Peter M. Topping
YOU?
Author Swipe
View article: All two‐dimensional expanding Ricci solitons
All two‐dimensional expanding Ricci solitons Open
The second author and H. Yin [ Ars Inveniendi Analytica . DOI 10.15781/4x5c-9q97 ] have developed a Ricci flow existence theory that gives a complete Ricci flow starting with a surface equipped with a conformal structure and a non‐atomic R…
View article: Monotonicity of the modulus under curve shortening flow
Monotonicity of the modulus under curve shortening flow Open
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 su…
View article: Delayed parabolic regularity for curve shortening flow
Delayed parabolic regularity for curve shortening flow Open
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 …
View article: Uniqueness of Ricci flows from non‐atomic Radon measures on Riemann surfaces
Uniqueness of Ricci flows from non‐atomic Radon measures on Riemann surfaces Open
In previous work (Topping and Yin, https://arxiv.org/abs/2107.14686 ), we established the existence of a Ricci flow starting with a Riemann surface coupled with a non‐atomic Radon measure as a conformal factor. In this paper, we prove uniq…
View article: Ricci flow and PIC1
Ricci flow and PIC1 Open
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. Fin…
View article: All two-dimensional expanding Ricci solitons
All two-dimensional expanding Ricci solitons Open
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 disco…
View article: Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces
Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces Open
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 smo…
View article: Manifolds with PIC1 pinched curvature
Manifolds with PIC1 pinched curvature Open
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.…
View article: Nontrivial breathers for Ricci flow
Nontrivial breathers for Ricci flow Open
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
\nallowed to be noncom…
View article: Global regularity of three-dimensional Ricci limit spaces
Global regularity of three-dimensional Ricci limit spaces Open
Miles Simon and the second author, in their recent work [Geom. Topol. 25 (2021), pp. 913–948], established a local bi-Hölder correspondence between weakly noncollapsed Ricci limit spaces in three dimensions and smooth manifolds. In particu…
View article: Three-manifolds with non-negatively pinched Ricci curvature
Three-manifolds with non-negatively pinched Ricci curvature Open
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 wi…
View article: Metric limits of manifolds with positive scalar curvature
Metric limits of manifolds with positive scalar curvature Open
We show that any Riemannian metric conformal to the round metric on $S^n$, for $n\geq 4$, arises as a limit of a sequence of Riemannian metrics of positive scalar curvature on $S^n$ in the sense of uniform convergence of Riemannian distanc…
View article: Time zero regularity of Ricci flow
Time zero regularity of Ricci flow Open
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 pos…
View article: Smoothing a measure on a Riemann surface using Ricci flow
Smoothing a measure on a Riemann surface using Ricci flow Open
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, in…
View article: Nontrivial breathers for Ricci flow
Nontrivial breathers for Ricci flow Open
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 noncompa…
View article: A rigidity estimate for maps from $S^2$ to $S^2$ via the harmonic map flow
A rigidity estimate for maps from $S^2$ to $S^2$ via the harmonic map flow Open
We show how a rigidity estimate introduced in recent work of Bernand-Mantel, Muratov and Simon can be derived using the harmonic map flow.
View article: Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow
Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow Open
The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of si…
View article: Rate of curvature decay for the contracting cusp Ricci flow
Rate of curvature decay for the contracting cusp Ricci flow Open
We prove that the Ricci flow that contracts a hyperbolic cusp has curvature decay maxK∼1t2. In order to do this, we prove a new Li–Yau type differential Harnack inequality for Ricci flow.
View article: Uniqueness and nonuniqueness of limits of Teichmueller harmonic map flow
Uniqueness and nonuniqueness of limits of Teichmueller harmonic map flow Open
The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of si…
View article: Ricci flow and Ricci Limit Spaces
Ricci flow and Ricci Limit Spaces Open
I survey some of the developments in the theory of Ricci flow and its applications from the past decade. I focus mainly on the understanding of Ricci flows that are permitted to have unbounded curvature in the sense that the curvature can …
View article: Global Regularity of Three-Dimensional Ricci Limit Spaces
Global Regularity of Three-Dimensional Ricci Limit Spaces Open
We construct a global homeomorphism from any 3D Ricci limit space to a smooth manifold, that is locally bi-Holder. This extends the recent work of Miles Simon and the second author, and we build upon their techniques. A key step in our pro…
View article: Teichmüller harmonic map flow into nonpositively curved targets
Teichmüller harmonic map flow into nonpositively curved targets Open
The Teichmüller harmonic map flow deforms both a map from an oriented closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $M$, so as to reduce the energy of the map as quickly as possible [16…
View article: Sharp Decay Estimates for the Logarithmic Fast Diffusion Equation and the Ricci Flow on Surfaces
Sharp Decay Estimates for the Logarithmic Fast Diffusion Equation and the Ricci Flow on Surfaces Open
We prove the sharp local smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known estimate for . It…
View article: Local control on the geometry in 3D Ricci flow
Local control on the geometry in 3D Ricci flow Open
The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a defini…
View article: Refined asymptotics of the Teichmüller harmonic map flow into general targets
Refined asymptotics of the Teichmüller harmonic map flow into general targets Open
The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve.…
View article: Rate of curvature decay for the contracting cusp Ricci flow
Rate of curvature decay for the contracting cusp Ricci flow Open
We prove that the Ricci flow that contracts a hyperbolic cusp has curvature decay like one over time squared. In order to do this, we prove a new Li-Yau type differential Harnack inequality for Ricci flow on surfaces.
View article: Sharp decay estimates for the logarithmic fast diffusion equation and the Ricci flow on surfaces
Sharp decay estimates for the logarithmic fast diffusion equation and the Ricci flow on surfaces Open
We prove the sharp local L^1 - L^\infty smoothing estimate for the logarithmic fast diffusion equation, or equivalently, for the Ricci flow on surfaces. Our estimate almost instantly implies an improvement of the known L^p - L^\infty estim…