arXiv subjects
Publications and source records attributed to Eddygledson S. Gama.
We prove the existence of horizontal Jenkins-Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds $M\times\mathbb{R}$. Moreover, we give examples of these graphs in the cases of $\mathbb{R}^3$ and $\mathbb{H}^2\times\mathbb{R}$.
The so called Jenkins-Serrin problem is a kind of Dirichlet problem for graphs with prescribed mean curvature that combines, at the same time, continuous boundary data with regions of the boundary where the boundary values explodes either to $+\infty$ or to $-\infty.$ We give a survey on the development of Jenkins-Serrin type problems over domains on Riemannian manifolds. The existence of this type of graphs imposes restrictions on the geometry of the boundary of these domains. We also improve some earlier results by proving Theorem 1.8, and prove the existence of translating Jenkins-Serrin graphs (Theorem 1.9).
A translating soliton is a hypersurface $M$ in $\mathbb{R}^{n+1}$ such that the family $M_t= M- t \,\mathbf{e}_{n+1}$ is a mean curvature flow, i.e., such that normal component of the velocity at each point is equal to the mean curvature at that point $\mathbf{H}=\mathbf{e}_{n+1}^{\perp}.$ In this paper we obtain a characterization of hyperplanes which are parallel to $\mathbf{e}_{n+1}$ and the family of tilted grim reaper cylinders as the only translating solitons in $\mathbb{R}^{n+1}$ which are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder. This result was proven for translators in $\mathbb{R}^3$ by the second author, Perez-Garcia, Savas-Halilaj and Smoczyk under the additional hypotheses that the genus of the surface was locally bounded and the cylinder was perpendicular to the translating velocity.