https://neckpinch.com/authors/peter-topping/New arXiv papers by Peter Topping, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT 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/2601.13767arxiv:2601.13767Tue, 20 Jan 2026 00:00:00 GMTArjun Sobnack, Peter M. Topping — In 1995, Hamilton introduced a Harnack inequality for convex solutions of the mean curvature flow. In this paper we prove an alternative Harnack inequality for curve shortening flow, i.e. one-dimensional mean curvature flow, that does not require any assumption of convexity. For an initial proper curve in the plane whose ends are radial lines but which is otherwise arbitrarily wild, we use the Harnack inequality to give an explicit time by which the curve shortening flow evolution must become graphical. This gives a new instance of delayed parabolic regularity. The Harnack inequality also gives estimates describing how a polar graphical flow with radial ends settles down to an expanding solution. Finally, we relate our Harnack inequality to Hamilton's by identifying a pointwise curvature estimate implied by both Harnack inequalities in the special case of convex flows.Mean curvature flowCurve shorteningHarnack inequalities https://arxiv.org/abs/2409.03098arxiv:2409.03098Wed, 04 Sep 2024 00:00:00 GMTArjun Sobnack, Peter M. Topping — Given two disjoint nested embedded closed curves in the plane, both evolving under curve shortening flow, we show that the modulus of the enclosed annulus is monotonically increasing in time. An analogous result holds within any ambient surface satisfying a lower curvature bound.Curve shortening https://arxiv.org/abs/2408.04049arxiv:2408.04049Wed, 07 Aug 2024 00:00:00 GMTArjun Sobnack, Peter M. Topping — Given two curves bounding a region of area A that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time A/π. We prove several results of this form and demonstrate that no estimate can hold before that time. As an example application, we construct solutions to graphical curve shortening flow starting with initial data that is merely an L¹ function.Curve shortening https://arxiv.org/abs/2309.00596arxiv:2309.00596Fri, 01 Sep 2023 00:00:00 GMTPeter M. Topping — We survey several problems concerning Riemannian manifolds with positive curvature of one form or another. We describe the PIC1 notion of positive curvature and argue that it is often the sharp notion of positive curvature to consider. Finally we explain how recent Ricci flow theory is particularly well adapted to solve these problems.Ricci flowSphere theorems https://arxiv.org/abs/2307.05306arxiv:2307.05306Tue, 11 Jul 2023 00:00:00 GMTLuke T. Peachey, Peter M. Topping — The second author and H. Yin have developed a Ricci flow existence theory that gives a complete Ricci flow starting with a surface equipped with a conformal structure and a nonatomic Radon measure as a volume measure. This led to the discovery of a large array of new expanding Ricci solitons. In this paper we use the recent uniqueness theory in this context, also developed by the second author and H. Yin, to give a complete classification of all expanding Ricci solitons on surfaces. Along the way, we prove a converse to the existence theory that is not constrained to solitons: every complete Ricci flow on a surface over a time interval (0,ε) admits a tdownarrow 0 limit within the class of admissible initial data. This makes surfaces the first nontrivial setting for Ricci flow in which a bijection can be given between the entire set of complete Ricci flows over maximal time intervals (0,T), and a class of initial data that induces them.Ricci flowRicci solitonsSurfacesStability & uniqueness https://arxiv.org/abs/2306.08398arxiv:2306.08398Wed, 14 Jun 2023 00:00:00 GMTPeter M. Topping, Hao Yin — In previous work we established the existence of a Ricci flow starting with a Riemann surface coupled with a nonatomic Radon measure as a conformal factor. In this paper we prove uniqueness. Combining these two works yields a canonical smoothing of such rough surfaces that also regularises their geometry at infinity.Ricci flowSurfacesStability & uniqueness https://arxiv.org/abs/2211.07623arxiv:2211.07623Mon, 14 Nov 2022 00:00:00 GMTMan-Chun Lee, Peter M. Topping — Recently it has been proved (Lee-Topping 2022, Deruelle-Schulze-Simon 2022, Lott 2019) that three-dimensional complete manifolds with non-negatively pinched Ricci curvature must be flat or compact, thus confirming a conjecture of Hamilton. In this paper we generalise our work on the existence of Ricci flows from non-compact pinched three-manifolds in order to prove a higher-dimensional analogue. We construct a solution to Ricci flow, for all time, starting with an arbitrary complete non-compact manifold that is PIC1 pinched. As an application we prove that any complete manifold of non-negative complex sectional curvature that is PIC1 pinched must be flat or compact.Ricci flow3-manifoldsSphere theoremsBoundaries & non-compact https://arxiv.org/abs/2204.00504arxiv:2204.00504Fri, 01 Apr 2022 00:00:00 GMTMan-Chun Lee, Peter M. Topping — We show that every complete non-compact three-manifold with non-negatively pinched Ricci curvature admits a complete Ricci flow solution for all positive time, with scale-invariant curvature decay and preservation of pinching. Combining with recent work of Lott and Deruelle-Schulze-Simon gives a proof of Hamilton's pinching conjecture without additional hypotheses.Ricci flowCurvature pinching3-manifoldsBoundaries & non-compact https://arxiv.org/abs/2202.12564arxiv:2202.12564Fri, 25 Feb 2022 00:00:00 GMTMan-Chun Lee, Peter M. Topping — We consider the problem of when a smooth Ricci flow, for positive time, that attains smooth initial data in a weak sense must be smooth down to the initial time. We obtain curvature estimates for an example where this fails. We prove a positive result in the case that the flow satisfies a lower IC1 curvature bound, equivalent to a lower Ricci bound in three dimensions. As an application, we prove that Gromov-Hausdorff limits of WPIC1 manifolds are WPIC1.Ricci flowLimit spaces & RCD https://arxiv.org/abs/2107.14686arxiv:2107.14686Fri, 30 Jul 2021 00:00:00 GMTPeter M. Topping, Hao Yin — We formulate and solve the existence problem for Ricci flow on a Riemann surface with initial data given by a Radon measure as volume measure. The theory leads us to a large class of new examples of nongradient expanding Ricci solitons, including the first example of a nongradient Kaehler Ricci soliton. It also settles the question of whether a smooth flow for positive time that attains smooth initial data in a distance metric sense must be smooth down to the initial time. We disprove this by giving an example of a complete Ricci flow starting with the Euclidean plane that is not the static solution.Ricci flowRicci solitonsKähler–Ricci solitonsSurfacesKähler geometry https://arxiv.org/abs/2107.14676arxiv:2107.14676Fri, 30 Jul 2021 00:00:00 GMTPeter M. Topping — Perelman has proved that there cannot exist a nontrivial breather for Ricci flow on a closed manifold. Here we construct nontrivial expanding breathers for Ricci flow in all dimensions when the underlying manifold is allowed to be noncompact.Ricci flowBoundaries & non-compact