https://neckpinch.com/authors/matheus-hudson/New arXiv papers by Matheus Hudson, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT 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/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