A metric Ricci flow for surfaces and its applications

Motivated largely by Perleman’s work, the Ricci flow has become lately an object of interest and study in Graphics and Imaging. Various approaches have been suggested previously, ranging from classical approximation methods of smooth differential operators to discrete, combinatorial methods.

In this paper we introduce a metric Ricci flow for surfaces and we investigate its properties: existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for the metric one and that, moreover, the same results hold for a more general, metric notion of curvature. Furthermore, using the metric curvature approach, we show the existence of the Ricci flow for polyhedral 2-manifolds of piecewise constant curvature. We also study the problem of the realizability of the said flow in $\mathbb{R}^3$.

Keywords

combinatorial surface Ricci flow, metric curvature

2010 Mathematics Subject Classification

Primary 52C26, 53C44, 68U05. Secondary 51K10, 57R40, 65D18.

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Published 13 May 2014