Publications (11)0 Total impact
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Peter M. Topping
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ABSTRACT: 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.
05/2013;
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ABSTRACT: Given a completely arbitrary surface, whether or not it is complete and
whether or not it has bounded curvature, there exists a Ricci flow evolution of
that surface that exists for a specific amount of time and is unique within a
certain class of solutions [GT11]. In the special case that the initial surface
is complete and of bounded curvature, this Ricci flow agrees with the classical
flow of Hamilton and Shi [GT11]. However, in this work we show that our flow
exists strictly beyond the classical flow. Indeed, the Hamilton-Shi flow stops
when the curvature blows up, but our flow can always be extended beyond that
time unless the volume decreases to zero. In this paper we construct examples
of complete, bounded curvature initial surfaces whose subsequent (unique) Ricci
flows exist for all time, with the property that their curvature is bounded on
some initial time interval, unbounded on some intermediate time interval, and
bounded again beyond some later time.
02/2013;
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ABSTRACT: We prove a uniform estimate, valid for every closed Riemann surface of genus
at least two, that bounds the distance of any quadratic differential to the
finite dimensional space of holomorphic quadratic differentials in terms of its
antiholomorphic derivative.
11/2012;
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ABSTRACT: The Teichm\"uller 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.
09/2012;
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ABSTRACT: 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 $\pi_1$, and this
minimal immersion will be homotopic to the original map in the case that
$\pi_2=0$.
05/2012;
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ABSTRACT: 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.
07/2010;
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ABSTRACT: 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.
05/2010;
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ABSTRACT: Schur's lemma states that every Einstein manifold of dimension $n\geq 3$ has
constant scalar curvature. Here $(M,g)$ is defined to be Einstein if its
traceless Ricci tensor $$\Rico:=\Ric-\frac{R}{n}g$$ is identically zero. In
this short note we ask to what extent the scalar curvature is constant if the
traceless Ricci tensor is assumed to be \emph{small} rather than identically
zero.
03/2010;
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ABSTRACT: We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle.
11/2009;
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ABSTRACT: We show uniqueness of Ricci flows starting at a surface of uniformly negative
curvature, with the assumption that the flows become complete instantaneously.
Together with the more general existence result proved in [10], this settles
the issue of well-posedness in this class.
06/2009;
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ABSTRACT: To every Ricci flow on a manifold M over a time interval I, we associate a shrinking Ricci soliton on the space-time M x I. We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the results of the second author, and McCann and the second author; we briefly survey the link between these subjects. Comment: slight changes in hypotheses theorem 1.1, a sign corrected in Proposition 2.2, section 6 made more precise
07/2008;
