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Peter M. Topping

Publications and source records attributed to Peter M. Topping.

At least 37 records · Page 2Linked to original sources
arXiv2019

Pyramid Ricci Flow in Higher Dimensions

In this paper, we construct a pyramid Ricci flow starting with a complete Riemannian manifold $(M^n,g_0)$ that is PIC1, or more generally satisfies a lower curvature bound $K_{IC_1}\geq -α_0$. That is, instead of constructing a flow on $M\times [0,T]$, we construct it on a subset of space-time that is a union of parabolic cylinders $B_{g_0}(x_0,k)\times [0,T_k]$ for each natural number $k$, where $T_k\downarrow 0$, and prove estimates on the curvature and Riemannian distance. More generally, we construct a pyramid Ricci flow starting with any noncollapsed $IC_1$-limit space, and use it to establish that such limit spaces are globally homeomorphic to smooth manifolds via homeomorphisms that are locally bi-Hölder.

math.DG↗
arXiv2018

Global Regularity of Three-Dimensional Ricci Limit Spaces

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 proof is the construction of local "pyramid Ricci flows", existing on uniform regions of spacetime, that are inspired by Hochard's partial Ricci flows.

math.DG↗
arXiv2018

Horizontal curves of hyperbolic metrics

We analyse the fine convergence properties of one parameter families of hyperbolic metrics, on a fixed underlying surface, that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms.

math.DG↗
arXiv2017

Global weak solutions of the Teichmüller harmonic map flow into general targets

We analyse finite-time singularities of the Teichmüller harmonic map flow -- a natural gradient flow of the harmonic map energy -- and find a canonical way of flowing beyond them in order to construct global solutions in full generality. Moreover, we prove a no-loss-of-topology result at finite time, which completes the proof that this flow decomposes an arbitrary map into a collection of branched minimal immersions connected by curves.

math.DG↗
arXiv2017

Local control on the geometry in 3D Ricci flow

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 definite amount of time under three-dimensional Ricci flow, and leads to local C/t decay of the full curvature tensor, irrespective of what is happening beyond the local region. As a by-product, our results generalise the Pseudolocality theorem of Perelman and Tian-Wang in this dimension by not requiring the Ricci curvature to be almost-positive, and not asking the volume growth to be almost-Euclidean.

math.DG↗
arXiv2016

Flowing maps to minimal surfaces

We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus 0 case, our flow is just the harmonic map flow, and it tries to find branched minimal 2-spheres as in Sacks-Uhlenbeck and Struwe etc. In the genus 1 case, we show that our flow is exactly equivalent to that considered by Ding-Li-Lui. In general, we recover the result of Schoen-Yau and Sacks-Uhlenbeck that an incompressible map from a surface can be adjusted to a branched minimal immersion with the same action on $π_1$, and this minimal immersion will be homotopic to the original map in the case that $π_2=0$.

math.DG↗
arXiv2016

Teichmüller harmonic map flow into nonpositively curved targets

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]. The flow then tries to converge to a branched minimal immersion when it can [16,18]. The only thing that can stop the flow is a finite-time degeneration of the metric on $M$ where one or more collars are pinched. In this paper we show that finite-time degeneration cannot happen in the case that the target has nonpositive sectional curvature, and indeed more generally in the case that the target supports no bubbles. In particular, when combined with [16,18,9], this shows that the flow will decompose an arbitrary such map into a collection of branched minimal immersions.

math.DG↗
arXiv2015

Sharp decay estimates for the logarithmic fast diffusion equation and the Ricci flow on surfaces

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 estimate for p larger than 1. It also has several applications in geometry, providing the missing step in order to pose the Ricci flow with rough initial data in the noncompact case, for example starting with a general noncompact Alexandrov surface, and giving the sharp asymptotics for the contracting cusp Ricci flow, as we show elsewhere.

math.AP↗
arXiv2015

Refined asymptotics of the Teichmüller harmonic map flow into general targets

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. Given a weak solution of the flow that exists for all time $t\geq 0$, we find a sequence of times $t_i\to\infty$ at which the flow at different scales converges to a collection of branched minimal immersions with no loss of energy. We do this by developing a compactness theory, establishing no loss of energy, for sequences of almost-minimal maps. Moreover, we construct an example of a smooth flow for which the image of the limit branched minimal immersions is disconnected. In general, we show that the necks connecting the images of the branched minimal immersions become arbitrarily thin as $i\to\infty$.

math.DG↗
arXiv2014

Ricci flows with bursts of unbounded curvature

Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metric, this Ricci flow always exists for all time and converges (after scaling by a factor $\frac1{2t}$) to this hyperbolic metric [GT11], i.e. our Ricci flow geometrises the surface. In this paper we show that there exist complete, bounded curvature initial metrics, including those conformal to a hyperbolic metric, which have subsequent Ricci flows developing unbounded curvature at certain intermediate times. In particular, when coupled with the uniqueness from [Top13], we find that any complete Ricci flow starting with such initial metrics must develop unbounded curvature over some intermediate time interval, but that nevertheless, the curvature must later become bounded and the flow must achieve geometrisation as $t\to\infty$, even though there are other conformal deformations to hyperbolic metrics that do not involve unbounded curvature. Another consequence of our constructions is that while our Ricci flow from [GT11] must agree initially with the classical flow of Hamilton and Shi in the special case that the initial surface is complete and of bounded curvature, by uniqueness, it is now clear that our flow lasts for a longer time interval in general, with Shi's flow stopping when the curvature blows up, but our flow continuing strictly beyond in these situations. All our constructions of unbounded curvature developing and then disappearing are in two dimensions. Generalisations to higher dimensions are then immediate.

math.AP↗
arXiv2014

Ricci flows with unbounded curvature

Until recently, Ricci flow was viewed almost exclusively as a way of deforming Riemannian metrics of bounded curvature. Unfortunately, the bounded curvature hypothesis is unnatural for many applications, but is hard to drop because so many new phenomena can occur in the general case. This article surveys some of the theory from the past few years that has sought to rectify the situation in different ways.

math.DG↗
arXiv2013

Uniqueness of Instantaneously Complete Ricci flows

We prove uniqueness of instantaneously complete Ricci flows on surfaces. We do not require any bounds of any form on the curvature or its growth at infinity, nor on the metric or its growth (other than that implied by instantaneous completeness). Coupled with earlier work, particularly [23, 11], this completes the well-posedness theory for instantaneously complete Ricci flows on surfaces.

math.DG↗
arXiv2012

Asymptotics of the Teichmüller harmonic map flow

The Teichmüller harmonic map flow, introduced in [9], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time. We obtain a branched minimal immersion from the degenerate domain.

math.DG↗
arXiv2011

On Type I Singularities in Ricci flow

We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of Naber. As a by-product we conclude that the volume of a finite-volume singular set vanishes at the singular time. We also define a notion of density for Type I Ricci flows and use it to prove a regularity theorem reminiscent of White's partial regularity result for mean curvature flow.

math.DG↗
arXiv2011

Ricci flows with unbounded curvature

We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).

math.AP↗
arXiv2011

Existence of Ricci flows of incomplete surfaces

We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.

math.AP↗