5 problems

Let $\mathcal{C}_{\theta}$ be the spherical cap, let $\alpha>\frac{1}{n+2}$ and let $\theta\in(0,\pi)$. Suppose that $h$ is a positive, strictly convex function on…

Let $\widehat{\Sigma}\in\mathcal{K}_{\theta}$ and let $\theta\in(0,\pi)$. For $z\in\operatorname{int}(\widehat{\partial\Sigma})$, write $\widehat{\Sigma}^{*}_{z}$ for the correspon…

Let $\mathcal{C}_{\theta}$ be the spherical cap and consider the capillary soliton equation referred to as Eq. … . The preceding theorem shows convergence of the normalized capilla…

Let $n\geq 2$, let $\Sigma\subset -{\mathbb R}^{n+1}_{+}$ be a convex hypersurface with capillary boundary supported on $\partial-{\mathbb R}_{+}^{n+1}$ at a contact angle…

Let $M^n$, with $n\geq 3$, be a capillary hypersurface in the unit Euclidean ball $B^{n+1}$. Classification conjecture. The only stable such hypersurfaces are a totally geodesic hy…