For all $N\ge 9$, we find smooth entire epigraphs in ${\mathbb{R}}^N$, namely, smooth domains of the form $\Omega :=\{x\in {\mathbb{R}}^N|x_N>F(x_1,\dots ,x_{N-1})\}$, which are not half-spaces and in which a problem of the form $\Delta u+f\left(u\right)=0$ in $\Omega$ has a positive, bounded solution with $0$ Dirichlet boundary data and constant Neumann boundary data on $\partial \Omega$. This answers negatively for large dimensions a question by Berestycki, Caffarelli, and Nirenberg. In 1971, Serrin proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy to a famous result by Alexandrov that states that an embedded compact surface with constant mean curvature (CMC) in Euclidean space must be a sphere. In lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given CMC surface where Serrin’s overdetermined problem is solvable.

Among other things, we prove the following two topological statements about closed hyperbolic $3$-manifolds. First, every rational second homology class of a closed hyperbolic $3$-manifold has a positive integral multiple represented by an oriented connected closed ${\pi }_1$-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, ${\pi }_1$-injectively immersed oriented closed $1$-submanifold in a closed hyperbolic $3$-manifold has an equidegree finite cover which bounds an oriented connected compact ${\pi }_1$-injectively immersed quasi-Fuchsian subsurface. In, we exploit techniques developed by Kahn and Markovic but we only distill geometric and topological ingredients from those papers, so no hard analysis is involved in this article.