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Regularity of minimal surfaces with capillary boundary conditions
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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 $1$.
Forward citations
Cited by 3 Pith papers
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Weiss monotonicity and capillary hypersurfaces
Renormalized capillary area density converges to the Weiss energy, giving angle-independent curvature estimates and a Bernstein theorem for capillary minimizers.
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Some remarks on singular capillary cones with free boundary
Minimizing capillary cones are flat in dimension 4 when the free-boundary mean curvature has one sign, and axially symmetric ones are flat up to dimension 6.
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The weighted isoperimetric inequality and Sobolev inequality outside convex sets
The paper's main isoperimetric and Sobolev claims rest on a weight assumption that is impossible as written, so the outside-convex-set theorems are vacuous.
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