Symmetries in some extremal problems between two parallel hyperplanes

TL;DR

This paper proves symmetry properties of certain hypersurfaces with boundary between two parallel hyperplanes, under various boundary conditions and assumptions on mean curvature dependence.

Contribution

It establishes new symmetry results for hypersurfaces with boundary in a slab, considering different boundary conditions and mean curvature dependencies.

Findings

Hypersurfaces are symmetric under constant contact angle conditions.

Symmetry holds when the boundary derivative of the height function is non-increasing.

Symmetry is also proven when the boundary is symmetric to a perpendicular orthogonal to the hyperplanes.

Abstract

Let $M$ be a compact hypersurface with boundary $\partial M=\partial D_1 \cup \partial D_2$, $\partial D_1 \subset \Pi _1$, $\partial D_2 \subset \Pi _2$, $\Pi_1$ and $\Pi _2$ two parallel hyperplanes in $\mathbb{R}^{n+1}$ ($n \geq 2$). Suppose that $M$ is contained in the slab determined by these hyperplanes and that the mean curvature $H$ of $M$ depends only on the distance $u$ to $\Pi _i$, $i=1,2$. We prove that these hypersurfaces are symmetric to a perpendicular orthogonal to $\Pi _i$, $i=1,2$, under different conditions imposed on the boundary of hypersurfaces on the parallel planes: (i) when the angle of contact between $M$ and $\Pi _i$, $i=1,2$ is constant; (ii) when $\partial u / \partial \eta$ is a non-increasing function of the mean curvature of the boundary, $\partial \eta$ the inward normal; (iii) when $\partial u / \partial \eta$ has a linear dependency on the distance to a fixed point inside the body that hypersurface englobes; (iv) when $\partial D_i$ are symmetric to a perpendicular orthogonal to $\Pi _i$, $i=1,2$.