minimizing the squared mean curvature integral for ...sequin/cs284/text/hsu.pdfminimizing the...

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Minimizing the Squared Mean Curvature Integral for Surfaces in Space Forms Lucas Hsu, Rob Kusner and John Sullivan CONTENTS 1. Introduction 2. Experimental Setup 3. Minimizing Spheres 4. Minimizing Tori 5. Higher-Genus Minimizing Surfaces 6. Further Work Acknowledgements References We minimize a discrete version of the squared mean curvature integral for polyhedral surfaces in three-space, using Brakke’s Surface Evolver [Brakke 1992]. Our experimental results sup- port the conjecture that the smooth minimizers exist for each genus and are stereographic projections of certain minimal sur- faces in the three-sphere. 1. INTRODUCTION

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Page 1: Minimizing the Squared Mean Curvature Integral for ...sequin/CS284/TEXT/hsu.pdfMinimizing the Squared Mean Curvature Integral for Surfaces in Space Forms Lucas Hsu, Rob Kusner and

Minimizing the Squared Mean Curvature Integralfor Surfaces in Space FormsLucas Hsu, Rob Kusner and John Sullivan

CONTENTS

1. Introduction2. Experimental Setup3. Minimizing Spheres4. Minimizing Tori5. Higher-Genus Minimizing Surfaces6. Further WorkAcknowledgementsReferences

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We minimize a discrete version of the squared mean curvatureintegral for polyhedral surfaces in three-space, using Brakke’sSurface Evolver [Brakke 1992]. Our experimental results sup-port the conjecture that the smooth minimizers exist for eachgenus and are stereographic projections of certain minimal sur-faces in the three-sphere.

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Page 2: Minimizing the Squared Mean Curvature Integral for ...sequin/CS284/TEXT/hsu.pdfMinimizing the Squared Mean Curvature Integral for Surfaces in Space Forms Lucas Hsu, Rob Kusner and

192 Experimental Mathematics, Vol. 1 (1992), No. 3

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Page 3: Minimizing the Squared Mean Curvature Integral for ...sequin/CS284/TEXT/hsu.pdfMinimizing the Squared Mean Curvature Integral for Surfaces in Space Forms Lucas Hsu, Rob Kusner and

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Page 4: Minimizing the Squared Mean Curvature Integral for ...sequin/CS284/TEXT/hsu.pdfMinimizing the Squared Mean Curvature Integral for Surfaces in Space Forms Lucas Hsu, Rob Kusner and

194 Experimental Mathematics, Vol. 1 (1992), No. 3

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198 Experimental Mathematics, Vol. 1 (1992), No. 3

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Hsu, Kusner and Sullivan: Minimizing the Squared Mean Curvature Integral for Surfaces in Space Forms 199

FIGURE 5. �R� � �!�1���2�3c������� �:�3L����" ���)$;������-������1��#��C����,� 9C� A���" >T����" ���5$6���B� ����7T-/.L(p�#�J�3 � �X�0�3�� � � .qra�npi@e�b [Dw�w�b jP�&� Z5j|l % Òj|c¤c>[d_�e�]#\ ÎNq�e�qNem\�\ n�[P�-��[Rq�l|Z0eÊ^�]#e�b([D]#^0jdq�\

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FIGURE 8. � �4�B <p�!�#>�"j�3!� ��" �%�:���5$6 ���C��-�H��1�����������&<p7X7��#��"&<�����������E��#�*�!�&g%� HQ� �8�#H�� ���X� �!" �6�3�#� � % � µ �I�3�X����" ������>���>��A �3��>@" ��=�����>WP!EB� �p�C��E������G� ��" �%�3�#�G� �D�����G7X� >�>�C��C�#�2�Z�m�3����h.V����� �ZH�� �������D�#��B ���G����>���"Z�����G[Zg��!��H �3��>n�2��"&���!�#"" ��=�����7��#�*���J$67X� >�>� ��" � E���R�3��>@" � E����-/Pp<�� �� >�� ��EX���# ��� �5�:�!��" �ISp�C7,�:��� ���T��� � % � µ .9(p�#�J�3 � �X�0�3�� � � .

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