https://neckpinch.com/authors/alvaro-kruger-ramos/New arXiv papers by Álvaro Krüger Ramos, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2602.02424arxiv:2602.02424Mon, 02 Feb 2026 00:00:00 GMTRonaldo F. de Lima, Álvaro K. Ramos — We develop the theory of translating solitons for the Mean Curvature Flow (MCF) in hyperbolic space of dimension n+1≥ 3. More specifically, we establish that horospheres are dynamically stable as radial graphical solutions to MCF. To that end, we construct rotationally invariant translators analogous to the winglike solitons introduced by Clutterbuck, Schnürer and Schulze, which serve as barriers in an argument based on White's avoidance principle and the strong maximum principle for parabolic PDEs.Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessRotational symmetry & warped products https://arxiv.org/abs/2402.04757arxiv:2402.04757Wed, 07 Feb 2024 00:00:00 GMTRonaldo F. de Lima, Álvaro K. Ramos, João Paulo dos Santos — We establish the existence of one-parameter families of helicoidal surfaces of ℍ²×ℝ which, under mean curvature flow, simultaneously rotate about a vertical axis and translate vertically.Mean curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2307.14136arxiv:2307.14136Wed, 26 Jul 2023 00:00:00 GMTR. F. de Lima, A. K. Ramos, J. P. dos Santos — We consider translators (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic 3-space ℍ³, providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete rotational translators in ℍ³, one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in ℍ³ and apply it to prove that properly immersed translators to MCF in ℍ³ are not cylindrically bounded. As a further application of the tangency principle, we prove that any horoconvex translator which is complete or transversal to the x₃-axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in ℍ³ which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in ℍ³ and, after classifying the rotators of constant mean curvature, we show that there exists a one-parameter family of complete rotators which are all helicoidal, bringing to the hyperbolic context a distinguished result by Halldorsson, set in ℝ³.Mean curvature flowSelf-shrinkers & translatorsStability & uniqueness https://arxiv.org/abs/2210.09781arxiv:2210.09781Tue, 18 Oct 2022 00:00:00 GMTIzabella Muraro de Freitas, Álvaro Krüger Ramos — So far, the most magnificent breakthrough in mathematics in the 21st century is the Geometrization Theorem, a bold conjecture by William Thurston (generalizing Poincaré's Conjecture) and proved by Grigory Perelman, based on the program suggested by Richard Hamilton. In this survey article, we will explain the statement of this result, also presenting some examples of how it can be used to obtain interesting results in differential geometry.Geometrization