A new monotone quantity along the inverse mean curvature flow in ℝⁿ

Abstract

We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in R n . As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypersurface. Statement of the ResultMonotone quantities along hypersurfaces evolving under the inverse mean flow have many applications in geometry and relativity. In [3], Huisken and Ilmanen applied the monotone increasing property of Hawking mass to give a proof of the Riemannian Penrose Inequality. In a recent paper [1], Brendle, Hung and Wang discovered a monotone decreasing quantity along the inverse mean curvature flow in Anti-Desitter-Schwarzschild manifolds and used it to establish a Minkowski-type inequality for star-shaped hypersurfaces.In this note, we provide a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in R n :Theorem 1. Let Σ be a smooth, closed, embedded hypersurface with positive mean curvature in R n . Let I be an open interval and X : Σ × I → R n be a smooth map satisfying ∂X ∂t = 1 H ν, (1.1)

An Extension of Hsiung–Minkowski Formulas and Some Applications

“…This generalizes the result in [20,21]. (We remark that in [21], a special case of Corollary 1.1 is proved using the inverse mean curvature flow approach instead of using integral formulas. It is interesting to compare the two approaches.)…”

Solitons for the inverse mean curvature flow

“…This singular shooting problem is well-defined (see [4] and [9]), and the solution satisfies h ′′ (0) = −1/(nCh 0 ) > 0. Differentiating (17) and analyzing the equation for h ′′′ (r) shows that, under the above conditions, we have h ′′ (r) > 0 and h ′ (r) > 0, for r > 0, as long as the solution is defined. The global behavior of the solution ultimately depends on the value of C.…”

“…To see this, suppose to the contrary that h ′ increases to ∞ at a point r top < ∞. Then, since C > 1/(n − 1), equation (17) forces h ≥ ǫrh ′ when r is close to r top , for some ǫ > 0. However, integrating this inequality shows that h ′ does not blow-up at a finite point; hence the solution exist for all r > 0.…”

“…To establish upper and lower bounds for the solution r(h), we study the profile curve C by writing it as a graph over the axis of rotation: (r, h(r)). Then, we have the following second order non-linear differential equation (17) h…”

“…We begin by considering the initial value problem where we shoot perpendicularly to the axis of rotation. For C > 0, let h(r) be a solution to (17) with h(0) = h 0 < 0 and h ′ (0) = 0. This singular shooting problem is well-defined (see [4] and [9]), and the solution satisfies h ′′ (0) = −1/(nCh 0 ) > 0.…”