Peter Topping

I have been working mainly on geometric flows. More generally, I study nonlinear partial differential equations, with an emphasis on those arising in the calculus of variations, geometric analysis, applied analysis and differential geometry.

Particular areas of specialisation currently include
1) Harmonic maps and their heat flow; bubbling phenomena.
2) Ricci flow.
3) Compensation properties of Jacobian determinants
4) Isoperimetric inequalities.
5) Minimal surfaces and mean curvature flow.
6) Willmore surfaces.
7) Fluid dynamics.

Papers: Here are some ABSTRACTS of selected articles, (no longer maintained) many of which are available below to download:

1) `Rigidity in the harmonic map heat flow.' J. Differential Geometry, 45 (1997) 593-610. PDF or Postscript or DVI.
2) `The Optimal Constant in Wente's $L^\infty$ Estimate.' Comment. Math. Helvetici, 72 (1997) 316-328. pdf , Postscript (preprint) or DVI (preprint).
3) `Mean curvature flow and geometric inequalities.' J. Reine Angew. Math., 503 (1998) 47-61. Postscript or pdf.
4) `The isoperimetric inequality on a surface.' Manuscripta Math., 100 (1999) 23-33. PDF or Postscript or DVI.
5) `An Example of a Nontrivial Bubble Tree in the Harmonic Map Heat Flow.' Appeared in `Harmonic Morphisms, Harmonic Maps and Related Topics,' J. C. Wood et al ed., CRC Press (1999). Postscript or DVI.
6) `Pressure estimates in two dimensional incompressible fluid flow.' Physica D, 137 (2000) 143-156. Postscript or DVI.
7) `Towards the Willmore conjecture.' Calc. Var., 11 (2000) 361-393. PDF or Postscript or DVI.
8) `Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow.' Annals of Math., 159 (2004) 465-534. PDF or Postscript or DVI.
9) `Reverse bubbling and nonuniqueness in the harmonic map flow.' IMRN, 10 (2002) 505-520. PDF or Postscript or DVI.
10) `An approach to the Willmore conjecture.' In `Global theory of minimal surfaces,' ed. David Hoffman. Clay math. proc., vol. 2, AMS (2005) 769-772. PDF or Postscript or DVI.
11) `Winding behaviour of finite-time singularities of the harmonic map heat flow.' Math. Zeit., 247 (2004) 279-302. PDF, Postscript or DVI.
12) `Bubbling of almost-harmonic maps between 2-spheres at points of zero energy density.' In `Variational problems in Riemannian geometry: Bubbles, scans and geometric flows,' eds Baird et. al. Progress in nonlinear differential equations and their applications, 59 (2004) 33-42. Postscript , DVI or pdf .
13) `Improved regularity of harmonic map flows with Hölder continuous energy.' Calc. Var., 21 (2004) 47-55. PDF
14) `Diameter control under Ricci flow.' C.A.G., 13 (2005) 1039-1055. PDF.
15) `Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics.' J.E.M.S., 12 (2010) 1429--1451. PDF.
16) joint with Klaus Ecker, Dan Knopf and Lei Ni: `Local monotonicity and mean value formulas for evolving Riemannian manifolds.' J. Reine Angew. Math., 616 (2008) 89-130. PDF.
17) `Relating diameter and mean curvature for submanifolds of Euclidean space.' Comment. Math. Helvetici, 83 (2008) 539-546. PDF.
18) joint with Robert McCann: `Ricci flow, entropy and optimal transportation.' American Journal of Math., 132 (2010) 711-730. PDF.
19) `L-optimal transportation for Ricci flow.' J. Reine Angew. Math., 636 (2009) 93--122. PDF.
20) joint with Esther Cabezas-Rivas: `The canonical shrinking soliton associated to a Ricci flow.' Calc. Var. and PDE, 43 (2012) 173-184. PDF.
21) joint with Gregor Giesen: `Ricci flow of negatively curved incomplete surfaces.' Calc. Var. and PDE, 38 (2010) 357--367. PDF.
22) joint with Esther Cabezas-Rivas: `The Canonical Expanding Soliton and Harnack inequalities for Ricci flow.' Trans. Amer. Math. Soc. 364 (2012) 3001-3021. PDF.
23) `Ricci flow: The foundations via optimal transportation.' Lecture notes (2009) to appear in `Seminaires et congres', S.M.F. PDF.
24) joint with Camillo De Lellis: `Almost-Schur lemma.' Calc. Var. and PDE, 43 (2012) 347--354. PDF.
25) joint with Joerg Enders and Reto Mueller: `On Type I Singularities in Ricci flow.' Communications in Analysis and Geometry, 19 (2011) 905--922. PDF.
26) `Reverse bubbling in geometric flows.' Surveys in Geometric Analysis and Relativity. Advanced Lectures in Mathematics 20, International Press (2011) eds H. Bray and W.P. Minicozzi II. Volume dedicated to Richard Schoen on the occasion of his sixtieth birthday. PDF.
27) joint with Gregor Giesen: `Existence of Ricci flows of incomplete surfaces.' Comm. Partial Differential Equations, 36 (2011) 1860--1880. PDF.
28) `Uniqueness and nonuniqueness for Ricci flow on surfaces: Reverse cusp singularities.' I.M.R.N., 2012 (2012) 2356--2376. PDF.
29) joint with Ben Sharp: `Decay estimates for Riviere's equation, with applications to regularity and compactness.' Preprint (2011) to appear in T.A.M.S. PDF.
30) joint with Gregor Giesen: `Ricci flows with unbounded curvature.' Math. Zeit., 273 (2013) 449--460. PDF.
31) `Remarks on Hamilton's Compactness Theorem for Ricci flow.' Preprint (2011) to appear J. Reine Angew. Math. PDF.
32) joint with Melanie Rupflin: `Flowing maps to minimal surfaces.' Preprint (2012) PDF.
33) joint with Melanie Rupflin and Miaomiao Zhu: `Asymptotics of the Teichmueller harmonic map flow.' Preprint (2012) PDF.
34) joint with Melanie Rupflin: `A uniform Poincaré estimate for quadratic differentials on closed surfaces.' Preprint (2012) PDF.
35) joint with Sebastian Helmensdorfer: `The geometry of Differential Harnack Estimates.' Preprint (2013). PDF.
36) joint with Sebastian Helmensdorfer: `Bouncing of charged droplets: An explanation using mean curvature flow.' Preprint (2013). PDF.
37) joint with Gregor Giesen: `Ricci flows with bursts of unbounded curvature.' Preprint (2013). PDF.
38) `Uniqueness of Instantaneously Complete Ricci flows.' Preprint (2013). PDF.

Geometric flows and related topics, 26-30 March 2007, Warwick.

Minimal surfaces and constant mean curvature surfaces . Tuesday 20 January 2009, Warwick.

Nonlinear PDE and free boundary problems . A series of minicourses and invited talks. 15-19 June 2009, Warwick. Poster

Mini-Workshop on Geometric Analysis. 23-24 June 2010, Warwick.

Workshop on Geometric Analysis. 20-24 June 2011, ICMS, Edinburgh.

South West PDE Winter school, 5-6 January 2012 at Warwick.

Warwick-Imperial-Cambridge Geometric Analysis seminar, 10 May 2012, at Warwick.

At the Frontier of Analysis and Probability, Sept 2012, at Warwick.

International Conference on Geometric Analysis, Sept 2012, at Seillac, Loire valley, France.

International conference on Nonlinear PDE, 10-13 September 2012, at Oxford.

Mathematics Institute, University of Warwick, Coventry, CV4 7AL, England
Phone: TBA
Fax: +44 24 7652 4182
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