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Papers: Here are some ABSTRACTS of selected articles, (no longer maintained) many of which are available below to download: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.' To appear in J.E.M.S. 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. Get from the web page of Dan Knopf.
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.' To appear, American Journal of Math. PDF. This paper supercedes the earlier paper `Diffusion is a 2-Wasserstein contraction on any manifold evolving by reverse Ricci flow. (2006)'
19) `L-optimal transportation for Ricci flow.' Preprint (2007). J. reine angew. Math., to appear. PDF.
20) joint with Esther Cabezas-Rivas: `The canonical shrinking soliton associated to a Ricci flow.' Preprint (2008) PDF.
21) joint with Gregor Giesen: `Ricci flow of negatively curved incomplete surfaces.' Preprint (2009) to appear in Calc. Var. and PDE. PDF.
22) joint with Esther Cabezas-Rivas: `The Canonical Expanding Soliton and Harnack inequalities for Ricci flow.' Preprint (2009).

Mathematics Institute, University of Warwick, Coventry, CV4 7AL, England
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