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Mathematics Institute

Papers of Peter Topping

If you spot that any of the links below are broken (or are not open access where claimed) then I would be very grateful if you would let me know. Thanks!

  • 65) joint with Arjun Sobnack: `Monotonicity of the modulus under curve shortening flow.' Preprint (2024). arxiv.

  • 64) joint with Arjun Sobnack: `Delayed parabolic regularity for curve shortening flow.' Preprint (2024). arxiv.

  • 63) `Ricci flow and PIC1.' Published in `Essays on Geometric Flows: Celebrating 40 Years of Ricci Flow.' Surveys in Differential Geometry XXVII. International Press. Eds: H.D.Cao, R.Hamilton, S.T.Yau (2022) 189-211. arxiv.

  • 62) joint with Luke Peachey: `All two-dimensional expanding Ricci solitons.' To appear, Journal of the London Mathematical Society. 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) Journal, open access. and arxiv.

  • 60) joint with Man-Chun Lee: `Manifolds with PIC1 pinched curvature.' Preprint (2022). arxiv.

  • 59) joint with Man-Chun Lee: `Three-manifolds with non-negatively pinched Ricci curvature.' To appear, Journal of Differential Geometry. arxiv.

  • 58) joint with Man-Chun Lee: `Metric limits of manifolds with positive scalar curvature.' To appear, American Journal of Mathematics. arxiv.

  • 57) joint with Man-Chun Lee: `Time zero regularity of Ricci flow.' International Mathematics Research Notices, 2022;, rnac300, arxiv.

  • 56) joint with Hao Yin: `Smoothing a measure on a Riemann surface using Ricci flow.' To appear, Ars Inveniendi Analytica. arxiv.

  • 55) `Nontrivial breathers for Ricci flow.' Bulletin of the L.M.S. 55 (2023) 344--348. Journal, 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. arxiv.

  • 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. Journal, open access.

  • 52) joint with Andrew McLeod: `Pyramid Ricci Flow in Higher Dimensions.' Math. Zeit. Vol. 296 (2020) 511--523. Journal, open access.

  • 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). arxiv.

  • 50) `Ricci flow and Ricci limit spaces.' `Geometric Analysis.' [Cetraro 2018] Springer Lecture Notes in Mathematics, CIME Foundation Subseries. Vol. 2263 (2020). arxiv.

  • 49) joint with Andrew McLeod: `Global Regularity of Three-Dimensional Ricci Limit Spaces.' Trans. Amer. Math. Soc. Ser. B. 9 (2022) 345--370. Journal, open access.

  • 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. 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.

  • 46) joint with Miles Simon: `Local control on the geometry in 3D Ricci flow.' J. Differential Geometry, Vol. 122, No. 3 (2022) 467-518. arxiv.

  • 45) joint with Hao Yin: `Rate of curvature decay for the contracting cusp Ricci flow.' Comm. Anal. Geom. Vol. 28, (2020) 1221-1250. Journal, with free PDF.

  • 44) joint with Melanie Rupflin: `Horizontal curves of hyperbolic metrics.' Calc. Var. (2018) 57:106. Journal, open access.

  • 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. Journal, open access.

  • 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. Journal, open access.

  • 41) `Ricci flows with unbounded curvature.' Proceedings of the International Congress of Mathematicians, Seoul 2014. arxiv.

  • 40) joint with Melanie Rupflin: `Teichmüller harmonic map flow into nonpositively curved targets.' J. Differential Geometry. 108 (2018) 135--184. Journal, with free 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.

  • 38) `Uniqueness of Instantaneously Complete Ricci flows.' Geometry and Topology 19-3 (2015), 1477--1492. Journal, open access.

  • 37) joint with Gregor Giesen: `Ricci flows with bursts of unbounded curvature.' Comm. Partial Differential Equations. 41 (2016) 854--876. arxiv.

  • 36) joint with Sebastian Helmensdorfer: `Bouncing of charged droplets: An explanation using mean curvature flow.' EPL (Europhysics letters) 104 (2013) 34001. `Editor's choice.' arxiv.

  • 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. Journal, open access.

  • 34) joint with Melanie Rupflin: `A uniform Poincaré estimate for quadratic differentials on closed surfaces.' Calc. Var. and PDE. 53 (2015) 587--604. arxiv.

  • 33) joint with Melanie Rupflin and Miaomiao Zhu: `Asymptotics of the Teichmueller harmonic map flow.' Advances in Math. 244 (2013) 874--893. arxiv.

  • 32) joint with Melanie Rupflin: `Flowing maps to minimal surfaces.' American Journal of Math. 138 (2016) 1095--1115. arxiv.

  • 31) `Remarks on Hamilton's Compactness Theorem for Ricci flow.' J. Reine Angew. Math. 692 (2014) 173--191. arxiv.

  • 30) joint with Gregor Giesen: `Ricci flows with unbounded curvature.' Math. Zeit., 273 (2013) 449--460. arxiv.

  • 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. arxiv.

  • 28) `Uniqueness and nonuniqueness for Ricci flow on surfaces: Reverse cusp singularities.' I.M.R.N., 2012 (2012) 2356--2376. arxiv.

  • 27) joint with Gregor Giesen: `Existence of Ricci flows of incomplete surfaces.' Comm. Partial Differential Equations, 36 (2011) 1860--1880. arxiv.

  • 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.

  • 25) joint with Joerg Enders and Reto Mueller: `On Type I Singularities in Ricci flow.' Communications in Analysis and Geometry, 19 (2011) 905--922. arxiv.

  • 24) joint with Camillo De Lellis: `Almost-Schur lemma.' Calc. Var. and PDE, 43 (2012) 347--354. arxiv.

  • 23) `Ricci flow: The foundations via optimal transportation.' In `Optimal Transportation, Theory and Applications.' LMS lecture notes series, vol. 413, (2014) CUP. 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. Journal, with free PDF.

  • 21) joint with Gregor Giesen: `Ricci flow of negatively curved incomplete surfaces.' Calc. Var. and PDE, 38 (2010) 357--367. arxiv.

  • 20) joint with Esther Cabezas-Rivas: `The canonical shrinking soliton associated to a Ricci flow.' Calc. Var. and PDE, 43 (2012) 173-184. arxiv.

  • 19) L-optimal transportation for Ricci flow.' J. Reine Angew. Math., 636 (2009) 93--122. Journal. PDF.

  • 18) joint with Robert McCann: `Ricci flow, entropy and optimal transportation.' American Journal of Math., 132 (2010) 711-730. Journal. PDF.

  • 17) `Relating diameter and mean curvature for submanifolds of Euclidean space.' Comment. Math. Helvetici, 83 (2008) 539-546. Journal, with free 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. arxiv.

  • 15) `Ricci flow compactness via pseudolocality, and flows with incomplete initial metrics.' J.E.M.S., 12 (2010) 1429--1451. Journal, with free PDF.

  • 14) `Diameter control under Ricci flow.' C.A.G., 13 (2005) 1039-1055. Journal, with free PDF.

  • 13) `Improved regularity of harmonic map flows with Hölder continuous energy.' Calc. Var., 21 (2004) 47-55. Journal. PDF.

  • 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. PDF.

  • 11) `Winding behaviour of finite-time singularities of the harmonic map heat flow.' Math. Zeit., 247 (2004) 279-302. Journal. PDF.

  • 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.

  • 9) `Reverse bubbling and nonuniqueness in the harmonic map flow.' IMRN, 10 (2002) 505-520. Journal. PDF.

  • 8) `Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow.' Annals of Math., 159 (2004) 465-534. Journal, with free PDF.

  • 7) `Towards the Willmore conjecture.' Calc. Var., 11 (2000) 361-393. Journal. PDF.

  • 6) `Pressure estimates in two dimensional incompressible fluid flow.' Physica D, 137 (2000) 143-156. Journal. PDF.

  • 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.

  • 4) `The isoperimetric inequality on a surface.' Manuscripta Math., 100 (1999) 23-33. Journal. PDF.

  • 3) `Mean curvature flow and geometric inequalities.' J. Reine Angew. Math., 503 (1998) 47-61. Journal. PDF.

  • 2) `The Optimal Constant in Wente's $L^\infty$ Estimate.' Comment. Math. Helvetici, 72 (1997) 316-328. Journal. PDF.

  • 1) `Rigidity in the harmonic map heat flow.' J. Differential Geometry, 45 (1997) 593-610. Journal, with free PDF.