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.
Min-max construction of minimal surfaces with a fixed angle at the boundary
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abstract
We prove the existence of minimal surfaces in a bounded convex subset of $\mathbb R^3$, $\mathcal M$, intersecting the boundary of $\mathcal M$ with a fixed contact angle. The proof is based on a min-max construction in the spirit of Almgren-Pitts for the capillarity functional.
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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.