NONNEGATIVE RICCI CURVATURE AND MINIMAL GRAPHS WITH LINEAR GROWTH Giulio Colombo, Eddygledson S. Gama, Luciano Mari, Marco Rigoli Analysis and Pde, 2024 We study minimal graphs with linear growth on complete manifolds $M^m$ with $\\mathrm{Ric} \\ge 0$. Under the further assumption that the $(m-2)$-th Ricci curvature in radial direction is bounded below by $C r(x)^{-2}$, we prove that any such graph, if non-constant, forces tangent cones at infinity of $M$ to split off a line. Note that $M$ is not required to have Euclidean volume growth. We also show that $M$ may not split off any line. Our result parallels that obtained by Cheeger, Colding and Minicozzi for harmonic functions. The core of the paper is a new refinement of Korevaar's gradient estimate for minimal graphs, together with heat equation techniques.
A barrier principle at infinity for varifolds with bounded mean curvature Eddygledson S. Gama, Jorge H.S. de Lira, Luciano Mari, Adriano A. de Medeiros Journal of the London Mathematical Society, 2022 Our work investigates varifolds Σ⊂M$\\Sigma \\subset M$ in a Riemannian manifold, with arbitrary codimension and bounded mean curvature, contained in an open domain Ω$\\Omega$ . Under mild assumptions on the curvatures of M$M$ and on ∂Ω$\\partial \\Omega$ , also allowing for certain singularities of ∂Ω$\\partial \\Omega$ , we prove a barrier principle at infinity, namely we show that the distance of Σ$\\Sigma$ to ∂Ω$\\partial \\Omega$ is attained on ∂Σ$\\partial \\Sigma$ . Our theorem is a consequence of sharp maximum principles at infinity for varifolds, of independent interest.
The Jenkins–Serrin problem for translating horizontal graphs in M × R Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira, Francisco Martín Revista Matematica Iberoamericana, 2021 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 .
Jenkins–Serrin Graphs in M× R Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira, Francisco Martín Springer Proceedings in Mathematics and Statistics, 2021
Translating Solitons of the Mean Curvature Flow Asymptotic to Hyperplanes in Rn+1 Eddygledson S Gama, Francisco Martín International Mathematics Research Notices, 2020 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, that is, 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 that are parallel to the velocity and the family of tilted grim reaper cylinders as the only translating solitons in $\\mathbb{R}^{n+1}$ that 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 2nd 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.
Translating solitons c1-asymptotic to two half-hyperplanes Eddygledson Gama Proceedings of the American Mathematical Society, 2020 We prove that the hyperplanes parallel to e n + 1 \\mathbf {e}_{n+1} are the unique examples of translating solitons C 1 − C^1- asymptotic to two half-hyperplanes outside a vertical cylinder in R n + 1 \\mathbb {R}^{n+1} . This result generalizes a previous result due to F. Martín and the author.
