https://neckpinch.com/authors/joao-paulo-dos-santos/New arXiv papers by João Paulo dos Santos, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2511.21545arxiv:2511.21545Wed, 26 Nov 2025 00:00:00 GMTTarcios Andrey Ferreira, João Paulo dos Santos — We consider the hyperbolic three-space in the half-space model endowed with a metric Lie group structure. In this setting, translation surfaces are defined as products of two curves α and β with respect to the Lie group operation. We investigate minimal surfaces and solitons to the mean curvature flow arising from specific types of products of these curves. In particular, we provide classification results for minimal surfaces, hyperbolic translators, and conformal solitons to the mean curvature flow.Mean curvature flowSelf-shrinkers & translatorsSurfacesHomogeneous spacesMinimal & CMC surfaces https://arxiv.org/abs/2510.13057arxiv:2510.13057Wed, 15 Oct 2025 00:00:00 GMTValter Borges, Matheus Andrade Ribeiro de Moura Horácio, João Paulo dos Santos — In this article, we investigate a gradient almost Ricci soliton with harmonic Weyl tensor. We first prove that its Ricci tensor has at most three distinct eigenvalues of constant multiplicities in a neighborhood of a regular point of the potential function. Then, we classify those with exactly two distinct eigenvalues. It is worth mentioning that the case with exactly one eigenvalue has already been settled elsewhere. Our results are based on a local representation of these manifolds as multiply warped products of a one-dimensional base, having at most two Einstein fibers, which we also obtain in this paper. These results extend a result by Catino, who assumes, in addition, that the Weyl tensor is radially flat, and a result by Kim, who considers the four-dimensional case.Ricci solitonsAlmost & η-Ricci solitons4-manifoldsRotational symmetry & warped productsEigenvalues & spectrum https://arxiv.org/abs/2510.11939arxiv:2510.11939Mon, 13 Oct 2025 00:00:00 GMTValter Borges, Matheus Andrade Ribeiro de Moura Horácio, João Paulo dos Santos — In this article, we give a new proof of a result due to J. Kim, which states that the Ricci tensor of a gradient Ricci soliton with dimension n ≥ 4 and harmonic Weyl tensor has at most three distinct eigenvalues. This result constitutes an essential step in the classification of such manifolds, originally established by J. Kim in dimension 4 and subsequently extended to dimensions n≥5. Our proof offers two notable advantages: it is shorter and does not require the use of any specialized moving frame.Ricci solitonsEigenvalues & spectrum 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/2206.02635arxiv:2206.02635Mon, 06 Jun 2022 00:00:00 GMTFelippe Guimarães, João Batista Marques dos Santos, João Paulo dos Santos — We show that the evolution of isoparametric hypersurfaces of Riemannian manifolds by the mean curvature flow is given by a reparametrization of the parallel family in short time, as long as the uniqueness of the mean curvature flow holds for the initial data and the corresponding ambient space. As an application, we provide a class of Riemannian manifolds that admit hypersurfaces with constant principal curvatures, which are not isoparametric hypersurfaces. Furthermore, for a class of ambient spaces, we show that the singularities developed by the mean curvature flow with isoparametric hypersurfaces as the initial data are Type I singularities. We apply our results to describe the evolution of isoparametric hypersurfaces by the mean curvature flow in ambient spaces with nonconstant sectional curvature, such as homogenous 3-manifolds E(κ, τ) with 4-dimensional isometry groups, and Riemannian products ℚ²_(c₁) × ℚ²_(c₂) of space forms.Mean curvature flowSingularities3-manifolds4-manifoldsSphere theoremsStability & uniqueness https://arxiv.org/abs/2107.13918arxiv:2107.13918Thu, 29 Jul 2021 00:00:00 GMTA. L. Martínez-Triviño, J. P. dos Santos — We establish a uniqueness result for the [φ,e₃]-catenary cylinders by their asymptotic behaviour. Well known examples of such cylinders are the grim reaper translating solitons for the mean curvature flow. For such solitons, F. Martín, J. Pérez-García, A. Savas-Halilaj and K. Smoczyk proved that, if Σ is a properly embedded translating soliton with locally bounded genus, and 𝓒^(∞)-asymptotic to two vertical planes outside a cylinder, then Σ must coincide with some grim reaper translating soliton. In this paper, applying the moving plane method of Alexandrov together with a strong maximum principle for elliptic operators, we increase the family of [φ,e₃]-minimal graphs where these types of results hold under different assumption of asymptotic behaviour.Mean curvature flowSelf-shrinkers & translatorsStability & uniqueness