https://neckpinch.com/authors/reto-buzano/New arXiv papers by Reto Buzano, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2606.12379arxiv:2606.12379Wed, 10 Jun 2026 00:00:00 GMTReto Buzano, Gianmichele Di Matteo — We continue our local singularity analysis for Ricci flow initiated in ArXiv:2006.16227. Building on that framework, we study Type I singular points in general Ricci flows, without assuming any global Type I curvature bound, and prove that the scalar curvature must blow up at a Type I rate at each such point in all dimensions. As a consequence, Ricci flows with bounded scalar curvature cannot develop Type I singular points. This extends earlier results of the first author with Enders and Topping and with Mantegazza that relied on a global Type I assumption. We then adapt the same local perspective to ancient Ricci flows and analyse the curvature behaviour as time goes to negative infinity, showing in particular that every ancient Type I point exhibits scalar curvature behaviour of ancient Type I order.Ricci flowAncient solutionsSingularitiesScalar curvature https://arxiv.org/abs/2508.10790arxiv:2508.10790Thu, 14 Aug 2025 00:00:00 GMTAlessandro Bertellotti, Reto Buzano — We study blow-up sequences of Ricci shrinkers without global curvature assumptions based at points q at which the scalar curvature satisfies a Type I bound, proving that their F-limits split a line. In the four-dimensional case these limits are smooth Ricci shrinkers and the convergence is in the pointed smooth Cheeger-Gromov sense. As a consequence, limits along the integral curve of ∇f starting at such a point q split a line. This generalises known results about the geometry of ends of Ricci shrinkers that relied on global curvature bounds. To obtain our results, we extend the F-convergence theory from Bamler and Li-Wang.Ricci solitonsSingularities4-manifoldsScalar curvature https://arxiv.org/abs/2407.04131arxiv:2407.04131Thu, 04 Jul 2024 00:00:00 GMTAlessandro Bertellotti, Reto Buzano — We estimate the number of ends of smooth and singular Ricci shrinkers focussing first on general ends and later on asymptotically conical ones. In particular, we obtain a variety of applications to sequences of Ricci shrinkers converging in a weak pointed sense to a possibly singular limit Ricci shrinker, for instance no new conical end can form in the limit.Ricci solitonsWeak & singular flowsBoundaries & non-compact https://arxiv.org/abs/2312.07232arxiv:2312.07232Tue, 12 Dec 2023 00:00:00 GMTReto Buzano, Sylvain Maillot — We prove that the moduli space of mean convex two-spheres embedded in complete, orientable 3-dimensional Riemannian manifolds with nonnegative Ricci curvature is path-connected. This result is sharp in the sense that neither of the conditions of (strict) mean convexity, completeness, and nonnegativity of the Ricci curvature can be dropped or weakened. We also study the number of path components of mean convex Heegaard tori, again in ambient manifolds with nonnegative Ricci curvature. We prove that there are always either one or two path components and this number does not only depend on the homotopy type of the ambient manifold. We give a precise characterisation of the two cases and also discuss what happens if the mean convexity condition is weakened to nonnegative mean curvature.Mean curvature flow3-manifolds https://arxiv.org/abs/2206.06791arxiv:2206.06791Tue, 14 Jun 2022 00:00:00 GMTReto Buzano, Louis Yudowitz — We prove bubble-tree convergence of sequences of gradient Ricci shrinkers with uniformly bounded entropy and uniform local energy bounds, refining the compactness theory of Haslhofer-Mueller. In particular, we show that no energy concentrates in neck regions, a result which implies a local energy identity for the sequence. Direct consequences of these results are an identity for the Euler characteristic and a local diffeomorphism finiteness theorem.Ricci solitonsLimit spaces & RCDDiscrete & graph Ricci https://arxiv.org/abs/2110.06027arxiv:2110.06027Tue, 12 Oct 2021 00:00:00 GMTReto Buzano, Huy The Nguyen, Mario B. Schulz — In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactMinimal & CMC surfacesNumerics & visualization