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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3725–3746 | Cite as

Shrinking Doughnuts via Variational Methods

  • Gregory Drugan
  • Xuan Hien Nguyen
Article

Abstract

We use variational methods and a modified curvature flow to give an alternative proof of the existence of an embedded topological \(\mathbb {S}^1 \times \mathbb {S}^{n-1}\) self-shrinking hypersurface under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.

Keywords

Mean curvature flow Self-shrinking Singularities Geodesics 

Mathematics Subject Classification

53C44 

Notes

Acknowledgements

The authors would like to thank Stephen Kleene and Sigurd Angenent. Stephen Kleene introduced the authors in the hope to solve a related problem. That problem is still open and morphed into this one. Sigurd Angenent generously shared his expertise on the existence, properties, and convergence of parabolic flows.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Oregon Episcopal SchoolPortlandUSA
  2. 2.Department of MathematicsIowa State UniversityAmesUSA

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