We give improved estimates for the size of the singular set of minimizing capillary hypersurfaces: the singular set is always of codimension at least $4$, and this estimate improves if the capillary angle is close to $0$, $\fracπ{2}$, or $π$. For capillary angles that are close to $0$ or $π$, our analysis is based on a rigorous connection between the capillary problem and the one-phase Bernoulli problem.
In this paper, we study a stability problem of free boundary hypersurfaces, and also capillary ones whose boundary supported on a horosphere in hyperbolic space. We prove that umbilical hypersurfaces are only stable immersed capillary hypersurfaces whose boundary supported on a horosphere. Using the same method, we show that a totally geodesic hyperplane is only stable immersed type-II hypersurface whose boundary supported on a horosphere.
We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known examples except the totally geodesic ones and spherical caps are unstable.
In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. This solves completely a long-standing open problem. In the proof one of crucial ingredients is a new Minkowski type formula. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.
Let $Σ$ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in $\mathbb R^{n+1}$. Suppose that $Σ$ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. It is proved that if $\partial Σ$ is embedded for $n=2$, or if $\partialΣ$ is convex for $n\geq3$, then $Σ$ is part of the sphere. And the same is true for $Σ$ in the half-space of $\mathbb R^{n+1}$ with connected boundary $\partialΣ$.
We study stable immersed capillary hypersurfaces in a domain $\mathcal B$ which is either a half-space or a slab in the Euclidean space $\Bbb R^{n+1}.$ We prove that such a hypersurface $Σ$ is rotationally symmetric in the following cases: (1) $n=2$, $\mathcal B$ is a slab and $Σ$ has genus zero, (2) $n\geq 2$, $\mathcal B$ is a slab, the angle of contact is $π/2$ and each component of $\partialΣ$ is embedded, (3) $n\geq 2,$ $\mathcal B$ is a half-space, the angle of contact is $<π/2$ and each component of $\partialΣ$ is embedded. Moreover, in case (2), if not a right circular cylinder, the hy…
We study stable immersed capillary hypersurfaces $Σ$ in domains B of R n+1 bounded by hyperplanes. When B is a half-space, we show $Σ$ is a spherical cap. When B is a domain bounded by k hyperplanes P 1 ,. .. , P k , 2 $\le$ k $\le$ n + 1, having independent normals, and $Σ$ has contact angle $θ$ i with P i and does not touch the vertices of B, we prove there exists $δ$ > 0, depending only on P 1 ,. .. , P k , so that if $θ$ i $\in$ ($π$ 2 -- $δ$, $π$ 2 + $δ$) for each i, then $Σ$ has to be a piece of a sphere.
We propose a notion of stability for capillary hypersurfaces with constant higher order mean curvature and we generalize some results of the classical stability theory for CMC capillary hypersurfaces.
In a round ball $B\subset\mathbb{R}^{n+1}$ endowed with an $O(n+1)$-invariant metric we consider a radial function that weights volume and area. We prove that a compact two-sided hypersurface in $B$ which is stable capillary in weighted sense and symmetric about some line containing the center of $B$ is homeomorphic to a closed $n$-dimensional disk. When combined with Hsiang symmetrization and other stability results this allows to deduce that the interior boundary of any isoperimetric region in $B$ for the Gaussian weight is a closed $n$-disk of revolution. For $n=2$ we also show that a compa…
We give an a priori bound on the (n-7)-dimensional measure of the singular set for an area-minimizing n-dimensional hypersurface, in terms of the geometry of its boundary.
Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.
Locally stable minimal hypersurface could have singularities in dimension $\geq 7$ in general, locally modeled on stable and area-minimizing cones in the Euclidean spaces. In this paper, we present different aspects of how these singularities may affect the local behavior of minimal hypersurfaces. First, given a non-degenerate minimal hypersurface with strictly stable and strictly minimizing tangent cone at each singular point, under any small perturbation of the metric, we show the existence of a nearby minimal hypersurface under new metric. For a generic choice of perturbation direction, we …
We extend the results of Hardt and Simon on area-minimizing cones to prove that isolated singularities of stationary one-sided area-minimizing hypersurfaces can be locally perturbed away on the side that they are minimizing.
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.
