4.6 Article

On an inverse curvature flow in two-dimensional space forms

期刊

MATHEMATISCHE ANNALEN
卷 384, 期 1-2, 页码 285-308

出版社

SPRINGER HEIDELBERG
DOI: 10.1007/s00208-021-02285-5

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资金

  1. National Key R and D Program of China [2020YFA0713100]
  2. University of Science and Technology of China [KY0010000052]
  3. DECRA of the Australian Research Council [DE190100379, DP180100431]
  4. Australian Research Council [DE190100379] Funding Source: Australian Research Council

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This study focuses on the evolution of compact convex curves in two-dimensional space forms, proving that solutions exist for all time and converge exponentially fast to a standard round geodesic circle. This has important implications for proving geometric inequalities and providing counterexamples to certain conjectures.
We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetric inequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the n = 2 case of a conjecture of Girao-Pinheiro.

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