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Two low dimension curvature flows are studied: the Ricci flow on surfaces and the curve shortening flow of embedded closed curves in the plane. The main theorems proven are that the corresponding normalised flows have solutions existing for all time and which converge to a minimising configuration, namely one with constant curvature. The theorems follow from comparison theorems for isoperimetric quantities. For the Ricci flow, the isoperimetric profile is used. For the curve shortening flow,doi:10.25911/5d611b8ddea1b fatcat:t25osy6pj5fktgcfvotm4jffqm