lemma 2.1.1, there exists a maximal radius 0 < r < r2,such that for any x ∈ M and restricted outside of Wx, the extended exponentialmap expx : Br(0) − Wx → Br(x) is a homeomorphism, where Br(0) is the ballof radius r around the origin in the tangent space TxM . We define this maximalradius to be the injectivity radius i(M) of M. From now on, we set the radii ofall geodesic balls smaller than i(M). Due to an estimate on the Jacobi fields, wehave the following estimate on the volume of a reflected geodesic ball Br(x) whichis defined as vol(Br(x)) = vol(Br(x)) + vol(Br(x)).

Proposition 2.1.3. Consider the reflected geodesic ball Br(x), where d(x, ∂M) <

r << 1. Its volume has the following estimate

|vol(Br(x))− νnrn| 6 C(n,K1, K2)r

n+1,

where νn is the volume of the unit ball in Rn. This is due to an estimate of the

15

Jacobian of expx at v ∈ Br(0), denoted by Jx(v):

|Jx(v)− 1| 6 C(n,K1, K2)|v|.

Proof. Suppose γ(t)(t ∈ [0, t1]) is a geodesic with the unit tangent vector field T

reflected at p = γ(t0) ∈ ∂M . Let’s calculate the length of the Jacobi field J(t)

defined on the whole geodesic, with the initial condition J(0) = 0, J ′(0) = w.Denote by J1 and J2 the Jacobi field J before and after the collision. Then J1(t)

is well defined on [0, t0). Define J1(t0) and J ′1(t0) as the left limit of J1(t) and

J ′1(t) at t = t0. Now we define the initial condition of J2 by J2(t0) = J1(t0) andJ ′2(t0) = J ′

1(t0), where · is the reflection of vectors associated to the boundary ∂Mby reversing the sign of the normal component. In this way, J2(t) is well definedon [t0, t1]. The length of the whole J(t) is obviously smooth outside t = t0, so let’scheck the regularity at t = t0. Due to the fact that the reflection preserves theinner product, we have

< J2, J2 > |t=t0 =< J1(t0), J1(t0) >=< J1, J1 > |t=t0 ,

< J2, J2 >′ |t=t0 = 2 < J1(t0), J ′

1(t0) >=< J1, J1 >′ |t=t0 ,

and

< J2, J2 >′′ |t=t0 = 2 < J2(t0), J

′′2 (t0) > +2 < J ′

2(t0), J′2(t0) >

= 2R(T (t+0 ), J2(t0), T (t+0 ), J2(t0)) + 2 < J ′

1(t0), J′1(t0) >

= 2R(T (t−0 ), J1(t0), T (t−0 ), J1(t0)) + 2 < J ′

1(t0), J′1(t0) >

= < J1, J1 >′′ |t=t0

where T (t−0 ) and T (t+0 ) are the left and right limit of T at the collision pointp = γ(t0), and the second last equality is due to the following sublemma.

Sublemma. R(a, b, c, d) = R(a, b, c, d), where a, b, c, d ∈ TpM and p ∈ ∂M is thecollision point, and R is the curvature tensor of M .

Proof. Decompose the vectors into normal components and tangential componentswith respect to ∂M . The reflection preserves tangential components and reversesthe sign of normal components. The lemma directly follows from the anti-symmetry

16

of the curvature tensor.

So far we have proved |J(t)|2 ∈ C2. Let’s estimate the third derivative. Bystraightforward calculations,

< J, J >′′′ = 6 < J ′′, J ′ > +2 < J ′′′, J >

= 6R(T, J, T, J ′) + 2 < ∇TR(T, J)T, J >

= 2(∇R)(T, J, T, J, T ) + 8R(T, J, T, J ′)

= 2(∇R)(T, J, T, J, T ) + 4R(T, J + J ′, T, J + J ′)

−4R(T, J, T, J)− 4R(T, J ′, T, J ′),

where tensor ∇R is the covariant derivative of the curvature tensor R. Assume|K| 6 K1 and |∇R| 6 K2, and thus

| < J, J >′′′ (t)| 6 2K2|J |2 + 4K1|J + J ′|2 + 4K1|J |2 + 4K1|J ′|2

6 C(K1 +K2)(|J |2 + |J ′|2)

6 C(K1, K2)(t2 + o(t2) + 2 +O(t))

= C(K1, K2) +O(t).

Therefore for any t ∈ [0, t1] and t1 << 1, we get the estimate for the length of theJacobi field up to the third order

||J(t)|2 − t2| 6 C(K1, K2)t3.

The Jacobian estimate immediately follows. Note that the calculations above allowsituations of multiple collisions, and this Jacobian estimate proves the volumeestimate if we allow multiple collisions. Due to the fact that the set of the multiplecollisions Wr(x) has small measure (Lemma 2.1.2), we obtain the volume estimatefor our reflected geodesic ball which only allows one collision.

As a comparison, if x is far from the boundary with no reflection involved, thefirst error term would be C(n,K1)r

n+2.

17

2.2 Proof of Theorem 1

We prove Theorem 1 by obtaining the upper and lower bound of the eigenvaluesof the graph Laplacian in Proposition 2.2.3 and 2.2.7, following the method of theoriginal proof [7].

Upper bound for λk(Γ)

Given a weighted graph Γ = Γ(X,µ, ρ). Define the discretization map P : L2(M) →L2(X) by

Pf(xi) = µ−1i

∫Vi

f(x)dx,

and P ∗ : L2(X) → L2(M) by

P ∗u =N∑i=1

u(xi)1Vi.

Immediately we have ||P ∗u||L2(M) = ||u||L2(X). For f ∈ L2(M) and r ∈ (0, 2ρ),define

Er(f, V ) =

∫V

∫Br(x)

|f(y)− f(x)|2dydx, (2.2.1)

and Er(f) = Er(f,M). Here Br(x) is considered to be the disjoint union of thestandard geodesic ball Br(x) and the reflected cap Br(x). We obtain upper boundsof the discrete norm and energy by their continuous counterparts in the followingtwo lemmas.

Lemma 2.2.1. For f ∈ C∞(M),

Er(f) 6 (1 + Cr)νn

n+ 2rn+2||∇f ||2L2(M).

Proof. Take two exact copies of M and glue along the boundary via the identitymap, denoted by M . Take the function f as two exact copies of f . The distancebetween points on different copies are measured by reflected geodesics, as we dis-cussed in Section 2.1. M can be considered as a manifold without boundary. Byvirtue of Proposition 2.1.3, the volume of (reflected) geodesic balls on M withradius r ≪ 1 is preserved along (reflected) geodesics, therefore Lemma 3.3 in [7]

18

applies: ∫M

∫Br(x)

|f(y)− f(x)|2dydx 6 (1 + Cr)νn

n+ 2rn+2||∇f ||2

L2(M),

where the factor (1 + Cr) comes from the Jacobian estimate in Proposition 2.1.3.Take half on both sides of the inequality and our lemma follows.

Lemma 2.2.2. For f ∈ C∞(M),∫Vi

|f(x)− Pf(xi)|2dx 6 C

νn(r − ε)nEr(f, Vi), for every r ∈ (ε, 2ρ).

Proof. Fix x, y ∈ Vi and r > ε, and consider U = Br(x) ∩ Br(y). Observe that Ucontains the reflected geodesic ball of radius r−ε centered at the midpoint (definedby the standard Riemannian distance) between x and y. Thus the volume of Uhas lower bound vol(U) > Cνn(r− ε)n by Proposition 2.1.3. The rest of the prooffollows from Lemma 3.4 in [7].

Now with these two lemmas, we are able to bound the discrete norm and energyby the continuous norm and energy. Set r = (n + 1)ε, and by Lemma 2.2.1 andLemma 2.2.2,

||f − P ∗Pf ||2L2 =∑i

∫Vi

|f(x)− Pf(xi)|2dx

6 C

νn(r − ε)n

∑i

Er(f, Vi)

6 C

νn(r − ε)nEr(f)

6 C

n+ 2(

r

r − ε)nr2||∇f ||2L2(M)

6 Cε2||∇f ||2L2(M). (2.2.2)

On the other hand,

||δ(Pf)||2 =n+ 2

νnρn+2

∑i

(∑

j:d(xi,xj)<ρ

+∑

j:d(xi,xj)<ρ

)µiµj|Pf(xj)− Pf(xi)|2

6 n+ 2

νnρn+2

∑i

(∑

j:d(xi,xj)<ρ

+∑

j:d(xi,xj)<ρ

)

∫Vi

∫Vj

|f(x)− f(y)|2dydx

19

=n+ 2

νnρn+2

∫M

∫∪

xj∼xiVj ,x∈Vi

|f(y)− f(x)|2dydx

6 n+ 2

νnρn+2Eρ+4ε(f)

6 (1 + Cρ+ Cε

ρ)||∇f ||2L2 . (2.2.3)

The last inequality is due to Lemma 2.2.1. The second last inequality is dueto

∪xj∼xi

Vj ⊂ Bρ+4ε(x) (multiplicity counted). Combining the two inequalities(2.2.2) and (2.2.3), we get the following estimate.

Proposition 2.2.3.

λk(Γ) 6 (1 + Cρ+ Cε

ρ+ Cε

√λk)λk(M).

Proof. The proof can be found in Proposition 4.4 in [7].

Lower bound for λk(Γ)

Next we deal with the lower bound. Fix r << 1 and consider two kernels kr, kr :M ×M → R+ defined by

kr(x, y) = r−nψ(d(x, y)

r),

kr(x, y) = r−nψ(d(x, y)

r),

where ψ(t) = n+22νn

(1 − t2) if t ∈ [0, 1], otherwise 0. Note that the constant ischosen so that

∫Rn ψ(|x|)dx = 1. Now define the associated integral operator

Λr : L2(M) → C0,1(M) given by

Λrf(x) =

∫Br(x)

f(y)(kr(x, y) + kr(x, y))dy.

From Lemma 2.1.1, we know that for any r < i(M), the extended exponentialmap expx : Br(0) − Wr(x) → Br(x) is a homeomorphism. Here we treat Br(x)

to be the disjoint union of the standard geodesic ball Br(x) and the reflected cap

20

Br(x) = y : d(x, y) < r. A direct computation yields the following

∇xkr(x, y) =n+ 2

νnrn+2exp−1

x (y), for y ∈ Br(x),

∇xkr(x, y) =n+ 2

νnrn+2exp−1

x (y), for y ∈ Br(x). (2.2.4)

Note that the exponential map in the second equality is defined via a reflectedgeodesic. These two identities are the consequences of the derivatives of d and d.(∇xd(x, y) = − exp−1

x (y)d(x,y)

and ∇xd(x, y) = − exp−1x (y)

d(x,y).)

We prove the following three lemmas estimating relevant terms associated withthe norm and energy of Λr(P

∗u) for some discrete function u, where P ∗u is thepiecewise constant function defined by the discrete function u at the beginningof this section. These lemmas will enable us to bound the continuous norm andenergy by their discrete counterparts. First we find out how much the imageof a constant discrete function via the operator Λr approximates the continuousconstant function on both value and derivative level.

Lemma 2.2.4. Define θ(x) = Λr(1M). For almost every x ∈M , one has

|θ(x)− 1| 6 Cr,

and|∇θ(x)| 6 C.

Proof. By the definition of the operator Λr and the kernel kr,

θ(x) =

∫Br(x)

(kr(x, y) + kr(x, y))dy

= r−n

∫Br(x)

ψ(d(x, y)

r)dy + r−n

∫Br(x)

ψ(d(x, y)

r)dy

= r−n

∫Br(0)−Wr(x)

ψ(r−1|v|)Jx(v)dv,

where Wr(x) ⊂ TxM is the set of multiple collisions defined in Section 2.1. Since∫Br(0)

ψ(r−1|v|)dv = rn by our choice of ψ, the first inequality in the lemma followsfrom the Jacobian estimate in Proposition 2.1.2 and the estimate on Wr(x) inLemma 2.1.3. As for the second inquality, we differentiate θ(x) using (2.2.4). Keep

21

in mind that we treat Br(x) to be the disjoint union of the standard geodesic ballBr(x) and the reflected cap Br(x).

∇θ(x) =n+ 2

νnrn+2

∫Br(x)

exp−1x (y)dy +

n+ 2

νnrn+2

∫Br(x)

exp−1x (y)dy

=n+ 2

νnrn+2

∫Br(x)

exp−1x (y)dy

=n+ 2

νnrn+2

∫Br(0)⊂TxM

vJx(v)dv −n+ 2

νnrn+2

∫Wr(x)

vJx(v)dv.

Since∫Br(0)

vdv = 0, we can replace Jx(v) by Jx(v)− 1 for the first term. Then theJacobian estimate (Proposition 2.1.3) yields the estimate of the first term.

n+ 2

νnrn+2|∫Br(0)

vJx(v)dv| 6n+ 2

νnrn+2

∫Br(0)⊂TxM

|v|Crdv 6 C.

The second term is also bounded by a constant due to the estimate on the measureof Wr(x) in Lemma 2.1.2.

Note that the extra power of r generated by the symmetry is crucial. Theproof will not work without it. Now we define Ir(f) = θ−1Λr(f) for a functionf ∈ L2(M) which we will later replace by the piecewise constant function P ∗u fora discrete function u. The following two lemmas estimate the norm and energy ofIr(f) in terms of the norm of f and Er(f).

Lemma 2.2.5. For f ∈ L2(M),

||Irf − f ||2L2(M) 6C

νnrnEr(f).

Proof. The proof is straightforward. One can refer to similar arguments in [7].

Lemma 2.2.6. For f ∈ L2(M),

||∇(Irf)||2L2(M) 6 (1 + Cr)n+ 2

νnrn+2Er(f).

Proof. For any fixed x0 ∈M , by the definition of θ we have

θ−1Λrf(x) = f(x0) + θ−1

∫M

(f(y)− f(x0))(k(x, y) + kr(x, y))dy. (2.2.5)

22

Differentiating and evaluating it at the point x0 yields

∇(θ−1Λrf)(x0) = θ−1(x0)

∫M

(f(y)− f(x0))(∂

∂xk(x0, y) +

∂

∂xkr(x0, y))dy

+ ∇(θ−1)(x0)

∫M

(f(y)− f(x0))(k(x0, y) + kr(x0, y))dy

= θ−1(x0)A1(x0) + A2(x0),

whereA1(x) =

∫Br(x)

(f(y)− f(x))(∂

∂xk(x, y) +

∂

∂xkr(x, y))dy,

andA2(x) = ∇(θ−1)

∫Br(x)

(f(y)− f(x))(k(x, y) + kr(x, y))dy.

Since |θ − 1| 6 Cr, we have

||∇(θ−1Λrf)||L2 6 (1 + Cr)||A1||L2 + ||A2||L2 . (2.2.6)

First, we estimate ||A1||L2 . Fix x ∈M , set ω = A1(x)|A1(x)| and we have

|A1(x)| = < A1(x), w >

=n+ 2

νnrn+2

∫Br(x)

(f(y)− f(x)) < exp−1x (y), w > dy

+n+ 2

νnrn+2

∫Br(x)

(f(y)− f(x)) < exp−1x (y), w > dy

=n+ 2

νnrn+2

∫Br(x)

(f(y)− f(x)) < exp−1x (y), w > dy

=n+ 2

νnrn+2

∫Br(0)−Wr(x)

(f(expx(v))− f(x)) < v,w > Jx(v)dv.

By the Cauchy-Schwartz inequality,

|A1(x)|2 6 (n+ 2

νnrn+2)2(

∫Br(0)

|f(expx(v))− f(x)|2|Jx(v)|2dv)(∫Br(0)

< v,w >2 dv)

=n+ 2

νnrn+2

∫Br(0)

|f(expx(v))− f(x)|2|Jx(v)|2dv.

The last equality is due to the following argument. Expand w = w1 to an or-thonormal basis wi of TxM , 1 6 i 6 n. Since |v|2 =

∑i < v,wi >

2, we have

23

∫|v|<r

|v|2dv =∑

i

∫|v|<r

< v,wi >2 dv. Due to the symmetry of Br(0) ⊂ TxM , we

have for any i, ∫|v|<r

< v,wi >2 dv =

1

n

∫|v|<r

|v|2dv =νn

n+ 2rn+2. (2.2.7)

Hence by Lemma 2.2.4,

||A1(x)||2L2 6 n+ 2

νnrn+2

∫M

∫Br(x)

|f(y)− f(x)|2|Jx(exp−1x (y))|2dydx

6 (1 + Cr)n+ 2

νnrn+2Er(f).

Next we estimate ||A2||L2 . Since θ 6 C, |∇θ| 6 C(Lemma 2.2.4) and |kr|, |kr| 6C

νnrn, we get

|A2(x)|2 6 |∇(θ−1)|2(∫Br(x)

(kr + kr)dy)(

∫Br(x)

|f(y)− f(x)|2(kr + kr)dy)

6 |∇(θ−1)|2θ(x) C

νnrn

∫Br(x)

|f(y)− f(x)|2dy

6 C

νnrn

∫Br(x)

|f(y)− f(x)|2dy.

Therefore,

||A2(x)||2L2 6C

νnrn

∫M

∫Br(x)

|f(y)− f(x)|2dydx =C

νnrnEr(f).

Finally combining the estimates on the norm of A1 and A2, by (2.2.6) we obtain

||∇(θ−1Λrf)||L2 6 (1 + Cr)||A1||L2 + ||A2||L2

6 ((1 + Cr)32 + Cr)

√n+ 2

νnrn+2Er(f)

6 (1 + Cr)

√n+ 2

νnrn+2Er(f).

Now we replace the L2(M) function f in the previous lemmas by the piecewiseconstant function P ∗u for a discrete function u ∈ L2(X). The only thing left is

24

to estimate Er(P∗u) in terms of the discrete energy ||δu||2. By the definition of

discrete energy ||δu||2, we have

||δu||2 =n+ 2

νnρn+2

∑i

(∑

j:d(xi,xj)<ρ

+∑

j:d(xi,xj)<ρ

)µiµj|u(xj)− u(xi)|2

=n+ 2

νnρn+2

∫M

∫∪

xj∼xiVj ,x∈Vi

|P ∗u(y)− P ∗u(x)|2dydx

> n+ 2

νnρn+2Eρ−4ε(P

∗u),

The last inequality is due to Bρ−4ε(xi) ⊂∪

j:xj∼xiVj (multiplicity counted). There-

fore we get

Eρ−4ε(P∗u) 6 νnρ

n+2

n+ 2||δu||2. (2.2.8)

Now set r = ρ − 4ε, and define Iu = Iρ−4ε(P∗u) = θ−1Λρ−4ε(P

∗u), for u ∈L2(X). Insert (2.2.8) into the estimates in Lemma 2.2.5 and 2.2.6, and we finallybound the norm and energy of Iu by the discrete norm and energy of u.

||Iu− P ∗u||2L2(M) 6 C

νn(ρ− 4ε)nEρ−4ε(P

∗u)

6 C(ρ

ρ− 4ε)nρ2||δu||2

6 Cρ2||δu||2, (2.2.9)

and

||∇(Iu)||2L2(M) 6 (1 + Cρ)n+ 2

νn(ρ− 4ε)n+2Eρ−4ε(P

∗u)

6 (1 + Cρ)(ρ

ρ− 4ε)n+2||δu||2

6 (1 + Cρ+ Cε

ρ)||δu||2. (2.2.10)

With (2.2.9) and (2.2.10), a similar argument as Proposition 2.2.3 shows thefollowing estimate.

Proposition 2.2.7.

λk(Γ) > (1− Cρ− Cε

ρ− Cρ

√λk)λk(M).

25

This proposition together with Proposition 2.2.3 proves Theorem 1. The con-vergence of eigenfunctions is a direct consequence of the estimates on L2 norms inLemma 2.2.5, the almost orthogonal property of the operator I which is a conse-quence of Lemma 2.2.5, and the convergence of the eigenvalues.

26

Chapter 3 |Graph approximations for metric-measurespaces glued out of manifolds

3.1 The Laplacian eigenvalue problemIn this chapter, suppose M is a metric-measure space which is isometrically gluedout of compact Riemannian manifolds of the same dimension. More precisely,consider a metric-measure space M satisfying the following conditions:(1)M = ∪m

i=1Mi, where each Mi is an n-dimensional compact Riemannian manifoldwithout boundary or with smooth boundary;(2) For any j < k, each gluing locus Mj ∩Mk is a submanifold of both Mj and Mk

of codimension at least 1. Denote the whole gluing locus by S = ∪j<k(Mj ∩Mk).And the Riemannian metrics of Mi agree at tangent spaces Tx(Mj ∩Mk) for anyx ∈Mj ∩Mk = ∅.

We start by discussing the space of functions on M . If u is a function on M ,denote by ui the restriction of u onto Mi.

Definition 3.1.1. C∞(M) = u| ui ∈ C∞(Mi − S) ∩ C0(Mi), for any i ;W k,p(M) = u| ui ∈ W k,p(Mi);H1(M) = W 1,2(M) = C∞(M) ⊂ W 1,2(M).The Lp and W k,p norms are the sum of norms with respect to the canonical Rie-mannian volume form of every Mi. For example,

||u||2L2(M) =m∑i=1

∫Mi

u2i dvi,

where dvi is the canonical Riemannian volume form of Mi. The gradient of a point

27

on Ml − S ⊂M is the gradient with respect to the Riemannian structure of Ml.

Proposition 3.1.2. (1) Lp(M), W k,p(M) and H1(M) are all Banach spaces.(2) Sobolev embedding theorem holds in W k,p(M), so does it in H1(M).(3) If each nonempty gluing locus Mi ∩Mj(i = j) has codimension at least 2, thenH1(M) = W 1,2(M).

Proof. (3) For codimension at least 3 case, one can simply use the linear interpo-lation function to approximate any W 1,2(M) function. As for codimension 2 case,consider the function fR(r) = lnR

ln ron a unit 2-dimensional ball if r 6 R < 1, other-

wise 1. By direct computation, this family of functions can be used to approximatea point jump in the plane in W 1,2(M) norm.

Define ∆M pointwise on M − S, which is the Laplace-Beltrami operator ∆Mi

on the manifold part where the point lies. We define the eigenvalue problem of∆M as follows.

Definition 3.1.3. Consider the following equation in weak sense,

−∆Mu = λu,

more precisely,

m∑i=1

∫Mi

∇iu · ∇iϕ dvi = λm∑i=1

∫Mi

uϕ dvi, ∀ϕ ∈ C∞(M),

where dvi is the canonical Riemannian volume form in Mi. If there exists a non-trivial solution u ∈ H1(M) for some λ ∈ R, then λ is an eigenvalue of −∆M , andu is an eigenfunction with respect to the eigenvalue λ.

Same as the classical case, the spectrum of −∆M is discrete and non-negative.The definition 3.1.3 is equivalent to the following min-max formula

λk(M) = infQk+1

supu∈Qk+1−0

||∇u||2L2(M)

||u||2L2(M)

,

whereQk+1 ranges over all k+1-dimensional subspaces of the Sobolev spaceH1(M).And the minimizers are the solutions of the Laplacian eigenvalue problem. Actually

28

the min-max formula can be used to prove the existence of eigenfunctions thanksto the Sobolev embedding theorem. The boundary condition is Neumann, whichcoincides with the case we just proved for manifolds with boundary. We focus onthe case where each nonempty gluing locus Mi ∩ Mj(i = j) has codimension 1,because for codimension at least 2, the spectrum of M is essentially the spectrumsof individual manifold parts due to Proposition 3.1.2(3).

Proposition 3.1.4. Assume each nonempty gluing locus Mi ∩ Mj(i = j) hascodimension 1. Then the eigenfunctions of ∆M lie in C∞(M) and they (afterorthonormalization) form an orthonormal basis of H1(M). The eigenfunctions aresubject to the Neumann boundary condition and a Kirchhoff-type condition at thegluing locus:

∂u

∂n|∂M = 0,

and ∑ ∂u

∂n|S = 0,

where the sum is over all possible inward normal directions at S and∂M = (∪i∂Mi)− S.If Mi ∩Mj has codimension at least 2 for some i, j, one can consider Mi and Mj

to be disjoint for the sake of spectra.

Proof. The smoothness of the eigenfunctions is due to the standard regularityprocedure. Without loss of generality, suppose each gluing locus Mi ∩ Mj(i =j) is connected. We apply the test function being the usual choice inside onemanifold, say M1, and cut off in other manifolds to satisfy the continuity condition.The regularity procedure implies u1 ∈ C∞(M1 − S). Similarly, we will get ui ∈C∞(Mi − S) for all i. And the values must agree at S, since a function with acodimension 1 jump will fail to be a H1(M) function. The boundary conditionand the condition at the gluing locus directly come from integrating by parts.

Let me give a few examples to show what the eigenfunctions look like on themetric-measure spaces we are studying.

Example 1. Consider two identical circles of perimeter 1 glued at one point. Onecan think of this space as the interval [0, 2], with 0, 1 and 2 glued. The lowereigenvalues and respective eigenfunctions of the Laplacian eigenvalue problem are

29

as follows, which are supported by numerical computations.λ0 = 0, f0(x) = const;λ1 = π2, f1(x) = sin πx;

λ2 = 4π2, f2,1(x) = cos 2πx, f2,2(x) = sin 2πx, f2,3(x) =

sin 2πx x ∈ [0, 1]

0 x ∈ (1, 2].

Note that f2,3 is not smooth on [0, 2] with respect to the usual topology, but it doeslie in C∞(M).

Example 2. Consider two circles of perimeter 2, 1 glued at one point. Then theeigenvalues and eigenfunctions are:λ0 = 0, f0(x) = const;λ1 =

49π2, f1(x) = cos 2

3πx−

√3 sin 2

3πx;

λ2 = π2, f2(x) =

sin πx x ∈ [0, 2]

0 x ∈ (2, 3];

λ3 =169π2, λ4 = 4π2, with multiplicity 1 and 3.

Example 3. Consider two flat tori glued at one point. Since codimension 2 jumpsare permitted, all the eigenfunctions on this space are the combination of eigen-functions on each torus. So we will see double multiplicities in this case.λ0 = 0, with multiplicity 2;λ1 = 4π2, with multiplicity 8;λ2 = 8π2, with multiplicity 8.

Example 4. Consider a circle of perimeter 1 glued with a segment of length 1. Itis essentially the space of [0, 2], with 1 and 2 glued.λ0 = 0, f0(x) = const;

λ1 ≈ (0.6π)2, f1(x) ≈

cos 0.6πx x ∈ [0, 1]

−√33cos 0.6π(x− 1.5) x ∈ (1, 2]

;

λ2 ≈ (1.4π)2, f2(x) ≈

cos 1.4πx x ∈ [0, 1]√33cos 1.4π(x− 1.5) x ∈ (1, 2]

;

λ3 = 4π2, f3,1(x) = cos 2πx, f3,2(x) =

0 x ∈ [0, 1]

sin 2πx x ∈ (1, 2].

Note that f1 and f2 are not smooth on [0, 2] with respect to the usual topology,either. The derivative splits at point 1, with half appearing past 1 and the otherhalf appearing at 2 in the other direction.

30

3.2 Proof of Theorem 2 and 3

In this section, we prove Theorem 2 and 3. Suppose the Assumption is satisfied.Continuing with our discussions in Chapter 1, for x, y ∈ M , we define d(x, y) =

minl dl(xl, yl) and

d(x, y) = minl dl(xl, yl), where xl ∈ Ml is the mirror image of x

in Ml, and dl is the Riemannian distance of Ml. For simplicity, we just write themas d and d, and define d by

y : d(x, y) < r = y : d(x, y) < r ⊔ y : d(x, y) < r,

where the union is a disjoint union. Fix x ∈ M , denote the three sets above byBr(x), Br(x) and Br(x) respectively. Given a weighed graph Γ = Γ(X,µ, ρ) on M

as described before, our choice of graph Laplacian in this case would be

∆Γu(xi) =2(n+ 2)

νnρn+2[

∑j:d(xi,xj)<ρ

µj(u(xj)− u(xi)) +∑

j:d(xi,xj)<ρ

µj(u(xj)− u(xi))].

(3.2.1)Keep in mind that (3.2.1) involves all the mirror images. The discrete energyassociated with this graph Laplacian is

||δu||2 =n+ 2

νnρn+2

∑i

(∑

j:d(xi,xj)<ρ

+∑

j:d(xi,xj)<ρ

)µiµj|u(xj)− u(xi)|2

=n+ 2

νnρn+2

∑i

∑j:d(xi,xj)<ρ

µiµj|u(xj)− u(xi)|2. (3.2.2)

Lower bound for λk(Γ)

Now let’s turn to the proof. The main obstacle here is the lower bound of λk(Γ),which demands a suitable kernel to do the job, as explained in Chapter 1. Withoutloss of generality, assume every gluing locus is connected. For an arbitrary pointx ∈M , say in M1, suppose M1 ∩Mj = ∅ for j = 2, · · · ,m. For a fixed r satisfyingr < ρ ≪ 1 and a given function f ∈ L2(M), we define a Lipschitz function

31

Λrf ∈ H1(M) byΛrf(x) =

∫M

kr(x, y)f(y)dy,

where

kr(x, y) =1

1 +∑m

j=2 αj(x)

1

rn(ϕ(d1(x, y)

r)χM1(y) +

m∑j=2

αj(x)ϕ(dj(x

j, y)

r)χMj

(y)),

where χ is the characteristic function, and ψ(t) = n+22νn

(1− t2) if t ∈ [0, 1] otherwise0, and

αj(x) =

0, d1(x,M1 ∩Mj) > r

34 ;

1− 1

r34d1(x,M1 ∩Mj), otherwise

1, x ∈M1 ∩Mj.

;

Immediately, we have |∇αj| 6 C

r34

almost everywhere, where the gradient istaken with respect to the Riemannian structure of M1. This is due to the fact thatthe function G(x) = d1(x,M1∩Mj) is a Lipschitz function with Lipschitz constant1. Therefore it is differentiable almost everywhere and |∇xd1(x,M1 ∩Mj)| 6

√n.

The continuity of Λrf at the gluing locus is guaranteed by the Assumption andthe cut-off function αj(x). To ensure continuity at the exact ρ 3

4 -distance from thegluing locus, the distance within which αj(x) is nonzero has to be smaller than thedistance within which the mirror images are defined, which means exactly r < ρ.

Let’s consider a simple case for a moment. Consider three book pages gluedalong an edge. For a point close enough to the edge on the first plane, we use thesymmetry Assumption to define its mirror images on the other two planes. Nowif we choose a point far from the edge, the point should not be affected by otherplanes at all, and this is reflected by the cut off function αj being zero, whichresults in only information from the first plane being gathered by the kernel kr.As the chosen point gets closer to the edge, the function αj starts being nonzero,thus the kernel kr gathers information from two other planes, and their impactsbecome larger as αj grows. Finally, when the chosen point reaches the edge, αj is1, which means all three planes will have equal impact on the kernel, because themirror images of the chosen point also move to the same point on the edge thanksto the Assumption. This means if we travel on any other plane to that same pointon the edge, the terms will be exactly the same, which ensures the continuity at

32

the gluing locus.Keep in mind that we will later take r = ρ − 4ε. Define θ(x) = Λr(1M).

Similar to the second half of Section 2.2, we start by the estimate of the value andderivative of θ(x).

Lemma 3.2.1. For almost every x ∈M − S,(1) |θ(x)− 1| 6 Cr,(2) |∇θ(x)| 6 C

ro(1) + C

r34(1 + r), where o(1) is the rate of [∇Φ]r and ||∇Φ||r

converging to 1 as r → 0 in the Assumption.In particular, if each nonempty gluing locus is a submanifold of codimension 1

without boundary as assumed in Theorem 3, the second inequality improves to(2∗) |∇θ(x)| 6 C

r34(1 + r).

Proof. (1) is straightforward, with the fact that 1rn

∫Rn ϕ(

d(x,y)r

)dy = 1 for x ∈ Rn,and the factor comes from the Jacobian estimate (Proposition 2.1.3), very muchalike Lemma 2.2.4. For (2), given x ∈Mi, we have

∂

∂xdj(·j, y)(x) = −(∇Φij)(x)

exp−1xj (y)

dj(xj, y).

By direct computations,

∂

∂xkr(x, y) =

1

1 +∑

j αj

n+ 2

νnrn+2exp−1

x (y)χBr(x)

+1

1 +∑

j αj

n+ 2

νnrn+2

∑j

αj(∇Φ)exp−1xj (y)χBr(xj)

+1

1 +∑

j αj

n+ 2

2νnrn

∑j

(∇αj)ϕ(d(xj, y)

r)χMj

+ (∇ 1

1 +∑

j αj

)1

rn(ϕ(d(x, y)

r)χMi

+∑j

αjϕ(d(xj, y)

r)χMj

),

(3.2.3)

where Br(xj) is the reflected geodesic ball within Mj, which is the disjoint union of

the standard geodesic ball Br(xj) ⊂Mj and the reflected cap Br(x

j) ⊂Mj. Onceintegrating over M , the last two terms are bounded by |∇αj|, keeping in mind that1rn

∫Rn ϕ(

d(x,y)r

)dy = 1 for x ∈ Rn.

33

Now let’s deal with the first two terms. Remember we have a similar estimate inLemma 2.2.4, where we replace the Jacobian Jx by Jx−1, thanks to the symmetry∫Br(0)⊂TxM

vdv = 0. This gives us some extra power of r to compensate for thedenominator rn+2. We still have the same thing for the first term. But for thesecond term, the symmetry may be distorted under the map ∇Φ. However, byvirtue of the almost isometry property of Φ by the Assumption, after canceling outthe vectors of opposite directions, all the resulting vectors will be contained in aball of radius ro(1), where o(1) is the rate of [∇Φ]r and ||∇Φ||r converging to 1 asr → 0. The second inequality follows.

Note that the extra rate of r we get from the symmetry is crucial. One will seethe proof does not work without the extra rate, similar to the proof of Theorem 1.For the sake of argument, we will use the explicit rate o(1) = Cr2 satisfied by theassumptions of Theorem 3. One can check the lower bound holds as long as [∇Φ]r

and ||∇Φ||r converge to 1 as r → 0.Just like what we did in the second half of Section 2.2, define Irf = θ−1Λr(f).

We proceed to bound the value and energy of Irf by their discrete counterparts.

Irf(x)− f(x) = θ−1(Λrf)(x)− f(x)

= θ−1(Λrf)(x)− θ−1Λr(f(x)1M)

= θ−1Λr(f − f(x)1M)(x)

= θ−1

∫M

(f(y)− f(x))kr(x, y)dy.

Since |θ| < C and kr 6 Crn

, we have

|Irf(x)− f(x)|2 = |θ−1

∫M

(f(y)− f(x))kr(x, y)dy|2

6 θ−2(

∫M

kr(x, y)dy)(

∫M

|f(y)− f(x)|2kr(x, y)dy)

6 C

rn

∫y:d(x,y)<r

|f(y)− f(x)|2dy.

Integrate over M , and we get

||Irf − f ||2L2(M) 6C

rnEr(f), (3.2.4)

34

whereEr(f) =

∫M

∫y:d(x,y)<r

|f(y)− f(x)|2dydx.

Now we turn to the energy estimate. By (2.2.5) and (3.2.3),

∇(θ−1Λrf)(x) = θ−1

∫M

(f(y)− f(x))∂

∂xkr(x, y)dy +∇(θ−1)

∫M

(f(y)− f(x))kr(x, y)dy

=1

1 +∑αj

n+ 2

2νnrnθ−1

∑j

(∇αj)

∫Br(xj)⊂Mj

(f(y)− f(x))ϕ(d(xj, y)

r)dy

+ (∇ 1

1 +∑αj

)(1 +∑j

αj)θ−1

∫M

(f(y)− f(x))kr(x, y)dy

+ ∇(θ−1)

∫M

(f(y)− f(x))kr(x, y)dy + θ−1A1,

where

A1(x) =1

1 +∑αj

n+ 2

νnrn+2

( ∫Br(x)

(f(y)− f(x))exp−1x (y)dy

+∑j

αj(x)

∫Br(xj)

(f(y)− f(x))(∇Φ)exp−1xj (y)dy

),

and denote the first three terms A2, A3 and A4 respectively. Due to |ϕ| 6 1, theestimates in Lemma 3.2.1 and the Cauchy-Schwartz inequality, we get

|A2|2 =1

(1 +∑αj)2

(n+ 2)2

4ν2nr2n

θ−2|∑j

(∇αj)

∫Br(xj)

(f(y)− f(x))ϕ(d(xj, y)

r)dy|2

6 C

r2n|∇αj|2(

∫Br(xj)

|f(y)− f(x)|2dy)|∫Br(xj)

1dy|

6 C(1 + r)

rn+32

∫d(x,y)<r

|f(y)− f(x)|2dy.

For A4, since |∇θ| 6 C

r34(1 + r), by the Cauchy-Schwartz inequality we have,

|A4|2 6 C

r32

(1 + 2r + r2)( ∫

M

|f(y)− f(x)|kr(x, y)dy)2

6 C

r32

(1 + 2r + r2)( ∫

M

|f(y)− f(x)|2kr(x, y)dy)( ∫

M

kr(x, y)dy)

35

6 C

r32

(1 + r)

∫M

|f(y)− f(x)|2kr(x, y)dy,

where the last inequality is due to the definition of θ and |θ| 6 1 + Cr. Sincekr 6 C

rn, we get

|A4|2 6C(1 + r)

rn+32

∫d(x,y)<r

|f(y)− f(x)|2dy.

A similar estimate can be done for A3 in the same way. At last, we come to A1.Denote

A(x) =n+ 2

νnrn+2

∫Br(xj)

(f(y)− f(x))(∇Φ)exp−1xj (y)dy.

Following similar estimates in Lemma 2.2.6, each integral can be estimated asfollows, omitting a factor of (1+Cr). For w = A(x)

|A(x)| , we have |A(x)| =< A(x), w >.Then

|A(x)| 6 n+ 2

νnrn+2

∫Br(0)

|(f(expxj(v))− f(x)) < (∇Φ)v, w > Jxj(v)|dv.

By the Cauchy-Schwartz inequality,

|A(x)|2 6 (n+ 2

νnrn+2)2(

∫Br(0)

|f(expxj(v))− f(x)|2|Jxj(v)|2dv)(

∫Br(0)

< (∇Φ)v, w >2 dv)

6 (n+ 2

νnrn+2)2(

∫Br(0)

|f(expxj(v))− f(x)|2|Jxj(v)|2dv)(

∫Br+Cr3 (0)

< v,w >2 dv)

=n+ 2

νnrn+2(1 + Cr2)

∫Br(0)

|f(expxj(v))− f(x)|2|Jxj(v)|2dv

6 n+ 2

νnrn+2(1 + Cr + Cr2)

∫d(x,y)<r

|f(y)− f(x)|2dy,

where we used ||∇Φ||r 6 1 + Cr2, which causes the radius of the ball to enlargeby a factor 1 + Cr2, and the equality is due to (2.2.7). Therefore,

|A(x)| 6√

n+ 2

νnrn+2(1 + Cr)

√∫d(x,y)<r

|f(y)− f(x)|2dy.

Observe that A(x) is the universal term in A1(x), so every term in A1 can bebounded just like A(x). Considering all the factors αj(x) and averaging by 1 +

36

∑αj(x), we get exact the same estimate for A1(x).

|A1(x)| 6√

n+ 2

νnrn+2(1 + Cr)

√∫d(x,y)<r

|f(y)− f(x)|2dy.

Now combine all four terms, and we obtain

|∇(θ−1Λrf)(x)| 6 |A1|+ |A2|+ |A3|+ |A4|

6√

n+ 2

νnrn+2(1 + Cr

14 )

√∫d(x,y)<r

|f(y)− f(x)|2dy.

Therefore,||∇(Irf)||2L2(M) 6

n+ 2

νnrn+2(1 + Cr

14 )Er(f). (3.2.5)

Setting r = ρ − 4ε and f = P ∗u, where u is a discrete function, this termmatches the discrete energy (3.2.2), which will give the main term. Denote Iu =

Iρ−4ε(P∗u) = θ−1Λρ−4ε(P

∗u), and we just finished proving the following lemma.

Lemma 3.2.2. (1) ||Iu− P ∗u||2L2(M) 6 C(ρ−4ε)n

Eρ−4ε(P∗u),

(2) ||∇(Iu)||2L2(M) 6 n+2νn(ρ−4ε)n+2 (1 + Cρ

14 )Eρ−4ε(P

∗u),

where Er(f) =∫M

∫y:d(x,y)<r |f(y)− f(x)|2dydx.

The lower bound of λk(Γ) can be obtained from Lemma 3.2.2 and (2.2.8).

Upper bound for λk(Γ)

Now we deal with the easier part: the upper bound of λk(Γ). We write this partof proof under the Assumption that [∇Φ]r and ||∇Φ||r converge to 1 as r → 0.Lemma 2.2.2 stays true due to the requirement that each Vi is contained in onesingle manifold, so we only have to obtain the energy estimate. Denote by Ω = Ω

2ρ34

the region within 2ρ34 respective Riemannian distance from the gluing locus S.

Remember that ρ 34 is the distance within which we define mirror images. Outside

Ω, everything is normal as in the case of one single manifold. Therefore Lemma2.2.1 holds for this region, which is for f ∈ C∞(M),

Eρ+4ε(f,Ωc) 6 (1 + Cρ)

νn(ρ+ 4ε)n+2

n+ 2||∇f ||2L2(M), (3.2.6)

37

whereEr(f, V ) =

∫V

∫d(x,y)<r

|f(y)− f(x)|2dydx.

Inside Ω, we can estimate the energy in terms of ||∇f ||∞.

Eρ+4ε(f,Ω) 6 C

∫Ω

∫d(x,y)<ρ+4ε

||∇f ||2∞ρ32 (1 + o(1))dydx

6 C||∇f ||2∞ρ34 (ρ+ 4ϵ)nρ

32 (1 + o(1))

6 C||∇f ||2∞(ρ+ 4ϵ)nρ94 (1 + o(1)), (3.2.7)

where we used that the volume |Ω| 6 C(voln−1(S))ρ34 , and volume |y : d(x, y) <

ρ + 4ε| 6 C(1 + o(1))(ρ + 4ε)n (by the Assumption), and within Ω the distancebetween x and xj is controlled by ρ

34 (4 + o(1)) (again by the Assumption), and

the smooth function f is continuous everywhere. Note that voln−1(S) is the n− 1

dimensional volume of the gluing locus S. Combining all these, we obtain thefollowing lemma,

Lemma 3.2.3. (1) ||f − P ∗Pf ||2L2 6 C(||∇f ||∞)(1 + o(1))ρ32 ,

(2) Eρ+4ε(f,Ωc) 6 (1 + Cρ)νn(ρ+4ε)n+2

n+2||∇f ||2L2(M),

(3) Eρ+4ε(f,Ω) 6 C(||∇f ||∞)(1 + o(1))(ρ+ 4ϵ)nρ94 ,

where Ω = Ω2ρ

34

is the region within 2ρ34 distance from the gluing locus S, and o(1)

is the rate of [∇Φ]r and ||∇Φ||r converging to 1 as ρ→ 0.

With this lemma and (2.2.3), we obtain the upper bound of λk(Γ). We usethe technique as before, taking f to be the linear combination of first k smoothnormalized eigenfunctions. Note that given a metric-measure space we are studyingand the eigenvalue number k, the upper bound of ||∇f ||∞

||f ||L2is pre-determined. In

particular, in the case where the rate of [∇Φ]r and ||∇Φ||r converging to 1 isexplicit, one can trace the rates to prove Theorem 3.

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Chapter 4 |An isotopy between diffeomorphismsof T2

4.1 The curve shortening flow on T2

Let’s review a few things about the standard curve shortening flow on a smoothRiemannian surface. The curve shortening flow is a family of closed curves γ(y, t) :S1 × [0, Tmax) →M evolving according to the equation

∂tγ(y, t) = k(y, t)N(y, t) (4.1.1)

with initial condition γ(y, 0) = γ0(y), where N(y, t) is unit normal vector withrespect to the time t curve γ(·, t) at the point γ(y, t). The length of time t curveγ(·, t) (or γt) is denoted by L(γt) and satisfies the formula

d

dtL(γt) = −

∫γt

k2(s, t)ds, (4.1.2)

where s is the arc-length parameter. This shows the length of a family of evolvingcurves is monotone decreasing in time.

The finiteness of the maximum time of existence Tmax completely determinesthe geometric behavior near the maximum time by the following Grayson’s Theo-rem.

Theorem 4.1.1 ( [16]). If the initial closed curve is smooth and embedded, thecurve shortening flow either converges to a point in finite time, or converges to aclosed geodesic in C∞ norm in infinite time and never develops singularities.

39

From now on, consider the Riemannian surface to be the standard 2-torusT2 = S1 × S1. The diffeomorphism F in question is of Cm,α (m > 3, α ∈ (0, 1)).We only consider the initial curve γ0 being the image of a horizontal geodesicvia the diffeomorphism F which is homotopic to the identity, and we know thatthis family of evolving curves γt does not shrink to a point and instead it mustconverge to a horizontal geodesic in infinite time by Theorem 4.1.1. Since ourfamily of initial curves are uniformly bounded in certain norms by the norms ofthe diffeomorphism, we can ask for a uniform curvature bound under the evolutionwithin any finite fixed time T .

Proposition 4.1.2. Up to any finite fixed time T , the curvatures are uniformlybounded for all families of evolving curves with the initial curves being the imagesof horizontal geodesics via the Cm,α diffeomorphism F , and the bound depends onlyon ||F ||C3,α , ||F−1||C1 , T . Denote the uniform curvature bound by K.

Proof. For a fixed diffeomorphism F , the boundedness is due to a trivial compact-ness argument. Indeed, the space of the images of all horizontal geodesics via thefixed diffeomorphism is compact, and by the continuous dependence of the curveshortening flow on the initial condition, the curvatures are uniformly bounded upto time T .

However, proving the dependence of the bound requires a little stronger argu-ment. We range the diffeomorphism over all Cm,α diffeomorphisms with fixed C3,α

norm and C1 norm of the inverse, and consider all the curve shortening flows withinitial curves being the image of horizontal geodesics via all diffeomorphisms inthis class. We need to prove that the curvatures are uniformly bounded for allsuch flows up to any finite fixed time T .

We argue by contradiction. Suppose unbounded, so there exists a sequenceof curve shortening flows whose curvatures can get arbitrarily large. Take thesequence of their initial curves and by compactness there exists a subsequenceconverging to a limit curve in C3. Our choice of the class of diffeomorphismsalong with a simple compactness reasoning are sufficient to guarantee that thelimit curve is embedded, regular, and not homotopic to a point. So the limit curveunder the evolution exists for all times and has bounded curvature by Theorem4.1.1. By the continuous dependence of the curve shortening flow on the initialcondition( [6]), if the initial curve is close to the limit curve in C3, their curvatures

40

stay close to each other under the evolution in finite time. This is a contradictionto the assumption that the curvatures in the assumptive sequence of flows can getarbitrarily large.

4.2 Proof of Theorem 4

In this section, we prove the main regularity result and consequently Theorem 4.First, we need the following lemma which uniformly determines an explicit timeafter which the curves become graphs. We restrict our attention to the evolvingcurves γt with the initial curve γ0 being the image of some horizontal geodesic viathe diffeomorphism F .

Lemma 4.2.1. Starting from T0 = (||F ||C1+1)2, all the curves under the evolutionare graphs.

Proof. Suppose T0 is the earliest time when γt becomes a graph. So for any t < T0,γt has a segment βt which has integral of curvature at least π/2. By the Cauchy-Schwartz inequality,

L(γt)

∫γt

k2ds > (

∫βt

kds)2 > π2

4.

Since the length of γt is decreasing in time, L(γt) 6 L(γ0) 6 ||F ||C1 + 1. Byequation (4.1.2) for t < T0,

d

dtL(t) = −

∫γt

k2ds < − π2

4L(γt)< − 1

||F ||C1 + 1.

This implies at time T0, L(γT0) < L(γ0)− 1||F ||C1+1

T0. Therefore

T0 < L(γ0)(||F ||C1 + 1) 6 (||F ||C1 + 1)2.

Once a curve becomes a graph, its vertical translations do not intersect. Thus theirimages under the evolution do not intersect either due to the maximum principle.Hence the curves remain graphs forever.

Denote by Φ(x, t) : T2 ×R+ → T2 the flow generated by the evolution, namelyby the time-dependent vector field kN. The flow is well defined due to the maxi-

41

mum principle. In the end we are going to stop the flow at the time T0 which isdetermined in Lemma 4.2.1, while for the moment we state the following regularityresult within any finite fixed time T .

Proposition 4.2.2. For each t ∈ [0, T ], Φ(·, t) is a Cm−2,α diffeomorphism, andΦ(x, t) is a C [m−1

2],α map on T2 × [0, T ]. The norms are explicitly controlled de-

pending only on ||F ||Cm,α , ||F−1||C1 , K, T,m, α, where K is the uniform curvaturebound described in Proposition 4.1.2.

Proof. First for any fixed time t ∈ [0, T ], Φ(·, t) is a homeomorphism. Indeed, themap Φ(·, t) is injective by the maximum principle and continuous by the continu-ous dependence of the curve shortening flow on the initial condition. Due to theInvariance of Domain Theorem, the image of Φ(t) is open and Φ(t) is a homeo-morphism from T2 to its image. The image is on the other hand compact, henceclosed. Thus Φ(t) is surjective and is a homeomorphism of T2.

Now we discuss the differentiability of the flow. There are three directions toconsider: two spatial directions and one time direction. Two of them, tangentialdirection of the curves and time direction, are smooth among themselves due tothe analyticity of the solution of parabolic equation at positive times, and theirnorms are uniformly bounded since all derivatives of curvatures grow at most ex-ponentially in finite time with the exponent depending on the uniform curvaturebound K.

The only problem is the other spatial direction, namely across different fam-ilies of flows. This is essentially the smooth dependence on the initial condition.It is known that the solution of the curve shortening flow exhibits Cr-smoothdependence on any parameter if the equation depends Cr smoothly on that pa-rameter( [3]). In our case, we can absorb the initial condition into a parameter,carry out a similar argument as the proof of the smooth dependence on parameters,and obtain an explicit estimate of the regularity.

Consider the equation of the curve shortening flow

∂γϵ(y, t)

∂t= k(γϵ)N(γϵ), (4.2.1)

with varying initial conditions γϵ(0) = F (γ0 + ϵh) for sufficiently small ϵ, whereF is the initial Cm,α diffeomorphism, γ0 is a fixed horizontal geodesic and h isthe unit vertical vector. Here ϵ is the parameter representing the unclear spatial

42

direction in question. We aim for the differentiabiliy with respect to the parameterϵ.

First, we reduce the equation for curves (4.2.1) to an equation for functions.For sufficiently small ϵ, all the initial curves γϵ(0) lie in the normal coordinatesystem determined by equidistant curves around the fixed curve γ0(0) = F (γ0).Hence the local solutions in sufficiently small time also lie in the same coordinatesystem. We define a Cm function u for a Cm curve γ within the coordinate systemto be the distance function of the curve γ from the fixed curve F (γ0). In thisway the local solution curves γϵ(y, t) are represented by functions u(y, t, ϵ), so aretheir curvatures and unit normal vectors. Thus the equation for curves (4.2.1)

is reduced to an equation for functions u(y, t, ϵ). By straightforward calculationsmostly identical to the ones in [3], one can obtain the following evolution equationsatisfied by u = u(y, t, ϵ) for sufficiently small ϵ, t :

ut = G(y, u, uy, uyy), (4.2.2)

where G is a nonlinear function of the four arguments, with the initial conditionu(y, 0, ϵ) = dist(γϵ(y, 0), γ0(y, 0)) within the said normal coordinate system. Wedenote uy by p and uyy by q. The function G satisfies the following properties:(1) ∂G

∂qis positive, which implies the equation is parabolic;

(2) The function G is a Cm−2,α function with respect to all arguments;(3) More precisely, the Cm−3,α norms of all the partial derivatives of G are uni-formly bounded explicitly depending only on ||F ||Cm,α , ||F−1||C1 , K, T,m.

Now we absorb the varying initial conditions into the equation (4.2.2) by sub-tracting u(y, 0, ϵ), and we obtain the equation of u(y, t, ϵ) = u(y, t, ϵ) − u(y, 0, ϵ)

with vanishing initial condition, represented by another nonlinear function G:

ut = G(y, ϵ, u, p, q),

where p, q denote the first and second derivative of u with respect to y respectively.The parameter ϵ appears in terms of the initial condition u(y, 0, ϵ) and its deriva-tives with respect to y up to the second order, which implies G is Cm−2,α-smoothwith respect to ϵ. It follows that all the properties of G hold for G.

Denote the Frechet derivative of G at u of any v by(dG(u)

)v = Gqvyy+ Gpvy+

Guv. Thanks to the properties of G, by a standard argument (e.g. [4]), the equa-

43

tion at u:(∂t − dG(u)

)v = f has a solution v ∈ Cm−1,α with the initial condition

v(0) = 0 for every f ∈ Cm−3,α, which means ∂t − dG(u) is invertible. The ImplicitFunction Theorem in Banach spaces implies that u is Cm−2-smooth with respectto y and ϵ, so is u. Equipped with the differentiability of u, we can estimate itsderivatives directly from the equation (4.2.2).

Differentiate the equation (4.2.2) with respect to ϵ, and we get the equationsatisfied by uϵ:

uϵt =∂G

∂quϵyy +

∂G

∂puϵy +

∂G

∂uuϵ,

with the initial condition to be uϵ(y, 0, ϵ) which is Cm−1,α-smooth with respect to y.By the regularity estimate of the linear parabolic equation (Theorem 5.1.9 in [21]),up to a sufficiently small time t0,

||∂u∂ϵ

(y, t, ϵ)||Cm−1,α(S1×[0,t0]) 6 eC1t0||∂u∂ϵ

(y, 0, ϵ)||Cm−1,α(S1×[0,t0])

6 C(||F ||Cm,α , ||F−1||C1 , K, T,m, α)eC1t0 ,

where C1 = C1(||F ||Cm,α , ||F−1||C1 , K, T,m, α). Here we freeze the parameter ϵand consider differentiation only with respect to y. Similar estimate also holdsfor ||∂u

∂y||Cm−1,α . By a standard bootstrapping argument, we arrive at the following

estimate for higher order derivatives with respect to ϵ up to the order m− 2 :

||∂(i+j)u

∂yi∂ϵj||C0,α 6 C(||F ||Cm,α , ||F−1||C1 , K, T, t0,m, α), for i+j 6 m and j 6 m−2.

(4.2.3)Since we know in advance that our flow exists for all times, it is not difficult

to check the same procedure can be continued from t0, primarily because the regu-larity estimate (4.2.3) gives sufficient regularity at time t0 to repeat the procedure.Thus with some care in tracing time-involved terms at each step, one can extendthe regularity estimate to any finite fixed time T . The estimate on u implies theestimate on the original curve γϵ:

||∂(i+j)γϵ

∂yi∂ϵj||C0,α 6 C(||F ||Cm,α , ||F−1||C1 , K, T,m, α), for i+ j 6 m and j 6 m− 2.

(4.2.4)As a consequence, we also get the Cm−2,α differentiability of the unit tangent vectorand unit normal vector of the original curve γϵ with respect to ϵ.

44

The flow is Cm−2,α in spatial directions with respect to the parameters y, ϵ dueto (4.2.4), consequently with respect to the original spatial parameter x ∈ T2 attime 0 determined by the initial diffeomorphism F . The only issue is the continuityof time derivatives in the ϵ direction. Since we are dealing with a second orderequation (4.2.2), the regularity in the time direction is given by the regularity ofthe second derivative with respect to the tangential direction. It follows that everytime taking a time derivative decreases the regularity in the ϵ direction by 2 inview of the regularity estimate (4.2.4). Thus one can only take time derivativesat most [m−1

2] times before losing the differentiability in the ϵ direction. Therefore

the total regularity of the flow will be reduced to [m−12

].Up to this point we have proved that for each fixed time t ∈ [0, T ], Φ(·, t) is a

Cm−2,α homeomorphism. To argue it is a diffeomorphism, by the Inverse FunctionTheorem it suffices to prove that the differential dΦ(·, t) is nonsingular everywhere.Again we look at the two spatial directions. Along the tangential direction of thecurve, the length of the velocity is uniformly bounded away from 0 in finite time,due to the evolution equation satisfied by the velocity ∂t|γy| = −k2|γy| along withthe uniform curvature bound K. The non-singularity of the differential dΦ(·, t)across different families of flows follows from the fact that the distance between twocurves is nondecreasing in time via the curve shortening flow ( [14]). Furthermoreone can show that the length of the differential dΦ(·, t) acting on any unit vector isbounded away from 0 depending only on ||F ||C1 , ||F−1||C1 , K, T , which implies thedifferential is nonsingular. By the Inverse Function Theorem, the inverse Φ(·, t)−1

is a Cm−2 homeomorphism and hence Φ(·, t) is a diffeomorphism.

Now we prove Theorem 4. First let us closely examine what happens to onesingle curve. Take a horizontal geodesic, and after applying the initial diffeomor-phism F and the curve shortening flow up to time T0 = (||F ||C1 + 1)2, we geta closed curve which is a graph with respect to the original horizontal geodesicby Lemma 4.2.1. However the parametrization of the curve is distorted, and weapply a second procedure of reparametrization to make it align with the naturalparametrization of the horizontal geodesic. Then we apply a third procedure tomove the curve back to the horizontal geodesic via the “ height function”. The de-tails are discussed below. Although it is straightforward to handle one single curve,in our case we need to apply these procedures to all the curves at once, and torecover the regularity we need some consistencies when applying these procedures

45

across different curves, which is provided by Proposition 4.2.2 specifically the reg-ularity estimate (4.2.4).

We discuss the second procedure of reparametrizations here. The compositionΦ1 = Φ(·, T0) F is a Cm−2,α diffeomorphism of T2 by Proposition 4.2.2 and it ishomotopic to the identity. This diffeomorphism Φ1 maps the family of horizontalgeodesics to a family of graphs. For a fixed horizontal geodesic, we consider a mapf which sends the fixed horizontal geodesic to the vertical projection of its image(graph) via the diffemorphism Φ1 onto itself. This map f is a diffeomorphism of S1

and it is homotopically trivial. Hence the lift f of the diffeomorphism f of S1 onthe universal cover R of S1 has degree 1, i.e. f(z + 1) = f(z) + 1 for z ∈ R. Thusthe isotopy on the universal cover ϕ(z, t) = (1− t)f(z) + tz on R× [0, 1] betweenf and the identity (of R) descends to an isotopy on S1 between f and the identity(of S1). This isotopy on S1 gives a reparametrization for a graph. Apply this con-struction to all the horizontal geodesics, and we obtain a reparametrization for thewhole family of graphs, denoted by Φ2(x, t) on T2× [0, 1]. Due to Proposition 4.2.2,this flow of reparametrization Φ2 is Cm−2,α. To glue the flow Φ2 and the curveshortening flow together, we need modifications to both flows in order to preservethe regularity. Take a suitable smooth function f1 : [T0, 4T0] → [T0, 4T0] satisfyingthat f1(t) = t on [T0, 2T0], f1 = 4T0 on [3T0, 4T0] and f1 is strictly increasing on[2T0, 3T0]. Extend the curve shortening flow up to time 4T0, and within [T0, 4T0]

modify the flow as Φ(t) = Φ(f1(t)). The norm of such modified flow is enlargedby a scale factor of a constant depending only on m. One can glue this modifiedcurve shortening flow with a similarly modified flow of reparametrization withoutaffecting the regularity. On the level of vector fields, this is simply the standardprocedure to glue two time-dependent vector fields together along time by multi-plying smooth functions of time vanishing in the gluing region.

Then we apply a third procedure to move the curves (graphs) back to the hori-zontal geodesics. The composition of the first two procedures Φ2(·, 1)Φ1 (properlyglued) is a homotopically trivial Cm−2,α diffeomorphism of T2, and its restrictiononto a fixed vertical geodesic defines a homotopically trivial diffeomorphism of S1.We can simply repeat the construction of the second procedure to find an isotopybetween this diffeomorphism of S1 and the identity. Apply the construction tothe whole family of vertical geodesics, and we obtain an isotopy on T2 betweenΦ2(1) Φ1 and the identity. Again this isotopy is Cm−2,α. One can similarly glue

46

this isotopy with the first two procedures while preserving the regularity. Theorem4 is proved.

47

References

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[2] S. B. Alexander, I. D. Berg, and R. L. Bishop, Cut loci, minimizers,and wavefronts in Riemannian manifolds with boundary. Michigan MathJ. 40 (1993), 229-237.

[3] S. Angenent, Parabolic equations for curves on surfaces Part I. Curves withp-intergrable curvature, Ann. Math. 132 (1990), 451-483.

[4] S. Angenent, Nonlinear analytic semiflows, Proc. Royal Soc. Edin-burgh. 115A (1990), 91-107.

[5] S. Boldsen, Different versions of mapping class groups of sur-faces, arXiv:0908.2221.

[6] S. Boussandel, R. Chill, E. Fasangova, Maximal regularity, the local in-verse function theorem, and local well-posedness for the curve shorteningflow, Czech. Math. J. 62 (2012), 335-346.

[7] D. Burago, S. Ivanov, and Y. Kurylev, A graph discretization of the Laplace-Beltrami operator. J. Spectr. Theory. 4 (2014), 675-714.

[8] D. Burago, S. Ivanov, and Y. Kurylev, Spectral stability of metric-measureLaplacians. arXiv:1506.06781v2.

[9] D. Burago, S. Ivanov, Y. Kurylev, J. Lu, Approximation of connection Lapla-cians spectra, preprint.

[10] D. Burago, J. Lu, T. Ozuch, How large isotopy is needed to connect homotopicdiffeomorphisms (of T2), J. Topol. Anal., accepted.

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[11] J. Cheeger, D. Ebin, Comparison theorems in Riemannian Geometry, Amer.Math Soc., 2008.

[12] B. Farb, D. Margalit, A primer on mapping class groups, Princeton Univ.Press, 2011.

[13] K. Fujiwara, Eigenvalues of Laplacians on a closed Riemannian manifold andits nets, Proc. Amer. Math. Soc. 123 (1995), 2585-2594.

[14] M. Gage, R. Hamilton, The heat equation shrinking convex plane curves, J.Diff. Geom. 23 (1986), 69-96.

[15] M. Grayson, The heat equation shrinks embedded plane curves to roundpoints, J. Diff. Geom. 26 (1987), 285-314.

[16] M. Grayson, Shortening embedded curves, Ann. Math. 129 (1989), 71-111.

[17] M. W. Hirsch, Differential Topology, Springer, 1976.

[18] J. Jost, Partial Differential Equations. Springer, 2007.

[19] G. Lebeau, L. Michel, Semi-classical analysis of a random walk on a mani-fold, Ann. Probab. 38 (2010), 277-315.

[20] J. Lu, A graph approximation to the spectrum of the Laplace operator, sub-mitted.

[21] A. Lunardi, Analytic semigroups and optimal regularity in parabolic prob-lems, Birkhauser, 1995.

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VitaJinpeng Lu

EDUCATION

The Pennsylvania State University, USAPh.D., Mathematics May 2019Advisor: Dmitri Burago

Tsinghua University, ChinaM.S., Mathematics June 2013B.E., Engineering Physics June 2011

PUBLICATIONS

1. J. Lu, A graph approximation to the spectrum of the Laplace operator, submit-ted.2. D. Burago, J. Lu, T. Ozuch, How large isotopy is needed to connect homotopicdiffeomorphisms (of T2), J. Topol. Anal., accepted.3. D. Burago, S. Ivanov, Y. Kurylev, J. Lu, Approximation of connection Lapla-cians spectra, preprint.