Schedule for: 24w5226 - Recent Advances in Comparison Geometry

A set dinner is served daily between 6:00 pm and 8:00 pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

A brief introduction with important logistical information, technology instruction, and opportunity for participants to ask questions.

The talk consists of two parts. In the first part of the talk, I will discuss a kind of open manifolds carries no complete positive scalar curvature metric, and in the second part of the talk, I will discuss Llarull type theorems on complete manifolds with positive scalar curvature. The talk based on my recent joint works with T.Hao, Y.Sun, R.Wu, J.Wang and J.Zhu.

In this talk, we study solutions to the minimal hypersurface equation $$\mathrm{div} \left( \frac{Du}{\sqrt{1+|Du|^2}} \right) = 0$$ defined on a complete Riemannian manifold $M$. The qualitative properties of such solutions are influenced by the geometry of $M$, and one may expect results similar to those holding in Euclidean space provided that $M$ has non-negative sectional or Ricci curvature. We focus on ${\rm Ric} \ge 0$, a case for which the analysis is subtler, especially because the lack of uniform ellipticity of the mean curvature operator makes comparison theory difficult to use. I will survey on recent splitting, Liouville and and half-space theorems obtained by the author in collaboration with G. Colombo, E.S. Gama, M. Magliaro and M. Rigoli. Techniques range from heat equation to potential theoretic arguments.

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

The Riemannian Penrose inequality affirms that the total mass of a gravitational system is at least the mass of the black hole inside it. In their 2001 paper, Huisken and Ilmanen proved the validity of this inequality using the monotonicity of the Hawking mass along the weak inverse mean curvature flow. In this talk, I will present an overview of a different approach which draws inspiration from the recent result by Agostiniani, Mantegazza, Mazzieri and Oronzio. The monotonicity of the Hawking mass turns out to be the mirrored image of a whole family of monotone quantities in nonlinear potential theory: they both reflect some information of each other while maintaining their specific properties. Going through the looking-glass, this family express its full potential and permits extending the validity of the Riemannian Penrose inequality in a more general setting. This talk is based on a series of works in collaboration with M. Fogagnolo (University of Padua), L. Mazzieri (University of Trento), A. Pluda (University of Pisa) and M. Pozzetta (University of Naples).

We present several new stability and rigidity results for scalar curvature. In particular, we prove stability of Llarull's theorem in all dimensions using spin geometry. Additionally, we discuss some related questions which are motivated by General Relativity. This is based upon joined work with Yiyue Zhang.

Substatic manifolds arise naturally in General Relativity as spatial slices of static spacetimes satisfying the Null Energy Condition. We will show that the substatic condition captures a large class of interesting model solutions and we will discuss its connection with the Bakry-Émery Ricci tensor. This will allow us to perform some known and new comparison arguments, leading to a Bishop-Gromov monotonicity, a splitting theorem and an isoperimetric inequality for substatic manifolds. Time permitting, we will also discuss an Heintze-Karcher inequality in this setting and how it can be exploited to improve on Brendle’s characterization of CMC hypersurfaces in substatic warped products. This talk is based on joint works with Mattia Fogagnolo and Andrea Pinamonti.

We introduce a new class of examples called drawstring, which provides further understanding on the geometry of scalar curvature in dimension three. Roughly speaking, drawstrings allow us to arbitrarily decrease the length of a curve by altering the metric near the curve in the way that only decrease the scalar curvature slightly. We will discuss the relation between drawstring and various other results and problems, including the stability problems in scalar curvature. We will also discuss the construction of drawstrings and some related thoughts. This talk is based on my joint work with Demetre Kazaras.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

Typical comparison results in Riemannian geometry, such as for volume or for spectrum of the Laplacian, require Ricci curvature lower bounds. In dimension three, we can prove several sharp comparison estimates assuming (mostly) a scalar curvature bound. The talk will present these results, their applications, and explain how dimension three is used in the proofs.

The concepts of Sobolev functions, elliptic operators and Banach spaces are central in modern geometric analysis. In the setting of Lorentzian geometry, however, unless one restricts the attention to Cauchy hypersurfaces these do not have a clear analogue, due to the signature of the metric tensor. Aim of the talk is to discuss some recent observations in this direction centered around the fact that for $p<1$ the $p$-D’Alambertian is elliptic on the space of time functions. The talk is mostly based on joint project with Beran, Braun, Calisti, McCann, Ohanyan, Rott, Saemann.

It is well-known that torus and spheres satisfy rigidity properties related to their scalar curvature geometry. There has been interest in generalizing it to metrics with weaker regularity. In this talk, we will discuss some results using the Ricci flow smoothing. This is based on joint work with L.-F. Tam.

The classical singularity theorems of General Relativity show that any spacetime with a smooth Lorentzian metric satisfying certain curvature and causality assumptions must be geodesically incomplete. From a mathematical perspective they can be viewed as analogues of Riemannian diameter estimates like the Bonnet-Myers theorem. Similarly to these diameter estimates the singularity theorems have a long history of refinements and proofs often rely on methods from comparison geometry. In my talk I'll present some recent developments in Lorentzian comparison geometry concerning two-sided area estimates with applications to the singularity theorems for metrics having almost non-negative timelike Ricci curvature in an integral sense. This is largely based on joint work with E.-A. Kontou, A. Ohanyan and B. Schinnerl.

The notion of $m$-intermediate curvature interpolates between Ricci curvature and scalar curvature. In this talk we describe extentions of classical results by Bonnet--Myers and Schoen--Yau to the setting of $m$-intermediate curvature: A non-existence result for metrics of positive $m$-intermediate curvature on manifolds with topology $N^n = \mathbb{T}^m \times \mathbb{S}^{n-m}$; a gluing result for manifolds with $m$-convex boundary; inheritance of spectral positivity along stable minimal hypersurfaces, and estimates for the $m$-diameter for uniform positive lower bounds. This talk is partially based on joint work with Simon Brendle and Sven Hirsch, and joint work with Aaron Chow and Jingbo Wan.

The curve shortening flow is an evolution equation in which a curve moves with normal velocity equal to its curvature, and can be interpreted as the gradient flow of the length. In this talk, I will consider its natural generalization to networks that are finite unions of sufficiently smooth curves whose endpoints meet at junctions. I will list the many technical challenges one has to face to give a puctual description of this evolution.

In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature. The conjecture has been made precise at an IAS Emerging Topics meeting: requiring that the sequence be three dimensional with uniform upper bounds on diameter and volume, and a positive uniform lower bound on MinA, which is the minimum area of a closed minimal surface in the manifold. In joint work with Wenchuan Tian and Changliang Wang, we present a sequence of warped product manifolds with warped circles over standard spheres, that have circular fibres over the poles whose length diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We prove this sequence converges in the $W^{1,p}$ sense for $p<2$ to an extreme limit space that has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch and that the total distributional scalar curvature converges. In joint work with Tian we prove GH and SWIF convergence to this extreme limit as well. Tian and Wang have also proven a $W^{1,p}$ compactness theorem in this setting. See https://sites.google.com/site/intrinsicflatconvergence/

In this talk, I will start with an introduction on asymptotically flat manifolds and classical mass inequalities including Riemannian positive mass theorem and Riemannian Penrose inequality. Then we make a review on recent developments on mass inequalities for asymptotically flat manifolds with arbitrary ends and mention the most general mass-systole conjecture raised by myself. Finally, I present my recent works on the mass-systole conjecture as evidence for its validity.

This talk is about the isoperimetric structure of spaces with lower bounds on either the Ricci or the scalar curvature. At first, I will describe a generalized existence result for the isoperimetric problem on noncompact spaces: every minimizing sequence either subconverges to an isoperimetric set, or part of the mass is lost in isoperimetric regions at infinity, in an appropriate sense. Then, I will explore two consequences of the latter principle. First, I will prove that on every $n$-dimensional noncompact manifold with nonnegative Ricci curvature the $n/(n-1)$-th power of the isoperimetric profile is concave. Second, I will prove that on $3$-manifolds endowed with continuous metrics that have nonnegative scalar curvature in an appropriate weak sense, isoperimetric sets exist for arbitrarily large volumes, provided the manifold is $C^0$-locally asymptotic to $\mathbb R^3$. If time allows, I will describe some links between this second result and weak versions of the positive mass theorem.

IASM will offer a free city tour including a round table dinner.

Using techniques in comparison geometry, we will show that there are vacuum static $3+1$ black hole solutions, metrically complete but with a non-standard spatial topology, that cannot be put into rotation, that is, there are no non-static stationary metrics close to them. To our knowledge, this is the first result of the kind in the literature. This is joint work with Javier Peraza.

We present the first examples of formally asymptotically flat black hole solutions with horizons of general lens space topology $L(p,q)$. These $5$-dimensional static/stationary spacetimes are regular on and outside the event horizon for any choice of relatively prime integers $p>q\geq 1$, in particular conical singularities are absent. They are supported by Kaluza-Klein matter fields arising from higher dimensional vacuum solutions through reduction on tori. The technique is sufficiently robust that it leads to the explicit construction of regular solutions, in any dimension, realising the full range of possible topologies for the horizon as well as the domain of outer communication, that are allowable with multi-axisymmetry. Lastly, as a by product, we obtain new examples of regular gravitational instantons in higher dimensions.

I will talk about a higher order scalar curvature, a generalized scalar curvature, and related geometric problems.

We consider the eigenvalue problem $\Delta^{\mathbb S^2}\xi + 2\xi=0$ in $\Omega$ and $\xi = 0$ along $\partial\Omega$, being $\Omega$ the complement of a disjoint and finite union of smooth and bounded simply connected regions in the two-sphere $\mathbb S^2$. Imposing that $|\nabla\xi|$ is locally constant along $\partial \Omega$ and that $\xi$ has infinitely many maximum points, we are able to classify positive solutions as the rotationally symmetric ones. As a consequence, we obtain a characterization of the critical catenoid as the only embedded free boundary minimal annulus in the unit ball whose support function has infinitely many critical points.

We prove new optimal symmetry results for solutions to the torsion problem on domains with disconnected boundaries. Time permitting, we discuss their relations with the uniqueness theorem for the Schwarzschild-de Sitter static black hole in general relativity. The results are obtained in collaboration with V. Agostiniani and S. Borghini.

In this talk, we will talk about a new kind of geometric inequalities in the Euclidean space, which are weighted Alexandrov-Fenchel type inequalities for star-shaped and $k$-convex hypersurfaces. As an application of the inequalities, we get a lower bound of the outer radius by the curvature integrals for star-shaped and $k$-convex hypersurfaces.

We show that there exists a quantity, depending only on $C^0$ data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the $C^0$ sense and has nonnegative scalar curvature in the sense of Ricci flow. Moreover, the $C^0$ mass at infinity is independent of choice of $C^0$-asymptotically flat coordinate chart, and the $C^0$ local mass has controlled distortion under Ricci-DeTurck flow when coupled with a suitably evolving test function.

Bartnik's quasilocal mass assigns to a given surface, coming with prescribed metric and mean curvature, the infimum of the ADM mass over a suitably restricted class of asymptotically flat extensions inducing the boundary data. I will present an asymptotic lower bound for the Bartnik mass of spheres with data close to that of the standard sphere in Euclidean space. The estimate relies on the construction of hyperannular fill-ins (spherical shells) which are approximately static vacuum.

In this talk, we present new monotone quantities and geometric inequalities associated with p-capacitary functions in asymptotically flat $3$-manifolds with simple topology and nonnegative scalar curvature. The inequalities become equalities on the spatial Schwarzschild manifolds outside rotationally symmetric spheres. This generalizes Miao's result from $p=2$ to $1 (Lecture Hall - Academic island（定山院士岛报告厅))