Differentiability of the stable norm in codimension one

Plane-like minimizers and differentiability of the stable norm

In this paper we investigate the strict convexity and the differentiability properties of the stable norm, which corresponds to the homogenized surface tension for a periodic perimeter homogenization problem (in a regular and uniformly elliptic case). We prove that it is always differentiable in totally irrational directions, while in other directions, it is differentiable if and only if the corresponding plane-like minimizers satisfying a strong Birkhoff property foliate the torus. We also discuss the issue of the uniqueness of the correctors for the corresponding homogenization problem.

高余维平均曲率流的边界正则性

Boundary regularity for mean curvature flows of higher codimension

该研究针对带任意余维边界的图形式平均曲率流，推导解梯度的Hölder范数全局界，为高余维平均曲率流的边界正则性分析提供了关键结果。

Constancy of the dimension in codimension one and locality of the unit normal on $\mathrm{RCD}(K,N)$ spaces

The aim of this paper is threefold. We first prove that, on $\mathrm{RCD}(K,N)$ spaces, the boundary measure of any set with finite perimeter is concentrated on the $n$-regular set $\mathcal{R}_n$, where $n\le N$ is the essential dimension of the space. After, we discuss localization properties of the unit normal providing representation formulae for the perimeter measure of intersections and unions of sets with finite perimeter. Finally, we study Gauss-Green formulae for essentially bounded divergence measure vector fields, sharpening the analysis in \cite{BuffaComiMiranda19}.\\ These tools a…

极小叶、余维一稳定范数与Bangert的一个问题

Minimal foliations, codimension-one stable norms, and a question of Bangert

该研究计算环面上等距群余维一度量的余维一稳定范数，对Bangert的问题给出否定答案，在三维环面构造无限维非平坦度量族，证明欧氏稳定范数与体积数据在立方平坦度量附近非局部单射。

Stability and curvature estimates for minimal graphs with flat normal bundles

It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minimal graphs of dimension no greater than six and any codimension is flat if the the normal bundle is flat and the density at infinity is finite. Such a Bernstein type theorem holds in any dimension if we assume additionally growth conditions on the volume element.

Self-shrinkers of the mean curvature flow in arbitrary codimension

For hypersurfaces of dimension greater than one, Huisken showed that compact self-shrinkers of the mean curvature flow with positive scalar mean curvature are spheres. We will prove the following extension: A compact self-similar solution in arbitrary codimension and of dimension greater than one is spherical, i.e. contained in a sphere, if and only if the mean curvature vector \be H\ee is non-vanishing and the principal normal \beν\ee is parallel in the normal bundle. We also give a classification of complete noncompact self-shrinkers of that type.