Peter Topping

I am working on various topics within geometric analysis, differential geometry, partial differential equations, calculus of variations and applied analysis. I am particularly interested in geometric flows, especially Ricci flow and harmonic map flows, bubbling phenomena, geometric inequalities, and regularity theory.

Quick links:

LECTURE NOTES (Graduate courses)

Geometric analysis and PDEs . Thursday 11 July 2024 - Friday 12 July 2024 at Warwick.

SLMath/MSRI, Berkeley, California, autumn 2024. New Frontiers in Curvature: Flows, General Relativity, Minimal Submanifolds, and Symmetry

Lecture notes:

`Lectures on the Ricci flow.' Published by CUP, 2006

Ricci flow and Ricci Limit Spaces. Cetraro, Italy, June 2018.

Teaching 2023/2024:

Term 1: MA3B8 Complex Analysis. Module web page, Autumn 2023.
Term 2: Foundations of Ricci flow. Module web page.

Warwick Geometric Analysis:

Current Analysis seminars. We also have an Analysis Seminar Archive (no longer maintained).

Opportunities to join the geometric analysis group at Warwick.
Details of my postdocs and students 2023/24


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. 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). PDF.
6) `Pressure estimates in two dimensional incompressible fluid flow.' Physica D, 137 (2000) 143-156. PDF.
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.' In `Optimal Transportation, Theory and Applications.' LMS lecture notes series, vol. 413, (2014) CUP. 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.' Transactions A.M.S. 365 (2013) 2317--2339. 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.' J. Reine Angew. Math. 692 (2014) 173--191. PDF.
32) joint with Melanie Rupflin: `Flowing maps to minimal surfaces.' American Journal of Math. 138 (2016) 1095--1115. PDF.
33) joint with Melanie Rupflin and Miaomiao Zhu: `Asymptotics of the Teichmueller harmonic map flow.' Advances in Math. 244 (2013) 874--893. PDF.
34) joint with Melanie Rupflin: `A uniform Poincaré estimate for quadratic differentials on closed surfaces.' Calc. Var. and PDE. 53 (2015) 587--604. PDF.
35) joint with Sebastian Helmensdorfer: `The geometry of Differential Harnack Estimates.' Actes du séminaire de Théorie spectrale et géométrie [Grenoble 2011-2012], 30 (2013) 77--89. PDF.
36) joint with Sebastian Helmensdorfer: `Bouncing of charged droplets: An explanation using mean curvature flow.' EPL (Europhysics letters) 104 (2013) 34001. ``Editor's choice'' PDF.
37) joint with Gregor Giesen: `Ricci flows with bursts of unbounded curvature.' Comm. Partial Differential Equations. 41 (2016) 854--876. PDF.
38) `Uniqueness of Instantaneously Complete Ricci flows.' Geometry and Topology 19-3 (2015), 1477--1492. DOI 10.2140/gt.2015.19.1477 PDF.
39) `Applications of Hamilton's Compactness Theorem for Ricci flow.' (Park City, 2013) IAS/Park City Mathematics Series, 22, AMS (2016) 15--50. PDF .
40) joint with Melanie Rupflin: `Teichmüller harmonic map flow into nonpositively curved targets.' J. Differential Geometry. 108 (2018) 135--184. PDF .
41) `Ricci flows with unbounded curvature.' Proceedings of the International Congress of Mathematicians, Seoul 2014. PDF .
42) joint with Tobias Huxol and Melanie Rupflin: `Refined asymptotics of the Teichmüller harmonic map flow into general targets.' Calc. Var. (2016) 55:85. doi:10.1007/s00526-016-1019-2 Preprint and Published version
43) joint with Hao Yin: `Sharp decay estimates for the logarithmic fast diffusion equation and the Ricci flow on surfaces.' Annals of PDE. (2017) 3:6. Preprint PDF . Open access
44) joint with Melanie Rupflin: `Horizontal curves of hyperbolic metrics.' Calc. Var. (2018) 57:106 Preprint. Open access.
45) joint with Hao Yin: `Rate of curvature decay for the contracting cusp Ricci flow.' Comm. Anal. Geom. Vol. 28, (2020) 1221-1250. PDF .
46) joint with Miles Simon: `Local control on the geometry in 3D Ricci flow.' J. Differential Geometry, Vol. 122, No. 3 (2022) 467-518. PDF , Journal , arxiv.
47) joint with Miles Simon: `Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces.' Geometry and Topology. Vol. 25, (2021) 913-948. arXiv version and preprint PDF.
48) joint with Melanie Rupflin: `Global weak solutions of the Teichmueller harmonic map flow into general targets.' Analysis and PDE. Vol 12 (2019) 815--842. Preprint PDF. Link to Journal.
49) joint with Andrew McLeod: `Global Regularity of Three-Dimensional Ricci Limit Spaces.' Trans. Amer. Math. Soc. Ser. B. 9 (2022) 345--370. Preprint PDF. Open access. arxiv .
50) `Ricci flow and Ricci limit spaces.' `Geometric Analysis.' [Cetraro 2018] Springer Lecture Notes in Mathematics, CIME Foundation Subseries. Vol. 2263 (2020) PDF. DOI
51) `Loss of initial data under limits of Ricci flows.' T. Hoffmann, M. Kilian, K. Leschke, F. Martin (Eds.) Minimal surfaces: integrable systems and visualisation, [Granada, 2018] Springer Proceedings in Mathematics & Statistics, 349 (2021). PDF.
52) joint with Andrew McLeod: `Pyramid Ricci Flow in Higher Dimensions.' Math. Zeit. Vol. 296 (2020) 511--523. Open Access PDF. Preprint PDF.
53) joint with James Kohout and Melanie Rupflin: `Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow.' Adv. Calc. Var. 15 (2022) 369--384. Preprint PDF. Open access.
54) `A rigidity estimate for maps from $S^2$ to $S^2$ via the harmonic map flow.' Bulletin of the L.M.S. 55 (2023) 338-343. PDF.
55) `Nontrivial breathers for Ricci flow.' Bulletin of the L.M.S. 55 (2023) 344--348. PDF , Journal, open access.
56) joint with Hao Yin: `Smoothing a measure on a Riemann surface using Ricci flow.' Preprint (2021). PDF.
57) joint with Man-Chun Lee: `Time zero regularity of Ricci flow.' International Mathematics Research Notices, 2022;, rnac300, Journal , PDF.
58) joint with Man-Chun Lee: `Metric limits of manifolds with positive scalar curvature.' Preprint (2022). PDF.
59) joint with Man-Chun Lee: `Three-manifolds with non-negatively pinched Ricci curvature.' Preprint (2022). PDF.
60) joint with Man-Chun Lee: `Manifolds with PIC1 pinched curvature.' Preprint (2022). arxiv.
61) joint with Hao Yin: `Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces.' Proceedings of the London Mathematical Society. 128: e12600 (2024) Open access. arxiv.
62) joint with Luke Peachey: `All two-dimensional expanding Ricci solitons.' Preprint (2023). arxiv.
63) `Ricci flow and PIC1.' Preprint (2023). arxiv.

Events I have co-organised:

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.

Warwick-Imperial-Cambridge Geometric Analysis Seminar. 30 May 2013, at Imperial.

Warwick-Imperial-Cambridge Geometric Analysis Postdoctoral meeting. 9/10 December 2013 at Warwick.

Warwick-Imperial-Cambridge Geometric Analysis Seminar. 19 March 2014 at Cambridge.

Lisbon 2014 Geometric Analysis conference. 7-11 July 2014 at ICT, Lisbon. Conference photo.

Junior Warwick-Imperial-Cambridge Geometric Analysis meeting. 12 December 2014 at De Morgan house, LMS, London.

Geometric flows: recent developments and applications. 12-17 April 2015 at BIRS, Banff, Canada.

Warwick-Imperial-Cambridge Geometric Analysis 3-day meeting. 15-17 July 2015 at Imperial college.

Partial Differential Equations. 2-8 August 2015 at Oberwolfach.

Junior Warwick-Imperial-Cambridge Geometric Analysis meeting. 6 January 2016 at QMU London.

Geometric PDE. 12-16 December 2016 at Warwick.

Geometry - in honour of Mario Micallef's 60th birthday. 8-10 May 2017 at Warwick.

Partial Differential Equations. 30 July - 5 August 2017 at Oberwolfach.

Junior Warwick-Imperial-Cambridge Geometric Analysis meeting. 3 November 2017 at Cambridge.

Geometric Analysis, ICMS Edinburgh, 28 May to 1 June 2018.

Geometric Analysis and General Relativity. A conference in honour of Gerhard Huisken's 60th birthday, 3 to 7 June 2019, ETH-Zuerich.

Partial Differential Equations. 21 July - 27 July 2019 at Oberwolfach.

Geometric PDE. 12-16 December 2022 at Warwick.

Ricci flow and related topics. 27-31 March 2023 at Warwick.

Differential Geometry and Geometric PDE. Friday 21 June 2024 at Warwick.


Ricci flows with rough initial data. Princeton IAS, 2019.

Sharp local decay estimates for the Ricci flow on surfaces. Institut des Hautes Études Scientifiques, 2016.

Global solutions of the Teichmueller harmonic map flow. MSRI, Berkeley 2016.

Ricci flows with unbounded curvature. International Congress of Mathematicians, Seoul, 2014.

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


>= Jan 2023