https://neckpinch.com/authors/miles-simon/New arXiv papers by Miles Simon, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2606.23272arxiv:2606.23272Mon, 22 Jun 2026 00:00:00 GMTMan-Chun Lee, Florian Litzinger, Miles Simon — In this work, we study the positive mass theorem under critical low regularity assumptions using Ricci flow smoothing. We show that asymptotically flat manifolds (Mⁿ,g) of regularity L^∞∩ W^(1,n) with non-negative distributional scalar curvature have non-negative ADM mass. Furthermore, when the ADM mass vanishes, the manifold is globally isometric to Euclidean space with respect to an integral distance introduced by De Cecco-Palmieri. This extends the recent work of Hafemann to the critical regularity case. Our approach is based on showing that Riemannian metrics of regularity L^∞∩ W^(1,n), whose scalar curvature is bounded from below in the distributional sense, admit a Ricci flow smoothing whose scalar curvature is bounded from below by the same initial lower bound in the classical sense. In contrast, Cecchini-Frenck-Zeidler constructed examples of metrics which are in L^∞∩ W^(1,p) for all 2<p<n, and whose distributional scalar curvature is bounded from below, that cannot be approximated by smooth metrics with the same scalar curvature lower bound. In this sense, our result is optimal.Ricci flowScalar curvatureBoundaries & non-compactPhysics https://arxiv.org/abs/2603.22086arxiv:2603.22086Mon, 23 Mar 2026 00:00:00 GMTAlix Deruelle, Man-Chun Lee, Felix Schulze, Miles Simon, Peter M. Topping — Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.Ricci flowHarnack inequalitiesCurvature pinching3-manifoldsSphere theoremsBoundaries & non-compact https://arxiv.org/abs/2411.13204arxiv:2411.13204Wed, 20 Nov 2024 00:00:00 GMTMiles Simon — In this paper we survey some results on Ricci flowing non-smooth initial data. Among other things, we give a non-exhaustive list of various weak initial data which can be evolved with the Ricci flow. We also survey results which show that various curvature lower bounds will, possibly up to a constant, be preserved, if we start with such possibly non-smooth initial data. Some proofs/proof sketches are given in certain cases. A list of some open problems related to these areas is given in the last section of the paper.Ricci flowWeak & singular flows https://arxiv.org/abs/2410.08667arxiv:2410.08667Fri, 11 Oct 2024 00:00:00 GMTJiawei Liu, Miles Simon — In this paper we study n-dimensional Ricci flows (Mⁿ,g(t))_(t∈ [0,T)), where T< ∞ is a potentially singular time, and for which the spatial L^p norm, p>n/2, of the scalar curvature is uniformly bounded on [0,T). In the case that M is closed and four dimensional, we explain why non-collapsing estimates hold and how they can be combined with integral bounds on the Ricci and full curvature tensor of the prequel paper of the authors, as well as non-inflating estimates (already known due to works of Bamler), to obtain an improved space time integral bound of the Ricci curvature. As an application of these estimates, we show that if we further restrict to n=4, then the solution convergences to an orbifold as t → T and that the flow can be extended using the Orbifold Ricci flow to the time interval [0,T+σ) for some σ>0. We also prove local versions of many of the results mentioned above.Ricci flow(Non)collapsing4-manifoldsScalar curvature https://arxiv.org/abs/2406.02351arxiv:2406.02351Tue, 04 Jun 2024 00:00:00 GMTJiawei Liu, Miles Simon — In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed n-dimensional Kähler Ricci flow. These integral estimates improve and extend the integral curvature estimates shown by the second author in an earlier paper. If the scalar curvature is uniformly bounded in the spatial L^p sense for some p>2, then the estimates imply a uniform bound on the spatial L² norm of the Riemannian curvature tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition, or we restrict to closed manifolds.Ricci flowKähler–Ricci flow4-manifoldsKähler geometryScalar curvature https://arxiv.org/abs/2403.00708arxiv:2403.00708Fri, 01 Mar 2024 00:00:00 GMTAlix Deruelle, Felix Schulze, Miles Simon — We study n-dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by C/t, starting at metric cones which are Reifenberg outside the tip. We show that any such flow behaves like a self-similar solution up to an exponential error in time. As an application, we show that smooth n-dimensional complete non-compact Riemannian manifolds which are uniformly PIC1-pinched, with positive asymptotic volume ratio, are Euclidean. This confirms a higher dimensional version of a conjecture of Hamilton and Lott under the assumption of non-collapsing. It also yields a new and more direct proof of the original conjecture of Hamilton and Lott in three dimensions.Ricci flow(Non)collapsingSphere theoremsScalar curvatureWeak & singular flowsBoundaries & non-compact https://arxiv.org/abs/2203.15313arxiv:2203.15313Tue, 29 Mar 2022 00:00:00 GMTAlix Deruelle, Felix Schulze, Miles Simon — This paper investigates the question of stability for a class of Ricci flows which start at possibly non-smooth metric spaces. We show that if the initial metric space is Reifenberg and locally bi-Lipschitz to Euclidean space, then two solutions to the Ricci flow whose Ricci curvature is uniformly bounded from below and whose curvature is bounded by c· t⁻¹ converge to one another at an exponential rate once they have been appropriately gauged. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott.Ricci flow3-manifoldsWeak & singular flowsStability & uniqueness https://arxiv.org/abs/2109.08541arxiv:2109.08541Fri, 17 Sep 2021 00:00:00 GMTTobias Lamm, Miles Simon — In this paper we construct solutions to Ricci DeTurck flow in four dimensions on closed manifolds which are instantaneously smooth but whose initial values g are (possibly) non-smooth Riemannian metrics whose components in smooth coordinates belong to W^(2,2) and satisfy 1/ah≤ g≤ a h for some 1<a<∞ and some smooth Riemannian metric h on M. A Ricci flow related solution is constructed whose initial value is isometric in a weak sense to the initial value of the Ricci DeTurck solution. Results for a related non-compact setting are also presented. Various L^p estimates for Ricci flow, which we require for some of the main results, are also derived. As an application we present a possible definition of scalar curvature ≥ k for W^(2,2) metrics g on closed four manifolds which are bounded in the L^(∞) sense by 1/ah≤ g≤ a h for some 1<a<∞ and some smooth Riemannian metric h on M.Ricci flow4-manifoldsScalar curvatureWeak & singular flowsBoundaries & non-compactHarmonic maps & DeTurck trick