https://neckpinch.com/authors/richard-bamler/New arXiv papers by Richard Bamler, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT 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/2609.18687arxiv:2609.18687Wed, 16 Sep 2026 00:00:00 GMTRichard H. Bamler, Felix Schulze, Lu Wang — We show that a compact mean curvature flow whose singularities have multiplicity-one cylindrical tangent flows admits a smooth Morse resolution. The resolution agrees with the spacetime track outside any prescribed neighborhood of its singular set and all its critical points have the expected index. No nondegeneracy or isolatedness assumption is required. In the mean-convex case the result follows by smoothing the global arrival function.Mean curvature flowSingularities https://arxiv.org/abs/2603.25082arxiv:2603.25082Thu, 26 Mar 2026 00:00:00 GMTRichard H. Bamler, Eric Chen, Ronan J. Conlon — We construct an example of an asymptotically conical (AC) non-Kähler expanding gradient Ricci soliton that has a Kähler tangent cone at infinity. This yields an example of a Kähler cone that can be desingularised by a smooth AC expanding gradient Ricci soliton but not by a smooth AC expanding gradient Kähler–Ricci soliton.Ricci solitonsKähler–Ricci solitonsKähler geometryBoundaries & non-compact https://arxiv.org/abs/2512.25050arxiv:2512.25050Wed, 31 Dec 2025 00:00:00 GMTRichard H. Bamler, Yi Lai — We introduce a new approach for analyzing ancient solutions and singularities of mean curvature flow that are locally modeled on a cylinder. Its key ingredient is a general mechanism, called the PDE–ODI principle, which converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This principle bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. As an application, we establish the uniqueness of the bowl soliton times a Euclidean factor among ancient, cylindrical flows with dominant linear mode. This extends previous results on this problem to the most general setting and is made possible by the stronger asymptotic control provided by our analysis. In the other case, when the quadratic mode dominates, we obtain a complete asymptotic expansion to arbitrary polynomial order, which will form the basis for a subsequent paper. Our framework also recovers and unifies several classical results. In particular, we give new proofs of the uniqueness of tangent flows (due to Colding-Minicozzi) and the rigidity of cylinders among shrinkers (due to Colding-Ilmanen-Minicozzi) by reducing both problems to a single ordinary differential inequality, without using the Łojasiewicz-Simon inequality. Our approach is independent of prior work and the paper is largely self-contained.Mean curvature flowRicci solitonsAncient solutionsSelf-shrinkers & translatorsSingularitiesStability & uniqueness https://arxiv.org/abs/2512.24524arxiv:2512.24524Wed, 31 Dec 2025 00:00:00 GMTRichard H. Bamler, Yi Lai — We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. We also obtain a more uniform version of the Mean Convex Neighborhood Conjecture, which only requires closeness to a cylinder at some initial time and yields a quantitative version of this structural description. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. Central to our method is a refined asymptotic analysis and a novel leading mode condition, together with a new "induction over thresholds" argument. In addition, our approach provides a full parameterization of the space of asymptotically cylindrical flows and gives a new proof of the existence of flying wing solitons. Our method is independent of prior work and, together with our prequel paper, this work is largely self-contained.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsSingularities(Non)collapsingRotational symmetry & warped products https://arxiv.org/abs/2312.02106arxiv:2312.02106Mon, 04 Dec 2023 00:00:00 GMTRichard H Bamler, Bruce Kleiner — We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in ℝ³. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining our work with results of Brendle and Choi-Haslhofer-Hershkovits-White, we show that any level set flow starting from an embedded surface diffeomorphic to a 2-spheres does not fatten. In fact, we obtain that the problem of evolving embedded 2-spheres via the mean curvature flow equation is well-posed within a natural class of singular solutions. Second, we use our result to remove an additional condition in recent work of Chodosh-Choi-Mantoulidis-Schulze. This shows that mean curvature flows starting from any generic embedded surface only incur cylindrical or spherical singularities. Third, our approach offers a new regularity theory for solutions of mean curvature flows that flow through singularities. Among other things, this theory also applies to the innermost and outermost flow of any embedded surface and shows that all singularity models of such flows must have multiplicity one. It also establishes equality of the fattening time with the discrepancy time. Lastly, we obtain a number of further results characterizing a separation phenomenon of mean curvature flows of surfaces.Mean curvature flowSingularitiesWeak & singular flows https://arxiv.org/abs/2206.10785arxiv:2206.10785Wed, 22 Jun 2022 00:00:00 GMTRichard H. Bamler, Charles Cifarelli, Ronan J. Conlon, Alix Deruelle — We prove the existence of a unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on the blowup of ℂ×ℙ¹ at one point. This completes the classification of such solitons in two complex dimensions.Ricci solitonsKähler–Ricci solitonsSingularitiesSurfacesKähler geometryScalar curvature https://arxiv.org/abs/2110.04661arxiv:2110.04661Sat, 09 Oct 2021 00:00:00 GMTRichard H. Bamler, Pak-Yeung Chan, Zilu Ma, Yongjia Zhang — In this paper, we prove a volume growth estimate for steady gradient Ricci solitons with bounded Nash entropy. We show that such a steady gradient Ricci soliton has volume growth rate no smaller than r^((n+1)/2). This result not only improves the estimate in [CMZ21b, Theorem 1.3], but also is optimal since the Bryant soliton and Appleton's solitons [Ap17] have exactly this growth rate.Ricci solitonsAncient solutionsPerelman entropy(Non)collapsing https://arxiv.org/abs/2110.02254arxiv:2110.02254Tue, 05 Oct 2021 00:00:00 GMTRichard H. Bamler — We show that any manifold admitting a non-collapsed, ancient Ricci flow must have finite fundamental group. This generalizes what was known for κ-solutions in dimensions 2, 3. We furthermore show that this fundamental group must be a quotient of the fundamental group of the regular part of any tangent flow at infinity.Ricci flowAncient solutionsSingularities(Non)collapsing https://arxiv.org/abs/2102.12615arxiv:2102.12615Thu, 25 Feb 2021 00:00:00 GMTRichard H Bamler — This is a survey on recent developments in Ricci flows.Ricci flow https://arxiv.org/abs/2009.03243arxiv:2009.03243Mon, 07 Sep 2020 00:00:00 GMTRichard H Bamler — In this paper we characterize non-collapsed limits of Ricci flows. We show that such limits are smooth away from a set of codimension ≥ 4 in the parabolic sense and that the tangent flows at every point are given by gradient shrinking solitons, possibly with a singular set of codimension ≥ 4. We furthermore obtain a stratification result of the singular set with optimal dimensional bounds, which depend on the symmetries of the tangent flows. Our methods also imply the corresponding quantitative stratification result and the expected L^p-curvature bounds. As an application of our theory, we obtain a description of the singularity formation of a Ricci flow at its first singular time and a thick-thin decomposition characterizing the long-time behavior of immortal flows. These results generalize Perelman's results in dimension 3 to higher dimensions. We also obtain a Backwards Pseudolocality Theorem and discuss several other applications.Ricci flowRicci solitonsSingularities(Non)collapsing3-manifoldsLimit spaces & RCD https://arxiv.org/abs/2008.07093arxiv:2008.07093Mon, 17 Aug 2020 00:00:00 GMTRichard H Bamler — In this paper we establish new geometric and analytic bounds for Ricci flows, which will form the basis of a compactness, partial regularity and structure theory for Ricci flows in [Bam20a, Bam20b]. The bounds are optimal up to a constant that only depends on the dimension and possibly a lower scalar curvature bound. In the special case in which the flow consists of Einstein metrics, these bounds agree with the optimal bounds for spaces with Ricci curvature bounded from below. Moreover, our bounds are local in the sense that if a bound depends on the collapsedness of the underlying flow, then we are able to quantify this dependence using the pointed Nash entropy based only at the point in question. Among other things, we will show the following bounds: Upper and lower volume bounds for distance balls, dependence of the pointed Nash entropy on its basepoint in space and time, pointwise upper Gaussian bound on the heat kernel and a bound on its derivative and an L¹-Poincaré inequality. The proofs of these bounds will, in part, rely on a monotonicity formula for a notion, called variance of conjugate heat kernels. We will also derive estimates concerning the dependence of the pointed Nash entropy on its basepoint, which are asymptotically optimal. These will allow us to show that points in spacetime that are nearby in a certain sense have comparable pointed Nash entropy. Hence the pointed Nash entropy is a good quantity to measure local collapsedness of a Ricci flow Our results imply a local ε-regularity theorem, improving a result of Hein and Naber. Some of our results also hold for super Ricci flows.Ricci flowEinstein metricsPerelman entropyHeat kernelScalar curvatureWeak & singular flowsLimit spaces & RCD https://arxiv.org/abs/1709.04122arxiv:1709.04122Wed, 13 Sep 2017 00:00:00 GMTRichard H. Bamler, Bruce Kleiner — We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman's conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three manifolds — in particular to the Generalized Smale Conjecture — which will appear in a subsequent paper.Ricci flowSingularitiesSurgery3-manifoldsSmale conjecture & diffeomorphismsStability & uniqueness