Regularity of minimal surfaces with capillary boundary conditions

TL;DR

This paper establishes regularity results for varifolds with capillary boundary conditions in Riemannian manifolds, showing that under certain conditions, such varifolds are smooth hypersurfaces near the boundary.

Contribution

It proves $ ext{ε}$-regularity theorems for capillary varifolds, including uniform first variation control and regularity near boundary points with density less than one.

Findings

Capillary varifolds with bounded mean curvature are smooth near the boundary.

The paper provides conditions under which varifolds coincide with $C^{1,eta}$ hypersurfaces.

Regularity results apply at generic boundary points where density is less than one.

Abstract

We prove $\varepsilon$-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya-Tonegawa \cite{KaTo}. We establish a uniform first variation control for all such varifolds (and free-boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half-plane with angle not equal to $\tfrac{\pi}{2}$, then it coincides with a $C^{1,\alpha}$ properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than $1$.