Uniformly Compressing Mean Curvature Flow

Abstract

Michor and Mumford showed that the mean curvature flow is a gradient flow on a Riemannian structure with a degenerate geodesic distance. It is also known to destroy the uniform density of gridpoints on the evolving surfaces. We introduce a related geometric flow which is free of these drawbacks. Our flow can be viewed as a formal gradient flow on a certain submanifold of the Wasserstein space of probability measures endowed with Otto's Riemannian structure. We obtain a number of analytic results concerning well-posedness and long-time stability which are however restricted to the 1D case of evolution of loops.

The gradient flow of the potential energy on the space of arcs

“…. We however claim that it is possible to construct some sequences η ǫ 0 ∈ A, v ǫ 0 ∈ T η ǫ 0 A, with α(η ǫ 0 ) = α 0 being a constant angle strictly between 0 and π, and with v ǫ 0 L 2 bounded away from 0, so that for the corresponding solutions to (21) one has ς ǫ (1) → 0. Indeed, assume for definiteness that d = 3 and g = (0, 0, −1).…”

“…Remark 6.5. It is worth noting that the geometric "uniformly compressing curve-shortening flow", which we have introduced and studied in the companion paper [21], after a suitable change of variables becomes very similar to (7):…”

On the Morse–Bott property of analytic functions on Banach spaces with Łojasiewicz exponent one half

“…For interpretations of mean curvature flow as a gradient system, we refer the reader to Bellettini [ [115], and Zaal [131]. Shi and Vorotnikov [106] provide a useful recent reference, with a view to applications. For introductions to mean curvature flow, we refer to Ecker [40], Mantegazza [82], Ritoré and Sinestrari [100].…”