Convexity Properties of Harmonic Functions on Parameterized Families of Hypersurfaces

Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion

We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. Several properties of solutions are analyzed. Emphasis is placed on to what extent the surface in our setting qualitatively evolves similar as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but give an example of a surface that exists for…

Cylindrical estimates for hypersurfaces moving by convex curvature functions

We prove a complete family of `cylindrical estimates' for solutions of a class of fully non-linear curvature flows, generalising the cylindrical estimate of Huisken-Sinestrari for the mean curvature flow. More precisely, we show that, for the class of flows considered, an $(m+1)$-convex ($0\leq m\leq n-2$) solution becomes either strictly $m$-convex, or its Weingarten map approaches that of a cylinder $\R^m\times S^{n-m}$ at points where the curvature is becoming large. This result complements the convexity estimate proved by the authors and McCoy for the same class of flows.

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non-compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth-order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well-posedness of both flows for initial surfaces that are $C^{1+α}$-regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long-term existen…

球面中horo-凸超曲面的保持quermassintegral的曲率流

The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere

本研究提出球面中horo-凸超曲面的完全非线性曲率流，可保持任意指定球面quermassintegral，证明了horo-凸性保持、长时间存在及指数收敛到对应测地球面。

Two-sided non-collapsing curvature flows

It was recently proved that embedded solutions of Euclidean hypersurface flows with speeds given by concave (convex), degree one homogeneous functions of the Weingarten map are interior (exterior) non-collapsing. These results were subsequently extended to hypersurface flows in the sphere and hyperbolic space. In the first part of the paper, we show that locally convex solutions are exterior non-collapsing for a larger class of speed functions than previously considered; more precisely, we show that the previous results hold when convexity of the speed function is relaxed to inverse-concavity.…

A Support Characterization for Functions on the Unit Sphere with Vanishing Integrals Arising from Tangent Planes to a Given Surface

Let $Σ$ be an axially symmetric, smooth, closed hypersurface in $\Bbb R^{n + 1}$ with a simply connected interior which is contained inside the unit sphere $\Bbb S^{n}$. For a continuous function $f$, which is defined on $\Bbb S^{n}$, the main goal of this paper is to characterize the support of $f$ in case where its integrals vanish on subspheres obtained by intersecting $\Bbb S^{n}$ with the tangent hyperplanes of a certain subdomain $\mathcal{U}\subsetΣ$ of $Σ$. We show that the support of $f$ can be characterized in case where its integrals also vanish on subspheres obtained by intersectin…