https://neckpinch.com/authors/jose-n-v-gomes/New arXiv papers by José N. V. Gomes, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2604.14366arxiv:2604.14366Wed, 15 Apr 2026 00:00:00 GMTJosé N. V. Gomes, Willian I. Tokura, Hikaru Yamamoto — We study the Ricci-Bourguignon flow on warped product manifolds with noncompact base. This setting leads naturally to a parabolic partial differential equation on the space of smooth warping functions, arising from the necessary and sufficient conditions for a warped metric to evolve under the flow. One of our main results establishes a gradient estimate for this equation, providing the analytic input for the geometric applications developed herein and, in particular, recovering classical gradient estimates for the heat equation under the Ricci flow. Furthermore, we develop a method for constructing explicit warped product solutions to the Ricci-Bourguignon flow and present examples that illustrate the scope and geometric relevance of our resultsRicci flowBoundaries & non-compactRotational symmetry & warped products https://arxiv.org/abs/2603.26903arxiv:2603.26903Fri, 27 Mar 2026 00:00:00 GMTJosé Nazareno Vieira Gomes, Marcus Antonio Mendonça Marrocos — We develop a method for constructing complete gradient Ricci solitons realized as fiber bundles endowed with warped metrics, and we establish necessary and sufficient conditions for their existence. As an application, we present new examples of complete gradient steady and shrinking Ricci solitons obtained via quotients by isometric group actionsRicci solitons https://arxiv.org/abs/2510.23239arxiv:2510.23239Mon, 27 Oct 2025 00:00:00 GMTJosé N. V. Gomes, Matheus Hudson, Carlos M. de Sousa — The main objective of this article is to study the mean curvature flow into an ambient compact smooth manifold M with boundary and with a Riemannian metric that evolves by a self-similar solution of the Ricci flow coupled with the harmonic map heat flow of a map from M to a Riemannian manifold N. In this context, we address a functional associated with this flow and calculate its variation along parameters that preserve the weighted volume measure. An extension of Hamilton's differential Harnack expression appears by considering the boundary of M evolving by mean curvature flow, which must vanish on the gradient steady soliton case. Next, we obtain a Huisken monotonicity-type formula for the mean curvature flow in the proposed background. We also show how to construct a family of mean curvature solitons and establish a characterization of such a family.Ricci flowMean curvature flowRicci solitonsHarnack inequalitiesHarmonic maps & DeTurck trick https://arxiv.org/abs/2412.19939arxiv:2412.19939Fri, 27 Dec 2024 00:00:00 GMTJosé N. V. Gomes, Matheus Hudson, Hikaru Yamamoto — We give an application of a Huisken monotonicity-type formula for the mean curvature flow in a compact smooth manifold with a Riemannian metric that evolves by a shrinking self-similar solution of the extended Ricci flow. Our investigation builds on previous articles by Huisken and the third author, as we apply their techniques to establish new results in this geometric setting. Moreover, under some natural geometric assumptions, the noncompact case is also solvedRicci flowMean curvature flowRicci solitonsBoundaries & non-compact https://arxiv.org/abs/2306.06504arxiv:2306.06504Sat, 10 Jun 2023 00:00:00 GMTCleiton Lira Cunha, José Nazareno Vieira Gomes, Marcus Antônio Mendonça Marrocos — We use variational methods to derive Hadamard-type formulae for the eigenvalues of a class of elliptic operators on a compact Riemannian manifold M. We then apply the latter in the following context. Consider a family of elliptic operators which is parametrized by either the set of all 𝓒^r–Riemannian metrics on M or the set of all 𝓒^r–diffeomorphisms on a domain into M. In either case, we prove that if a subset of the parametrizations set yields a simple spectrum of the operator, then it is necessarily a generic subset. We also analyse the behavior of the eigenvalues when the metric evolves along the Ricci flow on a closed Riemannian manifold, and we prove, under a suitable hypothesis, that they increaseRicci flowEigenvalues & spectrum https://arxiv.org/abs/2301.07073arxiv:2301.07073Tue, 17 Jan 2023 00:00:00 GMTJosé N. V. Gomes, Matheus Hudson — In this paper, we consider functionals related to mean curvature flow in an ambient space which evolves by an extended Ricci flow from the perspective introduced by Lott when studying a mean curvature flow in a Ricci flow background. One of them is a weighted extended version of the Gibbons-Hawking-York action on Riemannian metrics in compact manifolds with boundary. We compute its variational properties from which naturally arise boundary conditions to the analysis of its time-derivative under Perelman's modified extended Ricci flow. For instance, the boundary integrand term provides an extension of Hamilton's differential Harnack expression for mean curvature flows in Euclidean space. We also derive the evolution equations for both the second fundamental form and the mean curvature under mean curvature flow in an extended Ricci flow background. In the special case of gradient solitons to the extended Ricci flow, we discuss mean curvature solitons and establish a Huisken's monotonicity-type formula. We show how to construct a family of mean curvature solitons and establish a characterization of such a family. Also, we show how for constructing examples of mean curvature solitons in an extended Ricci flow background.Ricci flowMean curvature flowRicci solitonsHarnack inequalitiesBoundaries & non-compact https://arxiv.org/abs/2109.09870arxiv:2109.09870Mon, 20 Sep 2021 00:00:00 GMTJosé N. V. Gomes, Marcus A. M. Marrocos, Adrian V. C. Ribeiro — In this note, we prove triviality and nonexistence results for gradient Ricci soliton warped metrics. The proofs stem from the construction of gradient Ricci solitons that are realized as warped products, from which we know that the base spaces of these products are Ricci-Hessian type manifolds. We study this latter class of manifolds as the most appropriate setting to prove our results.Ricci solitonsRotational symmetry & warped products