https://neckpinch.com/authors/marcio-batista/New arXiv papers by Márcio Batista, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2610.14227/arxiv:2610.14227Fri, 09 Oct 2026 00:00:00 GMTHilário Alencar, Márcio Batista, Gregório Silva Neto — For 1≤ r≤ n-1 we study shrinking self-similar ends of the r-mean curvature flow, Sᵣ=c⟨ X,N⟩ with c<0, that are asymptotic to a cone. For every rotational cone of positive slope we construct a smooth embedded rotational end and prove that its profile is the only positive solution of the profile equation with the prescribed slope at infinity. Its angular defect xu'−u is strictly negative, which yields r-admissibility, and the profile has a complete asymptotic expansion for u(x)/x in powers of x^(−(r+1)), valid after arbitrary differentiation. The profiles depend real analytically on the slope and, on sufficiently distant exterior regions, foliate the image of the corresponding family. Our main result removes the symmetry assumption: two ends that converge smoothly, with multiplicity one, to the same circular cone coincide outside a compact set. The proof transforms the equation for their difference into an almost-eigenfunction inequality for a drift Laplacian on an auxiliary conical metric and applies Bernstein's asymptotic analysis on weakly conical ends. The argument extends to strictly r-admissible cones whose links have constant principal curvatures.Mean curvature flowStability & uniquenessBoundaries & non-compactLong-time existence & convergenceEigenvalues & spectrum https://neckpinch.com/papers/2203.10334/arxiv:2203.10334Sat, 19 Mar 2022 00:00:00 GMTHilário Alencar, Márcio Batista, Gregório Silva Neto — In this article, under mild constraints on the sectional curvature, we exploit a divergence formula for symmetric endomorphisms to deduce a general Poincaré type inequality. We apply such inequality to higher-order mean curvature of hypersurfaces of space forms and Einstein manifolds, to obtain several isoperimetric inequalities, as well as rigidity results for complete r-minimal hypersurfaces satisfying a suitable decay of the second fundamental form at infinity. Furthermore, using these techniques, we prove flatness and non-existence results for self-similar solutions to a large class of fully nonlinear curvature flows.Other curvature flowsEinstein metricsSphere theoremsStability & uniquenessGeometric inequalitiesMinimal & CMC surfaces