Abstract: I am going to discuss the evolution of compact convex curves in two-dimensional space forms under an inverse curvature type flow, first proposed by Brendle, Guan and Li. 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. This flow was studied by Brendle-Guan-Li and by Hu-Li-Wei in higher dimensions (hypersurfaces of dimension $n \ge 2$). The curve case ($n=1$) has been left open in earlier work and indeed the analysis of the flow must be done differently in this case. 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. This is joint work with Y. Wei, G. Wheeler and V. Wheeler.

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