https://neckpinch.com/authors/ronaldo-f-de-lima/New arXiv papers by Ronaldo F. de Lima, 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/2211.03918arxiv:2211.03918Tue, 08 Nov 2022 00:00:00 GMTRonaldo F. de Lima, Giuseppe Pipoli — We consider translators to the extrinsic flows in ℝⁿ×ℝ and ℍⁿ×ℝ (called r-mean curvature flows or r-MCF, for short) whose velocity functions are the higher order mean curvatures Hᵣ. We show that there exist rotational bowl-type and catenoid-type translators to r-MCF in both ℝⁿ×ℝ and ℍⁿ×ℝ, and also that there exist parabolic and hyperbolic catenoid-type translators to r-MCF in ℍⁿ×ℝ. In addition, we show that there exist Grim Reaper-type translators to Gaussian flow (n-MCF) in ℝⁿ×ℝ and ℍⁿ×ℝ. We also establish the uniqueness of all these translators (together with certain cylinders) among those which are invariant by either rotations or translations (Euclidean, parabolic or hyperbolic). We apply this uniqueness result to classify the translators to r-MCF in ℝⁿ×ℝ and ℍⁿ×ℝ whose r-th mean curvature is constant, as well as those which are isoparametric. Our results extend to the context of r-MCF in ℝⁿ×ℝ and ℍⁿ×ℝ the existence and uniqueness theorems by Altschuler–Wu (of the bowl soliton) and Clutterbuck–Schnürer–Schulze (of the translating catenoids) in Euclidean space.Mean curvature flowSelf-shrinkers & translatorsStability & uniqueness