Weak solutions of inverse mean curvature flow for hypersurfaces with boundary

Abstract

We consider the evolution of hypersurfaces with boundary under inverse mean curvature flow. The boundary condition is of Neumann type, i.e. the evolving hypersurface moves along, but stays perpendicular to, a fixed supporting hypersurface. In this setup, we prove existence and uniqueness of weak solutions. Furthermore, we indicate the existence of a monotone quantity which is the analog of the Hawking mass for closed hypersurfaces.

Capacity inequalities and rigidity of cornered/conical manifolds

“…Building on the ideas of Freire and Schwartz in their proof of mass-capacity inequalities [7], we obtain analogous results by considering now hypersurfaces with boundary evolving under inverse mean curvature flow, an approach introduced by Marquadt [21,22], who constructed solutions by rewriting the flow as an equation for the level set of a function whose advantage is to allow "jumps" in a natural way. For different approaches of this geometric flow, we refer the reader to [15,16,22].…”

“…Consider a foliation {Σ t } defined by the level sets of the function given by weak solution of IMCF in M \Ω 0 . By Lemma 5.1 and 5.3 of [22], each Σ t := ∂ M {ϕ < t}, ϕ ∈ C 0,1 loc (M ), is a C 1, 1 2 -hypersurface up to a set of dimension less than or equal to n − 8 which possesses a weak mean curvature in L ∞ given by…”

“…The hypersurface flows in the outward normal direction with speed N = 1 H and it is easy to see that there is a problem if either H = 0 or H changes the sign on Σ t . 1 To overcome theses problems, Marquadt [20,22] developed the notion of weak solutions for (2.2), proving the existence and uniqueness of such solutions guided by the ideas of Huisken and Ilmanen [12].…”

The inverse mean curvature flow perpendicular to the sphere

“…We need the spatial boundary derivatives of various curvature quantities, when the supporting hypersurface is a sphere. The calculations are quite similar to those in [15,18]. For the sake of completeness and for a better comprehensibility of the different notation, let us derive them in detail.…”

“…Thus we have extended the scalar function u. (ii) To obtain the full curvature flow from the scalar function u, we use the standard method applied in [15,Sec. 2.3], solving an ODE to allow for normal directed evolution.…”