Existence of free boundary disks with constant mean curvature in R3

Abstract

Given a surface Σ in $R^3$ diffeomorphic to $S^2$, Struwe [38] proved that for almost every H below the mean curvature of the smallest sphere enclosing Σ, there exists a branched immersed disk which has constant mean curvature H and boundary meeting Σ orthogonally. We reproduce this result using a different approach and improve it under additional convexity assumptions on Σ. Specifically, when Σ itself is convex and has mean curvature bounded below by $H_0$, we obtain existence for all $H\in (0,H_0)$. Instead of the heat flow in [38], we use a Sacks-Uhlenbeck type perturbation. As in previous joint work with Zhou [7], a key ingredient for extending existence across the measure zero set of H's is a Morse index upper bound.