About

Simon Brendle is a Professor of Mathematics at Columbia University in the City of New York. His research interests include Differential Geometry and Nonlinear Partial Differential Equations, focusing on these areas to advance mathematical understanding and solve complex problems within these fields.

Research topics

• Mathematics
• Geometry
• Pure mathematics
• Computer Science
• Mathematical analysis
• Artificial Intelligence
• Physics
• Algorithm

Selected publications

• Richard Streit Hamilton (1943–2024)

  Notices of the American Mathematical Society · 2025-10-01

  article1st authorCorresponding
  PublisherDOI

• On Gromov’s rigidity theorem for polytopes with acute angles

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2025-07-30

  articleOpen access1st authorCorresponding

  Abstract In his “Four Lectures”, Gromov conjectured a scalar curvature extremality property of convex polytopes. Moreover, Gromov outlined a proof of the conjecture in the special case when the dihedral angles are acute. Gromov’s argument relies on Dirac operator techniques together with a smoothing construction. In this paper, we give the details of such a smoothing construction, thereby providing a detailed proof of Gromov’s theorem.

  PublisherDOI

• On fill-ins with scalar curvature bounded from below and an inequality of Hijazi-Montiel-Roldán

  ArXiv.org · 2025-10-20

  preprintOpen access1st authorCorresponding

  We consider fill-ins of spin manifolds with scalar curvature bounded by $-n(n-1)$. Gromov proposed a conjecture relating the infimum of the mean curvature of such a fill-in to the hyperspherical radius. We observe that the inequality conjectured by Gromov follows by combining an inequality of Hijazi-Montiel-Roldán for the first Dirac eigenvalue with a recent theorem of Bär. Moreover, we give an alternative proof of the Hijazi-Montiel-Roldán inequality based on the work of Bär and Bär-Ballmann.

  PublisherOA PDFDOI

• Rigidity results for initial data sets satisfying the dominant energy condition

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2025-11-07

  articleOpen access

  Abstract Our work proves rigidity theorems for initial data sets associated with compact smooth spin manifolds with boundary and with compact convex polytopes, subject to the dominant energy condition. For manifolds with smooth boundary, this is based on the solution of a boundary value problem for Dirac operators. For convex polytopes, we use approximations by manifolds with smooth boundary.

  PublisherDOI

• The isoperimetric inequality

  arXiv (Cornell University) · 2024-02-08

  preprintOpen access1st authorCorresponding

  We discuss several classical and recent proofs of the isoperimetric inequality and the Sobolev inequality.

  PublisherOA PDFDOI

• Scalar Curvature Rigidity of Warped Product Metrics

  Symmetry Integrability and Geometry Methods and Applications · 2024-04-18 · 6 citations

  articleOpen access

  We show scalar-mean curvature rigidity of warped products of round spheres of dimension at least 2 over compact intervals equipped with strictly log-concave warping functions. This generalizes earlier results of Cecchini-Zeidler to all dimensions. Moreover, we show scalar curvature rigidity of round spheres of dimension at least 3 with two antipodal points removed. This resolves a problem in Gromov's ''Four Lectures'' in all dimensions. Our arguments are based on spin geometry.

  PublisherOA PDFDOI

• A geometric approach to apriori estimates for optimal transport maps

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2024-10-04

  articleOpen access1st authorCorresponding

  Abstract A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma–Trudinger–Wang condition is the Pogorelov second-derivative bound. This translates to an apriori interior <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> C^{1} estimate for smooth optimal maps. Here we give a new derivation of this estimate which relies in part on Kim, McCann and Warren’s observation that the graph of an optimal map becomes a volume maximizing spacelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost.

  PublisherDOI

• A local noncollapsing estimate for mean curvature flow

  American Journal of Mathematics · 2024-09-26 · 2 citations

  article1st authorCorresponding

  abstract: We prove a local version of the noncollapsing estimate for mean curvature flow. By combining our result with earlier work of X.-J. Wang, it follows that certain ancient convex solutions that sweep out the entire space are noncollapsed.

  PublisherDOI

• The Isoperimetric Inequality

  Notices of the American Mathematical Society · 2024-06-01 · 2 citations

  articleOpen access1st authorCorresponding

  The isoperimetric problem is one of the oldest and most famous problems in geometry.

  PublisherDOI

• Systolic inequalities and the Horowitz-Myers conjecture

  arXiv (Cornell University) · 2024-06-06

  preprintOpen access1st authorCorresponding

  Let $n$ be an integer with $3 \leq n \leq 7$, and let $g$ be a Riemannian metric on $B^2 \times T^{n-2}$ with scalar curvature at least $-n(n-1)$. We establish an inequality relating the systole of the boundary to the infimum of the mean curvature on the boundary. As a consequence, we obtain a new positive energy theorem where equality holds for the Horowitz-Myers metrics.

  PublisherOA PDFDOI

Recent grants

• Partial Differential Equations in Riemannian Geometry

  NSF · $256k · 2016–2019

• Parabolic flows in geometry

  NSF · $408k · 2009–2014

• PDE Problems in Geometry

  NSF · $208k · 2012–2016

• Parabolic problems in conformal geometry

  NSF · $131k · 2006–2009

• Geometric Flows, Geometric Inequalities, and Rigidity of Embeddings

  NSF · $222k · 2021–2024

Frequent coauthors

• Panagiota Daskalopoulos

  Columbia University

  11 shared

• Michael Eichmair

  University of Vienna

  9 shared

• Richard Schoen

  9 shared

• Nataša Šešum

  8 shared

• Kyeongsu Choi

  7 shared

• Keaton Naff

  7 shared

• Gerhard Huisken

  University of Tübingen

  6 shared

• André Neves

  University of Chicago

  5 shared

Education

• Ph.D., Mathematics

  University of Bonn

  2003

• M.S., Mathematics

  University of Bonn

  1999

• B.S., Mathematics

  University of Bonn

  1998