On an inverse curvature flow in two-dimensional space forms

Publication Name

Mathematische Annalen


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 Girão–Pinheiro.

Open Access Status

This publication may be available as open access

Funding Number

DECRA DE190100379

Funding Sponsor

Australian Research Council



Link to publisher version (DOI)