https://neckpinch.com/authors/esther-cabezas-rivas/New arXiv papers by Esther Cabezas-Rivas, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.20265arxiv:2610.20265Wed, 14 Oct 2026 00:00:00 GMTEsther Cabezas-Rivas, Carlo Sinestrari — We prove stability of geodesic spheres under volume-preserving mean curvature flow in the simply connected space forms under genuinely first-order geometric assumptions. The initial hypersurface is required to lie in a sufficiently thin annulus around a geodesic sphere and to have outward normal making a small angle with the radial direction. No convexity, mean convexity, initial curvature bound, or Hölder control of the normal is assumed. The main ingredient is a quantitative Alexandrov reflection argument that converts these initial assumptions into a strong reflection property preserved by the flow. The resulting radial and angular control allows us to derive estimates for the mean curvature and the full second fundamental form, and hence long-time existence, eventual convexity, and smooth exponential convergence to a geodesic sphere. In hyperbolic space, the eventual convexity can be strengthened to horospherical convexity. The reflection threshold also yields a quantitative estimate for the displacement of the limiting centre. For Lipschitz initial hypersurfaces, we develop a separate positive-time curvature estimate that is uniform along smooth approximations. This produces an instantaneously smooth evolution with the same eventual convexity and smooth exponential convergence.Mean curvature flowVolume-preserving & constrained flowsSphere theoremsStability & uniquenessLong-time existence & convergence https://arxiv.org/abs/2609.37549arxiv:2609.37549Tue, 29 Sep 2026 00:00:00 GMTRichard H. Bamler, Esther Cabezas-Rivas — We prove Petrunin's smoothing conjecture in all dimensions: every compact Euclidean polyhedral space without boundary and with nonnegative Alexandrov curvature is a Gromov-Hausdorff limit of smooth Riemannian orbifolds with geometrically nonnegative curvature. More strongly, the approximating metrics are positive-time slices of a single orbifold Ricci flow whose metric initial condition is the given polyhedral space. The proof rests on a new short-time existence and regularization theory for Ricci flow that, under two-sided volume bounds and a generalized segment inequality, replaces pointwise lower curvature control by small scale-invariant integral control of the defect from a preserved curvature cone. Although it allows arbitrarily large pointwise violations, this theory yields an existence time and positive-time curvature estimates independent of the initial upper curvature bound. As a further application, it gives rigidity consequences for manifolds with small integral curvature defect.Ricci flowLimit spaces & RCDStability & uniquenessHarmonic maps & DeTurck trick https://arxiv.org/abs/2608.10745arxiv:2608.10745Tue, 11 Aug 2026 00:00:00 GMTSara Albert-Niclòs, Esther Cabezas-Rivas, Shujing Pan — We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential C^∞ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.Other curvature flowsCurvature pinchingHomogeneous spacesLong-time existence & convergenceGeometric inequalities https://arxiv.org/abs/2604.13734arxiv:2604.13734Wed, 15 Apr 2026 00:00:00 GMTSara Albert-Niclòs, Esther Cabezas-Rivas — We study area- and length-preserving curvature flows for embedded closed curves on pinched Hadamard surfaces. In the variable-curvature setting, the evolution equations contain additional lower-order terms, so the PDE analysis requires refined comparison arguments and delicate curvature estimates. For smooth convex initial curves, we prove preservation and instantaneous strictness of convexity, long-time existence, and uniform bounds for the curvature and its higher derivatives. Under additional geometric assumptions, we obtain convergence of the curvature to a constant. In the rotationally symmetric case, the area-preserving flow exhibits a dichotomy: either the evolving curves converge exponentially to a geodesic circle, or they drift off to infinity and approach a constant-curvature limit curve. We also identify a geometric condition on the initial curve that prevents escape to infinity and guarantees convergence to a geodesic circle.Other curvature flowsVolume-preserving & constrained flowsLong-time existence & convergenceRotational symmetry & warped products https://arxiv.org/abs/2404.01668arxiv:2404.01668Tue, 02 Apr 2024 00:00:00 GMTEsther Cabezas-Rivas, Salvador Moll, Marcos Solera — We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in ℝ^N with no ellip\-ticity nor smoothness requirements, in order to include the crystalline case) and on the initial data. By means of an approximation procedure introduced by Moser, our solutions are limits of anisotropic p-harmonic functions or p-capacitary functions (after a change of variable), and we get uniqueness both for the approximating solutions (i.e., uniqueness of p-capacitary functions) and the limiting ones. Our notion of weak solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of BV-functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition (where a sliding Wulff shape plays the role of a ball), the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, examples of explicit solutions are built.Mean curvature flowOther curvature flowsInverse mean curvature flowAnisotropic & nonlocal flowsHarnack inequalitiesWeak & singular flowsStability & uniqueness https://arxiv.org/abs/2211.17040arxiv:2211.17040Wed, 30 Nov 2022 00:00:00 GMTEsther Cabezas-Rivas, Julian Scheuer — We introduce a mean curvature flow with global term of convex hypersurfaces in the sphere, for which the global term can be chosen to keep any quermassintegral fixed. Then, starting from a strictly convex initial hypersurface, we prove that the flow exists for all times and converges smoothly to a geodesic sphere. This provides a workaround to an issue present in the volume preserving mean curvature flow in the sphere introduced by Huisken in 1987. We also classify solutions for some constant curvature type equations in space forms, as well as solitons in the sphere and in the upper branch of the De Sitter space.Mean curvature flowVolume-preserving & constrained flowsSphere theoremsLong-time existence & convergenceGeometric inequalitiesPhysics