Anisotropic curvature measures and volume preserving flows

TL;DR

This paper develops anisotropic curvature measures for convex bodies, characterizes Wulff shapes through these measures, and studies volume-preserving flows that evolve convex hypersurfaces towards Wulff shapes, proving convergence results.

Contribution

It generalizes classical curvature measure results to the anisotropic setting and establishes convergence of volume-preserving flows to Wulff shapes.

Findings

Convex bodies with specific anisotropic curvature measure relations are scaled Wulff shapes.

Volume-preserving flows converge to Wulff shapes in Hausdorff sense.

Under certain conditions, convergence is smooth and exponential.

Abstract

In the first part of this paper, we develop the theory of anisotropic curvature measures for convex bodies in the Euclidean space. It is proved that any convex body whose boundary anisotropic curvature measure equals a linear combination of other lower order anisotropic curvature measures with nonnegative coefficients is a scaled Wulff shape. This generalizes the classical results by Schneider [Comment. Math. Helv. \textbf{54} (1979), 42--60] and by Kohlmann [Arch. Math. (Basel) \textbf{70} (1998), 250--256] to the anisotropic setting. The main ingredients in the proof are the generalized anisotropic Minkowski formulas and an inequality of Heintze--Karcher type for convex bodies. In the second part, we consider the volume preserving flow of smooth closed convex hypersurfaces in the Euclidean space with speed given by a positive power $\alpha $ of the $k$th anisotropic mean curvature plus a global term chosen to preserve the enclosed volume of the evolving hypersurfaces. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to the Wulff shape in the Hausdorff sense. The characterization theorem for Wulff shapes via the anisotropic curvature measures will be used crucially in the proof of the convergence result. Moreover, in the cases $k=1$, $n$ or $\alpha\geq k$, we can further improve the Hausdorff convergence to the smooth and exponential convergence.