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S. Brendle

Publications and source records attributed to S. Brendle.

At least 37 records · Page 2Linked to original sources
arXiv2019

Ricci flow with surgery on manifolds with positive isotropic curvature

We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates which ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension $n \geq 12$, we show that blow-up limits are weakly PIC2. This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension $3$. In the second part, we develop a theory of ancient solutions which have bounded curvature; are $κ$-noncollapsed; are weakly PIC2; and are uniformly PIC. This is an extension of Perelman's work; the additional ingredients needed in the higher dimensional setting are the differential Harnack inequality for solutions to the Ricci flow satisfying the PIC2 condition, and a rigidity result due to Brendle-Huisken-Sinestrari for ancient solutions that are uniformly PIC1. In the third part of this paper, we prove a Canonical Neighborhood Theorem for the Ricci flow with initial data with positive isotropic curvature, which holds in dimension $n \geq 12$. This relies on the curvature pinching estimates together with the structure theory for ancient solutions. This allows us to adapt Perelman's surgery procedure to this situation. As a corollary, we obtain a topological classification of all compact manifolds with positive isotropic curvature of dimension $n \geq 12$ which do not contain non-trivial incompressible $(n-1)$-dimensional space forms.

math.DG↗
arXiv2019

Uniqueness of convex ancient solutions to mean curvature flow in $\mathbb{R}^3$

A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension $3$ which have positive sectional curvature and are $κ$-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $\mathbb{R}^3$, and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in $\mathbb{R}^3$ which is strictly convex and noncollapsed.

math.DG↗
arXiv2017

Ricci flow with surgery in higher dimensions

We present a new curvature condition which is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's neck-like curvature pinching estimate. Using this estimate, we are able to prove a version of Perelman's Canonical Neighborhood Theorem in higher dimensions. This makes it possible to extend the flow beyond singularities by a surgery procedure in the spirit of Hamilton and Perelman. As a corollary, we obtain a classification of all diffeomorphism types of such manifolds in terms of a connected sum decomposition. In particular, the underlying manifold cannot be an exotic sphere. Our result is sharp in many interesting situations. For example, the curvature tensors of $\mathbb{CP}^{n/2}$, $\mathbb{HP}^{n/4}$, $S^{n-k} \times S^k$ ($2 \leq k \leq n-2$), $S^{n-2} \times \mathbb{H}^2$, $S^{n-2} \times \mathbb{R}^2$ all lie on the boundary of our curvature cone. Another borderline case is the pseudo-cylinder: this is a rotationally symmetric hypersurface which is weakly, but not strictly, two-convex. Finally, the curvature tensor of $S^{n-1} \times \mathbb{R}$ lies in the interior of our curvature cone.

math.DG↗
arXiv2017

A fully nonlinear flow for two-convex hypersurfaces

We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity $G_κ= \big ( \sum_{i < j} \frac{1}{λ_i+λ_j-2κ} \big )^{-1}$, where $λ_1 \leq \hdots \leq λ_n$ denote the curvature eigenvalues and $κ$ is a nonnegative constant. This defines a fully nonlinear parabolic equation, provided that $λ_1+λ_2>2κ$. In contrast to mean curvature flow, this flow preserves the condition $λ_1+λ_2>2κ$ in a general ambient manifold. Our main goal in this paper is to extend the surgery algorithm of Huisken-Sinestrari to this fully nonlinear flow. This is the first construction of this kind for a fully nonlinear flow. As a corollary, we show that a compact Riemannian manifold satisfying $\overline{R}_{1313}+\overline{R}_{2323} \geq -2κ^2$ with non-empty boundary satisfying $λ_1+λ_2 > 2κ$ is diffeomorphic to a $1$-handlebody. The main technical advance is the pointwise curvature derivative estimate. The proof of this estimate requires a new argument, as the existing techniques for mean curvature flow due to Huisken-Sinestrari, Haslhofer-Kleiner, and Brian White cannot be generalized to the fully nonlinear setting. To establish this estimate, we employ an induction-on-scales argument; this relies on a combination of several ingredients, including the almost convexity estimate, the inscribed radius estimate, as well as a regularity result for radial graphs. We expect that this technique will be useful in other situations as well.

math.DG↗
arXiv2017

Area bounds for minimal surfaces that pass through a prescribed point in a ball

Let $Σ$ be a $k$-dimensional minimal submanifold in the $n$-dimensional unit ball $B^n$ which passes through a point $y \in B^n$ and satisfies $\partial Σ\subset \partial B^n$. We show that the $k$-dimensional area of $Σ$ is bounded from below by $|B^k| \, (1-|y|^2)^{\frac{k}{2}}$. This settles a question left open by the work of Alexander and Osserman in 1973.

math.DG↗
arXiv2016

Gluing Eguchi-Hanson metrics and a question of Page

In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat Kähler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with $16$ orbifold points, and resolves the orbifold singularities by gluing in $16$ small Eguchi-Hanson manifolds which all have the same orientation. This construction was carried out rigorously by Topiwala, LeBrun-Singer, and Donaldson. In 1981, Page asked whether the above construction can be modified by reversing the orientations of some of the Eguchi-Hanson manifolds. This is a subtle question: if successful, this construction would produce Einstein metrics which are neither Kähler nor self-dual. In this paper, we focus on a configuration of maximal symmetry involving $8$ small Eguchi-Hanson manifolds of each orientation which are arranged according to a chessboard pattern. By analyzing the interactions between Eguchi-Hanson manifolds with opposite orientation, we identify a non-vanishing obstruction to the gluing problem, thereby destroying any hope of producing a metric of zero Ricci curvature in this way. Using this obstruction, we are able to understand the dynamics of such metrics under Ricci flow as long as the Eguchi-Hanson manifolds remain small. In particular, for the configuration described above, we obtain an ancient solution to the Ricci flow with the property that the maximum of the Riemann curvature tensor blows up at a rate of $(-t)^{\frac{1}{2}}$, while the maximum of the Ricci curvature converges to $0$.

math.AP↗
arXiv2015

Embedded self-similar shrinkers of genus 0

We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution to the mean curvature flow with genus $0$. More generally, we show that the only properly embedded self-similar shrinkers in $\mathbb{R}^3$ with vanishing intersection form are the sphere, the cylinder, and the plane. This answers two questions posed by T. Ilmanen.

math.DG↗
arXiv2015

Mean curvature flow with surgery of mean convex surfaces in $\mathbb{R}^3$

We define a notion of mean curvature flow with surgery for two-dimensional surfaces in $\mathbb{R}^3$ with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author.

math.DG↗
arXiv2015

A monotonicity formula for mean curvature flow with surgery

We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken's monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is, in fact, free of surgeries. This result has applications to the longtime behavior of mean curvature flow with surgery in Riemannian three-manifolds.

math.DG↗
arXiv2014

Embedded Weingarten tori in S^3

In this paper, we show that an embedded Weingarten surface in S^3 of genus 1 must be rotationally symmetric, provided that certain structure conditions are satisfied. The argument involves an adaptation of our proof of Lawson's Conjecture for minimal tori.

math.DG↗
arXiv2014

Two-point functions and their applications in geometry

The maximum principle is one of the most important tools in the analysis of geometric partial differential equations. Traditionally, the maximum principle is applied to a scalar function defined on a manifold, but in recent years more sophisticated versions have emerged. One particularly interesting direction involves applying the maximum principle to functions that depend on a pair of points. This technique is particularly effective in the study of problems involving embedded surfaces. In this survey, we first describe some foundational results on curve shortening flow and mean curvature flow. We then describe Huisken's work on the curve shortening flow where the method of two-point functions was introduced. Finally, we discuss several recent applications of that technique. These include sharp estimates for mean curvature flow, as well as the proof of Lawson's 1970 conjecture concerning minimal tori in S^3.

math.DG↗
arXiv2013

An inscribed radius estimate for mean curvature flow in Riemannian manifolds

We consider a family of embedded, mean convex hypersurfaces in a Riemannian manifold which evolve by the mean curvature flow. We show that, given any number $T>0$ and any $δ>0$, we can find a constant $C_0$ with the following property: if $t \in [0,T)$ and $p$ is a point on $M_t$ where the curvature is greater than $C_0$, then the inscribed radius is at least $\frac{1}{(1+δ) \, H}$ at the point $p$. The constant $C_0$ depends only on $δ$, $T$, and the initial data.

math.DG↗
arXiv2013

A sharp bound for the inscribed radius under mean curvature flow

We consider a family of embedded, mean convex hypersurfaces which evolve by the mean curvature flow. It follows from general results of White that the inscribed radius at each point on the surface is at least $\frac{c}{H}$, where $c$ is a constant that depends only on the initial data. Andrews recently gave a new proof of that fact using a direct monotonicity argument. In this paper, we improve this result and show that the inscribed radius is at least $\frac{1}{(1+δ) \, H}$ at each point where the curvature is large.

math.DG↗
arXiv2013

Minimal surfaces in S^3: a survey of recent results

In this survey, we discuss various aspects of the minimal surface equation in the three-sphere S^3. After recalling the basic definitions, we describe a family of immersed minimal tori with rotational symmetry. We then review the known examples of embedded minimal surfaces in S^3. Besides the equator and the Clifford torus, these include the Lawson and Kapouleas-Yang examples, as well as a new family of examples found recently by Choe and Soret. We next discuss uniqueness theorems for minimal surfaces in S^3, such as the work of Almgren on the genus 0 case, and our recent solution of Lawson's conjecture for embedded minimal surfaces of genus 1. More generally, we show that any minimal surface of genus 1 which is Alexandrov immersed must be rotationally symmetric. We also discuss Urbano's estimate for the Morse index of an embedded minimal surface and give an outline of the recent proof of the Willmore conjecture by Marques and Neves. Finally, we describe estimates for the first eigenvalue of the Laplacian on a minimal surface.

math.DG↗
arXiv2013

Alexandrov immersed minimal tori in S^3

We show that any minimal torus in $S^3$ which is Alexandrov immersed must be rotationally symmetric. An analogous result holds for surfaces of constant mean curvature.

math.DG↗
arXiv2013

Rotational symmetry of Ricci solitons in higher dimensions

Let (M,g) be a steady gradient Ricci soliton of dimension n \geq 4 which has positive sectional curvature and is asymptotically cylindrical. Under these assumptions, we show that (M,g) is rotationally symmetric. In particular, our result applies to steady gradient Ricci solitons in dimension 4 which are κ-noncollapsed and have positive isotropic curvature.

math.DG↗
arXiv2013

A volume comparison theorem for asymptotically hyperbolic manifolds

We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.

math.DG↗