Schedule for: 23w5072 - A unified view of Quasi-Einstein Manifolds

We will report on recent progress concerning the classification of homogeneous Einstein manifolds. In joint work with R. Lafuente we showed that a homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to Euclidean space, confirming the Alexseevskii conjecture from 1975. A Ricci flat homogeneous space is flat. The classification of homogeneous Einstein manifolds with positive scalar curvature is wide open, even though there exist several structure results concerning existence and non-existence.

We will talk about the geometry of four-dimensional gradient shrinking Ricci solitons. Initially, we will present some examples and classical results on gradient Ricci solitons in order to justify our special attention to dimension four. Next, we will present some recent classification results for four-dimensional gradient shrinking Ricci solitons. Moreover, we will discuss some open problems.

The fundamental gap is the difference between the first two eigenvalues of the Dirichlet problem for the Laplace operator. We will give a brief history of the problem, state the main conjecture, and give a survey of techniques and recent results for the subject.

Extremal black holes possess a well defined notion of a near-horizon geometry which describes the intrinsic geometry of the horizon. The spacetime Einstein equations imply that such near-horizon geometries correspond to a class of Riemannian quasi-Einstein metrics. As such, near-horizon geometries can be studied and classified independently of any parent black hole spacetime, and hence provide a useful arena to study the geometry and topology of black holes in a purely Riemannian setting. I will give an overview of the current state of research on near-horizon geometries in a variety of dimensions and theories, focusing on general constraints on their topology and symmetry, their explicit classification, and highlight open problems in this area. I will also describe a number of recent applications of near-horizon geometries to the classification programme for extremal black holes.

While Einstein's theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while and Ricci curvature is reformulated using entropic convexity along geodesics of probability measures. This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. We begin with a simplified approach to Kunzinger and Saemann's theory of (globally hyperboloid, regularly localizable) Lorentzian length spaces in which the time-separation function takes center stage. We show compatibility of two different notions of timelike geodesic used in the literature. We then propose a synthetic (i.e. nonsmooth) reformulation of the null energy condition by relating to the timelike curvature-dimension conditions of Cavalletti & Mondino (and Braun), and discuss its consistency and stability properties.

We have asked all junior participants to prepare a short 3 minute introduction with 3-5 slides to introduce themselves to everyone. If you have registered for the workshop recently and would like to present, please contact Eric Bahuaud.

We computed the \(L^p\) spectrum of Laplacians on \(k\)-forms on hyperbolic spaces. Moreover, we proved the \(L^p\) boundedness of certain resolvent of Laplacians by assuming the Ricci lower bound and manifold volume growth. This generalized a result of M. Taylor, in which bounded geometry of the manifold is assumed. This is a joint work of N. Charalambous.

I will present a characterization of the null energy condition for an (n+1)-dimensional, time-oriented Lorentzian manifold in terms of convexity of the relative (n-1)-Renyi entropy along null displacement interpolations on null hypersurfaces. More generally, we consider a Lorentzian manifold with a weight function and introduce the Bakry-Emery N-null energy condition that we characterize in terms of null displacement convexity of the relative N-Renyi entropy. As applications we prove a version of Hawking’s area monotonicity theorem for a black hole horizon and a Penrose singularity theorem in the context of weighted Lorentzian manifolds.

For \(n\)-dimensional Riemannian manifolds with Ricci curvature bounded below by \(-(n-1)\), the volume entropy is bounded above by \(n-1\). If \(M\) is compact, it is known that the equality holds if and only if \(M\) is hyperbolic. I will present a similar result for RCD spaces. Joint work with C. Connell, X. Dai, J. Nunez-Zimbrón, P. Suárez-Serrato and G. Wei.

I will report on joint work with Nelia Charalambous. Conformally compact manifolds are a class of non-compact manifolds with variable curvature that were introduced by Fefferman and Graham to study conformal invariants. They are a broad class and include many examples, including manifolds with Poincaré-Einstein metrics. Motivation to study the Laplace operator acting on \(L^p\) for \(p\) not equal to \(2\) comes from physics. For example, the most natural space on which to study heat diffusion is \(L^1\). Here, we show that for general values of \(p\), the \(L^p\) Laplace spectrum contains a certain parabolic region and is contained in a certain parabolic region. These regions are determined by the geometry of the conformally compact manifold.

A well-known sharp estimate of Cheng compares the bottom spectrum of a complete Riemannian manifold with Ricci curvature bounded from below by a negative constant with the counterpart bottom spectrum on a space form of constant negative curvature. The equality case in this estimate is understood in some cases, although not that much in general. We generalize this theory for manifolds admitting smooth densities, in particular for Ricci solitons, and in dimension three for manifolds with scalar curvature bounded from below.

The fundamental gap is the difference of the first two eigenvalues of the Laplace operator, which is important both in mathematics and physics and has been extensively studied. For the Dirichlet boundary condition the log-concavity estimate of the first eigenfunction plays a crucial role, which was established for convex domains in the Euclidean space and the round sphere. Joint with G. Khan, H. Nguyen, and G. Wei, we obtain log-concavity estimates of the first eigenfunction for convex domains in surfaces of positive curvature and consequently establish fundamental gap estimates. In a subsequent work, together with G. Kahn and G. Wei, we improve the log-concavity estimates and obtain stronger gap estimates which recover known results on round spheres.

We present simple conditions which ensure that an elliptic operator \(L\) generates an analytic semigroup on any complete manifold of bounded geometry. As applications we prove well-posedness of the Bach flow, and long-time continuous dependence of the Ricci flow, on manifolds of bounded geometry.

We define the embedding from the bound domain Ω in Cn into the collection of probability measures on Ω as a statistical manifold in the way that Fisher-Information metric restricted to Ω becomes the Bergman metric. We also show interesting results that could be proved under this framework. This is the Joint work with Jihun Yum.

This talk will be a review of the theory of weighted Lorentz-Finsler manifolds. A Lorentz-Finsler manifold is a generalization of a Lorentzian manifold in the same way that a Finsler manifold generalizes a Riemannian manifold. One can further equip a Lorentz-Finsler manifold with a time orientation as well as a weight, then we have a weighted Finsler spacetime. In this general framework, we can successfully develop the theory of Ricci curvature (singularity theorems, various comparison theorems, etc.). This is joint work (partly in progress) with Mathias Braun (Toronto), Yufeng Lu (Hong Kong), Ettore Minguzzi (Firenze).

On asymptotically hyperbolic manifolds, we consider a particular linear combination of the renormalized volume and the boundary integral for the usual ADM mass, which we call the volume-renormalized mass. It is well-defined and diffeomorphism invariant under weaker falloff conditions for the metric at infinity than one needs to define the renormalized volume and the hyperbolic ADM mass separately. We use the volume-renormalized mass to define a variant of the expander entropy on asymptotically hyperbolic manifolds which is monotonically increasing under the Ricci flow. Finally we use the expander entropy to prove a local positive mass theorem for Poincaré-Einstein metrics. This is joint work with Mattias Dahl and Stephen McCormick.

The famous Positive Mass Theorem of Schoen-Yau (and Witten) states that an asymptotically Euclidean manifold with nonnegative scalar curvature must have nonnegative ADM mass. Moreover, one has the rigidity statement that the mass is zero iff the manifold is the Euclidean space. Recently there has been a lot of interest and activity in Positive Mass Theorems for spaces which are asymptotically approaching an Euclidean space times a compact manifold, or more generally a fibration. We will discuss some of the work in this direction going back to our earlier work as well as some of our recent work and related work by Chen-Liu-Shi-Zhu and others. In particular there is a variant of the ADM mass introduced by Minerbe which plays an essential role in proving the rigidity statement here.

I will talk about recent works on regularity, compactness and uniqueness of asymptotically hyperbolic Einstein manifolds.

The quasi-Einstein equations with \(m = 2\) arise naturally within the study of the horizon geometries of `degenerate' black holes in general relativity. There has been considerable progress in constructing explicit solutions in the presence of symmetries. I will review these constructions and discuss the prospects of extending them for general m

We study uniqueness of the near horizon geometry that arises from degenerate AdS Kerr black holes in a neighbourhood of nearby near horizon geometries. The result is relevant in answering whether the only solutions to the near horizon geometry equations on a 2-sphere with negative cosmological constant arise from extremal AdS Kerr black holes.

In this talk I’ll survey some rigidity results for gradient solitons to geometric flows and the Quasi Einstein equation on homogenous spaces. The approach is to characterize when the hessian of a function can solve two general classes of equations involving a two-tensor that is invariant under a transitive group of isometries. This is joint work with Peter Petersen of UCLA.

In dimensions four and higher, the Ricci flow may encounter singularities modelled on cones with nonnegative scalar curvature. It may be possible to resolve such singularities and continue the flow using expanding Ricci solitons asymptotic to these cones, if they exist. I will discuss joint work with Richard Bamler in which we develop a degree theory for four-dimensional asymptotically conical expanding Ricci solitons, which in particular implies the existence of expanders asymptotic to a large class of cones.

In this talk I will describe two recent results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: they are hyperplane in R^4 while they do not exist in some positively curved closed Riemannian (n+1)--manifold when n ≤ 5. The first result was recently proved also by Chodosh and Li, and the second is a consequence of a more general result concerning minimal surfaces with finite index. Both theorems rely on a conformal method, inspired by a classical paper of Fischer-Colbrie. I will also present an application of these techniques to the study of critical metrics of a quadratic curvature functional. This is a joint work with P. Mastrolia (Università degli Studi di Milano) and A. Roncoroni (Politecnico di Milano).

The weighted Ricci curvature is one of the central objects in the study of smooth metric measure spaces. Recently, Lu-Minguzzi-Ohta have suggested a new approach that enables us to investigate the so-called curvature-dimension condition and the Wylie-Yeroshkin curvature condition for weighted affine connections in a unified way. In this talk, I will present various comparison geometric results under their curvature bound, and a characterization result via optimal transport. This talk is based on a joint work with Kazuhiro Kuwae (Fukuoka University).