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Richard H. Bamler

Publications and source records attributed to Richard H. Bamler.

At least 19 recordsLinked to original sources
arXiv2026

Smoothing polyhedral spaces via Ricci flow

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.

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arXiv2026

Morse resolution of mean curvature flows with cylindrical singularities

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.

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arXiv2026

A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

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.

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arXiv2026

Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture

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 \emph{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.

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arXiv2026

The PDE-ODI principle and cylindrical mean curvature flows

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 \emph{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.

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arXiv2025

Degree theory for 4-dimensional asymptotically conical gradient expanding solitons

We develop a new degree theory for 4-dimensional, asymptotically conical gradient expanding solitons. Our theory implies the existence of gradient expanding solitons that are asymptotic to any given cone over $S^3$ with non-negative scalar curvature. We also obtain a similar existence result for cones whose link is diffeomorphic to $S^3/Γ$ if we allow the expanding soliton to have orbifold singularities. Our theory reveals the existence of a new topological invariant, called the expander degree, applicable to a particular class of compact, smooth 4-orbifolds with boundary. This invariant is roughly equal to a signed count of all possible gradient expanding solitons that can be defined on the interior of the orbifold and are asymptotic to any fixed cone metric with non-negative scalar curvature. If the expander degree of an orbifold is non-zero, then gradient expanding solitons exist for any such cone metric. We show that the expander degree of the 4-disk $D^4$ and any orbifold of the form $D^4/Γ$ equals 1. Additionally, we demonstrate that the expander degree of certain orbifolds, including exotic 4-disks, vanishes. Our theory also sheds light on the relation between gradient and non-gradient expanding solitons with respect to their asymptotic model. More specifically, we show that among the set of asymptotically conical expanding solitons, the subset of those solitons that are gradient forms a union of connected components.

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arXiv2023

Diffeomorphism groups of prime 3-manifolds

Let $X$ be a compact orientable non-Haken 3-manifold modeled on the Thurston geometry $\text{Nil}$. We show that the diffeomorphism group $\text{Diff}(X)$ deformation retracts to the isometry group $\text{Isom}(X)$. Combining this with earlier work by many authors, this completes the determination the homotopy type of $\text{Diff}(X)$ for any compact, orientable, prime 3-manifold $X$.

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arXiv2023

Decomposing 4-manifolds with positive scalar curvature

We show that every closed, oriented, topologically PSC 4-manifold can be obtained via 0 and 1-surgeries from a topologically PSC 4-orbifold with vanishing first Betti number and second Betti number at most as large as the original one.

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arXiv2021

An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons

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 $r^{\frac{n+1}{2}}.$ 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.

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arXiv2021

On the fundamental group of non-collapsed ancient Ricci flows

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.

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arXiv2019

Ricci flow and contractibility of spaces of metrics

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry group. This proves the Generalized Smale Conjecture. Our argument is independent of Hatcher's theorem in the $S^3$ case and in particular it gives a new proof of the $S^3$ case.

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arXiv2019

On the rotational symmetry of 3-dimensional $κ$-solutions

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional $κ$-solutions. In this paper, we present an alternative proof for this fact and show that compact $κ$-solutions are rotational symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular Ricci flows.

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arXiv2018

Uniqueness and stability of Ricci flow through singularities

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.

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arXiv2018

Convergence of Ricci flows with bounded scalar curvature

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension $\geq 4$. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away from a set of codimension $\geq 4$. In the course of the proof, we will also establish $L^{p < 2}$-curvature bounds on time-slices of such flows.

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arXiv2017

Ricci flow and diffeomorphism groups of 3-manifolds

We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $RP^3$ and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold $X$, the inclusion $\text{Isom} (X,g)\to \text{Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.

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arXiv2017

The Ricci flow under almost non-negative curvature conditions

We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $-1$ can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than $-C$. Here the time of existence and the constant $C$ only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kaehler case. We also get a local version of the main theorem. As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds).

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arXiv2017

Long-time behavior of 3 dimensional Ricci flow -- C: 3-manifold topology and combinatorics of simplicial complexes in 3-manifolds

In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary observations in the topology of 3-manifolds. The main part is devoted to the construction of certain simplicial complexes in a given 3-manifold that exhibit useful intersection properties with embedded, incompressible solid tori. This paper is purely topological in nature and Ricci flows will not be used.

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