Parabolic Equations for Curves on Surfaces. 1. Curves with p-Integrable Curvature

This is the first of two papers in which the author develops a theory of parabolic equations for curves on surfaces which can be applied to the so- called curve shortening or flow by mean curvature problem, as well as to a number of models for phase transitions in two dimensions. This document introduces a class of equations for which the initial value problem is solvable for initial data with p-integrable curvature, and we also give estimates for the rate at which the p-norms of the curvature must blow up, if the curve becomes singular in finite time. A detailed discussion of the way in which solutions can become singular and a method for continuing the solution through a singularity will be the subject of the second part.