https://neckpinch.com/authors/simon-brendle/New arXiv papers by Simon Brendle, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.02325arxiv:2610.02325Thu, 01 Oct 2026 00:00:00 GMTSimon Brendle, Raphael Tsiamis — We extend the classification of manifolds with positive isotropic curvature to all dimensions n ≥ 5. This result also completes the classification of uniformly positive isotropic curvature on complete non-compact κ-noncollapsed ancient solutions and gradient shrinking Ricci solitons. The proof involves a construction of pinching cones in all dimensions, refining and extending to all dimensions the results of the first author. We have used MATHEMATICA to find viable parameters for the cone families. The final presentation is self-contained and can be verified manually. No stage of this work involved LLMs.Ricci flowRicci solitonsAncient solutions(Non)collapsingCurvature pinchingSphere theoremsBoundaries & non-compact https://arxiv.org/abs/2402.11803arxiv:2402.11803Mon, 19 Feb 2024 00:00:00 GMTS. Brendle, R. Tsiamis — We prove an eigenvalue estimate which holds on every properly embedded shrinker for mean curvature flow. This generalizes earlier work of Ding and Xin to the noncompact case.Mean curvature flowRicci solitonsBoundaries & non-compactEigenvalues & spectrum https://arxiv.org/abs/2201.02522arxiv:2201.02522Fri, 07 Jan 2022 00:00:00 GMTS. Brendle — The Ricci flow is a natural evolution equation for Riemannian metrics on a given manifold. The main goal is to understand singularity formation. In his spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of singularity formation in dimension 3. More precisely, Perelman showed that every finite-time singularity to the Ricci flow in dimension 3 is modeled on an ancient κ-solution. Moreover, Perelman proved a structure theorem for ancient κ-solutions in dimension 3. In this survey, we discuss recent developments which have led to a complete classification of all the singularity models in dimension 3. Moreover, we give an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension 3 (originally proved by the author in 2012).Ricci flowRicci solitonsAncient solutionsSingularities(Non)collapsing3-manifolds https://arxiv.org/abs/1811.02559arxiv:1811.02559Tue, 06 Nov 2018 00:00:00 GMTS. Brendle — It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient κ-solution. We prove that the every noncompact ancient κ-solution in dimension 3 is isometric to either the shrinking cylinders (or a quotient thereof), or the Bryant soliton.Ricci flowAncient solutionsSingularities3-manifoldsBoundaries & non-compact https://arxiv.org/abs/0705.0766arxiv:0705.0766Sun, 06 May 2007 00:00:00 GMTS. Brendle, R. M. Schoen — Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.Ricci flowSphere theorems