Searcharxiv⌕ Search

arXiv subjects

S. Brendle

Publications and source records attributed to S. Brendle.

At least 19 recordsLinked to original sources
arXiv2026

A metric on $S^2 \times S^2$ with positive sectional curvature

We construct a metric on $S^2 \times S^2$ with positive sectional curvature. Starting from the standard metric on $S^2 \times S^2$, we first perform a Cheeger deformation. The resulting metric has nonnegative sectional curvature. We refer to it as a Cheeger-M\"uter metric. We then consider a suitable third order perturbation of this Cheeger-M\"uter metric and show that the perturbed metrics have positive sectional curvature. The proof requires various calculations, some of which have been carried out with the help of MATHEMATICA. The MATHEMATICA code is attached to this submission.

math.DG↗
arXiv2026

On the spacetime positive energy theorem in arbitrary dimension

We describe how the spacetime positive energy theorem in dimension $n \geq 4$ follows from our recent work on the Riemannian version of the positive mass theorem. Our proof builds on the fundamental work of Schoen and Yau and the remarkable work of Eichmair, and uses the Jang equation with a capillary term. We also use the shielding principle from the work of Lesourd-Unger-Yau.

math.DG↗
arXiv2026

A dimension descent scheme for the positive mass theorem in arbitrary dimension

We describe how the Schoen-Yau proof of the positive mass theorem can be extended to arbitrary dimensions. To overcome the problem of singularities, we propose a new inductive scheme. To carry out the inductive step, we use a combination of several techniques, including the shielding principle of Lesourd-Unger-Yau, as well as a conformal blow-up argument in the spirit of Bi-Hao-He-Shi-Zhu. Our arguments also rely on the Cheeger-Naber bound for the Minkowski dimension of the singular set.

math.DG↗
arXiv2026

Geometric inequalities and the Alexandrov-Bakelman-Pucci technique

In this expository paper, we discuss a unified framework for proving various geometric inequalities, based on the so-called Alexandrov-Bakelman-Pucci technique. Examples include Cabr\'e's proof of the classical isoperimetric inequality in Euclidean space; the Fenchel-Willmore-Chen inequality for the mean curvature of a submanifold; the sharp version of the Michael-Simon Sobolev inequality for submanifolds; the sharp version of Ecker's logarithmic Sobolev inequality for submanifolds; and the Sobolev inequality for complete manifolds with nonnegative Ricci curvature and Euclidean volume growth. Finally, we discuss a connection to the work of Heintze and Karcher on the volume of a tubular neighborhood of a hypersurface in a manifold with nonnegative Ricci curvature.

math.DG↗
arXiv2025

The rigidity statement in the Horowitz-Myers conjecture

In this paper, we give an alternative proof of the Horowitz-Myers conjecture in dimension $3 \leq N \leq 7$. Moreover, we show that a metric that achieves equality in the Horowitz-Myers conjecture is locally isometric to a Horowitz-Myers metric.

math.DG↗
arXiv2024

Systolic inequalities and the Horowitz-Myers conjecture

Let $n$ be an integer with $3 \leq n \leq 7$, let $M$ be a compact manifold of dimension $n$ with boundary $\partial M$, and let $g$ be a Riemannian metric on $M$ with scalar curvature at least $-n(n-1)$. Under a topological assumption on $M$, we establish an inequality relating the infimum of the boundary mean curvature to the systole of the boundary $\partial M$. As a consequence, we obtain a new positive energy theorem, with equality being attained by the Horowitz-Myers metrics.

math.DG↗
arXiv2024

Eigenvalue estimates on shrinkers

We prove an eigenvalue estimate which holds on every properly embedded shrinker for mean curvature flow. This generalizes earlier work of Ding and Xin to the noncompact case.

math.DG↗
arXiv2023

Scalar curvature rigidity of convex polytopes

We prove a scalar curvature rigidity theorem for convex polytopes. The proof uses the Fredholm theory for Dirac operators on manifolds with boundary. A variant of a theorem of Fefferman and Phong plays a central role in our analysis.

math.DG↗
arXiv2023

On Gromov's rigidity theorem for polytopes with acute angles

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.

math.DG↗
arXiv2023

Minimal hypersurfaces and geometric inequalities

In this expository paper, we discuss some of the main geometric inequalities for minimal hypersurfaces. These include the classical monotonicity formula, the Alexander-Osserman conjecture, the isoperimetric inequality for minimal surfaces, and the Michael-Simon Sobolev inequality.

math.DG↗
arXiv2022

Singularity models in the three-dimensional Ricci flow

The Ricci flow is a natural evolution equation for Riemannian metrics on a given manifold. The main goal is to understand singularity formation. In his spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of singularity formation in dimension $3$. More precisely, Perelman showed that every finite-time singularity to the Ricci flow in dimension $3$ is modeled on an ancient $κ$-solution. Moreover, Perelman proved a structure theorem for ancient $κ$-solutions in dimension $3$. In this survey, we discuss recent developments which have led to a complete classification of all the singularity models in dimension $3$. Moreover, we give an alternative proof of the classification of noncollapsed steady gradient Ricci solitons in dimension $3$ (originally proved by the author in 2012).

math.DG↗
arXiv2022

Sobolev inequalities in manifolds with nonnegative curvature

We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic volume ratio of the ambient manifold.

math.DG↗
arXiv2020

The isoperimetric inequality for a minimal submanifold in Euclidean space

We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2.

math.DG↗
arXiv2020

Ancient solutions to the Ricci flow in dimension 3

It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient $κ$-solution. We prove that the every noncompact ancient $κ$-solution in dimension $3$ is isometric to either the shrinking cylinders (or a quotient thereof), or the Bryant soliton.

math.DG↗