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On an inverse curvature flow in two-dimensional space forms

Journal Article


Abstract


  • 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 Gir\~ao-Pinheiro.

Publication Date


  • 2021

Citation


  • Kwong, K. -K., Wei, Y., Wheeler, G., & Wheeler, V. -M. (2021). On an inverse curvature flow in two-dimensional space forms. Mathematische Annalen. doi:10.1007/s00208-021-02285-5

Web Of Science Accession Number


Abstract


  • 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 Gir\~ao-Pinheiro.

Publication Date


  • 2021

Citation


  • Kwong, K. -K., Wei, Y., Wheeler, G., & Wheeler, V. -M. (2021). On an inverse curvature flow in two-dimensional space forms. Mathematische Annalen. doi:10.1007/s00208-021-02285-5

Web Of Science Accession Number