Some results about constant mean curvature graphs in Killing submersions

Seminario de Geometría

Abstact: A Killing submersion is a Riemannian submersion from a 3-manifold \(\mathbb{E}\) to a surface \(M\), both connected and orientable, whose fibers are the integral curves of a non-vanishing Killing vector field \(\xi\in\mathfrak{X}(\mathbb{E})\). In this setting we give a suitable definition of the graph of a function \(f\in C^\infty(U)\), where \(U\) is an open subset of \(M\). We study the existence and uniqueness of solutions for the Jenkins-Serrin problem on relatively compact domains of \(M\) and we prove two general Collin-Krust type estimates for prescribed mean curvature graphs that extend classical result. Finally, we use these tools to prove that in the Heisenberg group there exists a unique minimal graph with prescribed bounded boundary values in every unbounded domain contained in a strip of the plane \(\left\{z=0\right\}\). This talk is partially based on a joint work with J.M. Manzano and B. Nelli.