Selected Publications
Simon Brendle
Research Monograph
Ricci Flow and the Sphere Theorem,
Graduate Studies in Mathematics, volume 111, American Mathematical Society (2010)
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, to appear in Notices of the American Mathematical Society (joint with M. Eichmair)
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 Minkowski inequality for hypersurfaces in the Anti-deSitter-Schwarzschild manifold, Communications on Pure and Applied Mathematics 69, 124--144 (2016) (joint with P.-K. Hung and M.-T. Wang)
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)
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
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)
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)
Eigenvalue estimates on shrinkers (joint with R. Tsiamis)
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)
Scalar Curvature Rigidity Problems
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)