4 problems

Let $N$ be a smooth Riemannian manifold, let $g\in C^{1,\alpha}$, and let $E\subset N$ be a Caccioppoli set satisfying the hypotheses of Corollary 2 with the stability assumption (…

Twin-Bubble Conjecture. There are at least two almost-embedded hypersurfaces in $M$ with mean curvature prescribed by $h$.

Let $(M^{n+1},g)$ be the ambient Riemannian manifold under consideration, let $h\in C^{\infty}(M)$, and let $H_{Y_p}$ denote the mean curvature of a hypersurface $Y_p\subset M$. Pr…

Entire-graph existence conjecture. There should exist an entire graph in $\mathbb{L}^3$ with prescribed mean curvature $H$.