Existence of constant mean curvature disks in $\mathbb{R}^3$ with capillary boundary condition

Abstract

We extend Struwe's result (Acta Math., 1988) on the existence of free boundary constant mean curvature disks to almost every prescribed boundary contact angle in $(0, π)$. Specifically, let $Σ$ be a surface in $\mathbb{R}^3$ diffeomorphic to the sphere, and let $Σ'$ be a convex surface enclosing $Σ$. Given $τ\in (-1, 1)$ and a constant $H \geq 0$ below the infimum of the mean curvature of $Σ'$, we show that for almost every $r \in (0, 1)$, in the region enclosed by $Σ'$ there exists a branched immersed disk with constant mean curvature $rH$ whose boundary meets $Σ$ at an angle with cosine value $rτ$. Moreover, the constant mean curvature disks we construct have index at most $1$.