https://neckpinch.com/authors/gregorio-silva-neto/New arXiv papers by Gregório Silva Neto, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.25833arxiv:2610.25833Mon, 19 Oct 2026 00:00:00 GMTGregŕio Silva Neto — We describe the self-expanders of the mean curvature flow in ℝ^(p+q+2) invariant under O(p+1)× O(q+1). Apart from the Clifford minimal cone, every such hypersurface is complete, proper and noncompact, and belongs to one of three families: two one-parameter families of embedded hypersurfaces diffeomorphic to ℝ^(p+1)×S^(q) or to ℝ^(q+1)×S^(p), whose profile curves leave an axis, and a two-parameter family diffeomorphic to ℝ×S^(p)×S^(q), parametrized by the point of the profile curve closest to the origin, which is the generic family: its set of initial conditions is open and of full measure in the phase space. Every end is asymptotically conical, with explicit decay rates, and every such hypersurface has Euclidean volume growth. A profile curve of the third family has at most one loop, and the point of the curve closest to the origin lies on that loop.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & CMC surfaces https://arxiv.org/abs/2610.25831arxiv:2610.25831Mon, 19 Oct 2026 00:00:00 GMTGregório Silva Neto — We describe the global structure of the self-shrinkers of the mean curvature flow in ℝ^(p+q+2) invariant under O(p+1)× O(q+1), without assuming properness or embeddedness. The main tool is a pair of piecewise monotone quantities along the profile curves, in the spirit of Hsiang's analysis of constant mean curvature hypersurfaces. We prove that every such self-shrinker other than the two invariant cylinders intersects both of them transversely, and all of them except the Clifford minimal cone are complete. Apart from those of the cylinders, each end is either asymptotic to a cone or meets both cylinders infinitely many times. Consequently, a complete noncompact example is proper if and only if all its ends are asymptotically conical, and this is equivalent to polynomial volume growth, to finite entropy, and to finitely many intersections with the cylinders. The reduced flow preserves a finite measure, and for almost every initial condition the profile curve is either closed or generates a complete nonproper self-shrinker of infinite entropy. We also bound the winding of the profile curve by its number of cylinder crossings and localize its self-intersections.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & CMC surfaces https://arxiv.org/abs/2610.14227arxiv: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://arxiv.org/abs/2601.08661arxiv:2601.08661Tue, 13 Jan 2026 00:00:00 GMTHilário Alencar, G. Pacelli Bessa, Gregório Silva Neto — In this paper, we establish nonexistence results for complete translating solitons of the r-mean curvature flow under suitable growth conditions on the (r-1)-mean curvature and on the norm of the second fundamental form. We first show that such solitons cannot be entirely contained in the complement of a right rotational cone whose axis of symmetry is aligned with the translation direction. We then relax the growth condition on the (r-1)-mean curvature and prove that properly immersed translating solitons cannot be confined to certain half-spaces opposite to the translation direction. We conclude the paper by showing that complete, properly immersed translating solitons satisfying appropriate growth conditions on the (r-1)-mean curvature cannot lie completely within the intersection of two transversal vertical half-spaces.Mean curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2402.09627arxiv:2402.09627Thu, 15 Feb 2024 00:00:00 GMTHilário Alencar, G. Pacelli Bessa, Gregório Silva Neto — In this paper, we prove gap results for complete self-shrinkers of the r-mean curvature flow involving a modified second fundamental form. These results extend previous results for self-shrinkers of the mean curvature flow due to Cao-Li and Cheng-Peng. To prove our results we show that, under suitable curvature bounds, proper self-shrinkers are parabolic for a certain second-order differential operator which generalizes the drifted Laplacian and, even if is not proper, this differential operator satisfies an Omori-Yau type maximum principle.Mean curvature flowSelf-shrinkers & translators https://arxiv.org/abs/2308.11679arxiv:2308.11679Tue, 22 Aug 2023 00:00:00 GMTGregório Silva Neto, Vanessa Silva — In this paper, we classify the nondegenerate ruled surfaces in the three-dimensional Lorentz-Minkowski space that are homothetic self-similar solutions for the inverse mean curvature flow. This classification shows the existence of two classes of non-cylindrical homothetic solitons: one with lightlike rulings and another one with non-lightlike rulings.Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifolds https://arxiv.org/abs/2308.11457arxiv:2308.11457Tue, 22 Aug 2023 00:00:00 GMTGregório Silva Neto, Vanessa Silva — In this paper, we classify the nondegenerate ruled surfaces in the three-dimensional Lorentz-Minkowski space that are translating solitons for the inverse mean curvature flow. In particular, we prove the existence of non-cylindrical ruled translating solitons, which contrast with the Euclidean setting.Mean curvature flowOther curvature flowsInverse mean curvature flowSelf-shrinkers & translators3-manifolds https://arxiv.org/abs/2203.10334arxiv: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 https://arxiv.org/abs/2203.06270arxiv:2203.06270Fri, 11 Mar 2022 00:00:00 GMTHilário Alencar, Gregório Silva Neto, Detang Zhou — In 1951, H. Hopf proved that the only surfaces, homeomorphic to the sphere, with constant mean curvature in the Euclidean space are the round (geometrical) spheres. In this paper we survey some contributions of Renato Tribuzy to generalize the result of Hopf as well as some recent results of the authors using these techniques for shrinking solitons of curvature flows and for surfaces in three-dimensional warped product manifolds, specially the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds.Other curvature flowsRicci solitons3-manifoldsRotational symmetry & warped productsPhysics https://arxiv.org/abs/2203.06214arxiv:2203.06214Fri, 11 Mar 2022 00:00:00 GMTHilário Alencar, Gregório Silva Neto — In this paper we prove that two-dimensional translating solitons in ℝ³ with finite L-index are homeomorphic to a plane or a cylinder and that a two-dimensional self-expander with finite L-index and sub exponential weighted volume growth has finite topology. We also prove that translating solitons and self-expanders have finite topology, provided the bottom of the spectrum of the L-stability operator is bounded from below and their weighted volume have subexponential growth.Mean curvature flowSelf-shrinkers & translatorsSurfacesStability & uniquenessEigenvalues & spectrum