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

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 $(0,\pi )$. Specifically, let $\mathrm{\Sigma }$ be a surface in ${\mathbb{R}}^3$ diffeomorphic to the sphere, and let ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ be a convex surface enclosing $\mathrm{\Sigma }$. Given $\tau \in (-1,1)$ and a constant $H\ge 0$ below the infimum of the mean curvature of ${\mathrm{\Sigma }}^{\mathrm{\prime }}$, we show that for almost every $r\in (0,1)$, in the region enclosed by ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ there exists a branched immersed disk with constant mean curvature $rH$ whose boundary meets $\mathrm{\Sigma }$ at an angle with cosine value $r\tau$. Moreover, the constant mean curvature disks we construct have index at most $1$.