Selected Publications

Simon Brendle

Research Monograph

Ricci Flow and the Sphere Theorem, Graduate Studies in Mathematics, volume 111, American Mathematical Society (2010)

Scalar Curvature

  • Deformations of the hemisphere that increase scalar curvature, Inventiones Mathematicae 185, 175--197 (2011) (joint with F. Marques and A. Neves)
  • Rigidity phenomena involving scalar curvature, Surveys in Differential Geometry, volume XVII, 179--202 (2012)
  • Scalar curvature rigidity of convex polytopes, Inventiones Mathematicae 235, 669--708 (2024)
  • On Gromov's rigidity theorem for polytopes with acute angles, Journal für die Reine und Angewandte Mathematik 827, 191--225 (2025 (joint with Y. Wang)
  • Systolic inequalities and the Horowitz-Myers conjecture (joint with P.-K. Hung)
  • The rigidity statement in the Horowitz-Myers conjecture (joint with P.-K. Hung)
  • A dimension descent scheme for the positive mass theorem in arbitrary dimension (joint with Y. Wang)
  • On the spacetime positive mass theorem in arbitrary dimension (joint with Y. Wang)
  • Lecture slides

    • Sharp Sobolev and isoperimetric inequalities

  • The isoperimetric inequality for a minimal submanifold in Euclidean space, Journal of the American Mathematical Society 34, 595--603 (2021)
  • The logarithmic Sobolev inequality for a submanifold in Euclidean space, Communications on Pure and Applied Mathematics 75, 449--454 (2022)
  • Sobolev inequalities in manifolds with nonnegative curvature, Communications on Pure and Applied Mathematics 76, 2192--2218 (2023)
  • Proof of the Michael-Simon-Sobolev inequality using optimal transport, Journal für die Reine und Angewandte Mathematik 804, 1--10 (2023) (joint with M. Eichmair)
  • Minimal hypersurfaces and geometric inequalities, Annales de la Faculté des Sciences de Toulouse 32, 179--201 (2023)
  • The isoperimetric inequality, Notices of the American Mathematical Society 71, 708--713 (2024) (joint with M. Eichmair)
  • Geometric inequalities and the Alexandrov-Bakelman-Pucci technique

    • Geometry of Hypersurfaces and the Lawson Conjecture

  • Embedded minimal tori in S^3 and the Lawson conjecture, Acta Mathematica 211, 177--190 (2013)
    • A discussion of Lawson's Conjecture by the Australian Mathematical Society: The 12th Problem
  • Embedded Weingarten tori in S^3, Advances in Mathematics 257, 462--475 (2014)
  • Two-point functions and their applications in geometry, Bulletin of the American Mathematical Society 51, 581--596 (2014)
  • Minimal surfaces in S^3: a survey of recent results, Bulletin of Mathematical Sciences 3, 133--171 (2013)
  • Constant mean curvature surfaces in warped product manifolds, Publications Mathématiques de l'IHÉS 117, 247--269 (2013)
  • Large outlying stable constant mean curvature spheres in initial data sets, Inventiones Mathematicae 197, 663--682 (2014) (joint with M. Eichmair)
  • A sharp bound for the area of minimal surfaces in the unit ball, Geometric and Functional Analysis 22, 621--626 (2012)
  • Area bounds for minimal surfaces that pass through a prescribed point in a ball, Geometric and Functional Analysis 235--239 (2017) (joint with P.-K. Hung)

    • Ricci Flow and the Differentiable Sphere Theorem

  • Manifolds with 1/4-pinched curvature are space forms, Journal of the American Mathematical Society 22, 287--307 (2009) (joint with R. Schoen)
  • Classification of manifolds with weakly 1/4-pinched curvatures, Acta Mathematica 200, 1--13 (2008) (joint with R. Schoen)
  • Sphere theorems in geometry, Surveys in Differential Geometry, volume XIII, 49--84 (2008) (joint with R. Schoen)
  • Curvature, sphere theorems, and the Ricci flow, Bulletin of the American Mathematical Society 48, 1--32 (2011) (joint with R. Schoen)
  • A general convergence theorem for the Ricci flow in higher dimensions, Duke Mathematical Journal 145, 585--601 (2008)
  • Einstein manifolds with nonnegative isotropic curvature are locally symmetric, Duke Mathematical Journal 151, 1--21 (2010)
  • A generalization of Hamilton's differential Harnack inequality for the Ricci flow, Journal of Differential Geometry 82, 207--227 (2009)
  • Ancient solutions to the Ricci flow with pinched curvature, Duke Mathematical Journal 158, 537--551 (2011) (joint with G. Huisken and C. Sinestrari)
  • Ricci flow with surgery on manifolds with positive isotropic curvature, Annals of Mathematics 190, 465--559 (2019)

    • Ancient Solutions to Geometric Flows

  • Rotational symmetry of self-similar solutions to the Ricci flow, Inventiones Mathematicae 194, 731--764 (2013)
  • Ancient solutions to the Ricci flow in dimension 3, Acta Mathematica 225, 1--102 (2020)
  • Unique asymptotics of compact ancient solutions to three-dimensional Ricci flow, Communications on Pure and Applied Mathematics 75, 1032--1073 (2022) (joint with S. Angenent, P. Daskalopoulos, and N. Sesum)
  • Uniqueness of compact ancient solutions to three-dimensional Ricci flow, Inventiones Mathematicae 226, 579--651 (2022) (joint with P. Daskalopoulos and N. Sesum)
  • Uniqueness of convex ancient solutions to mean curvature flow in R^3, Inventiones Mathematicae 217, 35--76 (2019) (joint with K. Choi)
  • Singularity models in the three-dimensional Ricci flow, Recent Progress in Mathematics, 87--118, Springer, Singapore, 2022

    • Geometric Flows for Hypersurfaces

  • A sharp bound for the inscribed radius under mean curvature flow, Inventiones Mathematicae 202, 217--237 (2015)
  • Mean curvature flow with surgery of mean convex surfaces in R^3, Inventiones Mathematicae 203, 615--654 (2016) (joint with G. Huisken)
  • Embedded self-similar shrinkers of genus 0, Annals of Mathematics 183, 715--728 (2016)
  • A fully nonlinear flow for two-convex hypersurfaces in Riemannian manifolds, Inventiones Mathematicae 210, 559--613 (2017) (joint with G. Huisken)
  • Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Mathematica 219, 1--16 (2017) (joint with K. Choi and P. Daskalopoulos)

    • The Yamabe Flow and the Compactness Conjecture for the Yamabe Problem

  • Convergence of the Yamabe flow for arbitrary initial energy, Journal of Differential Geometry 69, 217--278 (2005)
  • A short proof for the convergence of the Yamabe flow on S^n, Pure and Applied Mathematics Quarterly 3, 499--512 (2007)
  • Convergence of the Yamabe flow in dimension 6 and higher, Inventiones Mathematicae 170, 541--576 (2007)
  • Blow-up phenomena for the Yamabe equation, Journal of the American Mathematical Society 21, 951--979 (2008)
  • On the conformal scalar curvature equation and related problems, Surveys in Differential Geometry, volume XII, 1--19 (2007)