On the gradient flow for the anisotropic area functional

Abstract

We consider the anisotropic mean curvature flow for parametrized hypersurfaces and provide a new definition of gradient flow that takes into account the anisotropic nature of space. With this new approach we succeed in identifying the natural candidate for the anisotropic mean curvature vector using a variational method. Under the obtained flow Wulff shapes shrink self similarly. The new definition of gradient flow relies on understanding which Banach structure plays a fundamental role and performing a consistent choice of function spaces. The geometric setting we introduce also allows us to provide and justify a natural formulation for the anisotropic Willmore functional. Its first variation is computed.

Computational anisotropic Willmore flow

“…Here we briefly recall some notation and results that appeared in [5] and [18] and that are relevant to our discussion.…”

“…On the other hand the author has provided and motivated in [18] a geometrically consistent formulation of anisotropic Willmore functional (cf. also [2]).…”

“…also [2]). The point of view presented in [18] is the same we adopt in this work. The considered anisotropic Willmore functional (see (2.10) below) now differs from the one studied by Clarenz [5] and Palmer [14] in that the area element d V is replaced by its more natural anisotropic counterpart.…”

A Nested Variational Time Discretization for Parametric Anisotropic Willmore Flow

“…Now, based on the anisotropy γ and its dual where | · | denotes the usual Lebesgue volume in R d+1 . In particular, the underlying metric structure is dictated by the Wulff shape and its norm γ * (see [5], [42] and references therein). Based on these considerations let us first consider anisotropic mean curvature motion, which is defined as the gradient flow of the anisotropic surface area with respect to the above anisotropic metric.…”

“…Let us conclude this section with a study of boundaries ∂W of two-dimensional Wulff shapes W moving under anisotropic Willmore flow in the plane. To this end consider the parametrization x : (0, T ) × S 1 → R 2 , x(t, ν) = R(t)∇γ(ν) of the boundary of a (rescaled) Wulff shape R(t)W. Using the results given in [42] it is easily seen that x moves under anisotropic Willmore flow if R(t) solves the ODEṘ…”