Workshop on Ricci Curvature

Northwestern University
May 28-31, 2015

Program

Lectures to be held in Garrett Seminary 205 and Swift Hall 107, see schedule below. Click here for a campus map.

Abstracts:

Richard Bamler   (Berkeley). Title: On the scalar curvature blow up conjecture in Ricci flow

Abstract: It is a basic fact that the Riemannian curvature becomes unbounded at every finite-time singularity of the Ricci flow. Sesum showed that, more precisely, even the Ricci curvature becomes unbounded at every such singularity. Whether the same can be said about the scalar curvature has since remained a conjecture, which has resisted several attempts of resolution. In this talk, I will present new estimates for Ricci flows, which hold under a global or local scalar curvature bound, such as: distance distortion estimates, Gaussian bounds for the heat kernel and a backwards pseudolocality theorem. As an application, we partially confirm the scalar curvature blowup conjecture in dimension 4. This project is joint work with Qi Zhang.

Albert Chau   (UBC). Title: Long time existence for the Kähler Ricci flow on Cⁿ

Abstract: In the talk I will discuss the Kähler Ricci flow of a class of complete Kähler metrics on Cⁿ. In particular, the flow of unbounded curvature metrics, longtime existence and convergence of solutions will be discussed. Connections will be drawn to Yau's uniformization conjecture stating that a complete non-compact Kähler manifold with positive bisectional curvature is biholomorphic to Cⁿ. The talk will is based on joint work with Luen Fai Tam and Ka Fai Li.

Jeff Cheeger   (NYU). Title: Regularity of manifolds with bounded Ricci curvature and the codimension 4 conjecture

Abstract: This talk concerns joint work with Aaron Naber. We will indicate a proof of the conjecture that a noncollapsed Gromov-Hausdorff limit space of a sequence of manifolds Mⁿ_i with a uniform bound on Ricci curvature is smooth off a closed subset of Hasudorff (or Minkowski) codimension 4. We combine this result with quantitative stratification theory to prove local a priori L_q estimates on the full curvature tensor, for all q<2. In the case of Einstein manifolds, we improve this to estimates on the regularity scale. We also prove a conjecture of Anderson to the effect that the collection of 4-manifolds with |Ric_M⁴|≤ 3, Vol(M⁴) > v > 0 and diam(M⁴)< D, contains at most a finite number of diffeomorphism types. A local version of this is used to show that noncollapsed 4-manifolds with bounded Ricci curvature satisfy a priori L₂ Riemannian curvature estimates.

Robert Haslhofer   (NYU). Title: Weak solutions for the Ricci flow

Abstract: We introduce a new class of estimates for the Ricci flow, and use them both to characterize solutions of the Ricci flow and to provide a notion of weak solutions of the Ricci flow in the nonsmooth setting. Given a family (M,g_t) of Riemannian manifolds, we consider the path space of its space time. Our first characterization says that (M,g_t) evolves by Ricci flow if and only if a sharp infinite dimensional gradient estimate holds for all functions on path space. We prove additional characterizations in terms of the regularity of martingales on path space, as well as characterizations in terms of log-Sobolev and spectral gap inequalities for a family of Ornstein-Uhlenbeck type operators. Our estimates are infinite dimensional generalizations of much more elementary estimates for the linear heat equation on (M,g_t), which themselves generalize the Bakry-Emery-Ledoux estimates for spaces with lower Ricci curvature bounds. Based on our characterizations we can define a notion of weak solutions for the Ricci flow. This is joint work with Aaron Naber.

Hans-Joachim Hein   (Maryland). Title: Ricci-flat metrics on A_k singularities

Abstract: By work of Gauntlett-Martelli-Sparks-Yau, the 3-dimensional A_k singularity admits no Ricci-flat Kähler cone metrics for k > 2. We show that if k > 3, then A_k admits a Ricci-flat Kähler metric with an isolated point singularity whose Gromov-Hausdorff tangent cone has nonisolated singularities. These new metrics also give rise to counterexamples to various open questions in the geometry of Ricci curvature. Joint work with Aaron Naber.

Chi Li   (Stony Brook). Title: On rates and compactifications of Asymptotically Conical Calabi-Yau manifolds

Abstract: The rate of a Asymptotically Conical (AC) manifold measures how fast the AC manifold converges to its tangent cone at infinity. When the AC Calabi-Yau manifold comes from Tian-Yau's construction, I will relate the study of rate to the work of Grauert and Abate-Bracci-Toneva on how the divisor at infinity is embedded into the compactification. I will also show that the analytic compactification always exists in the quasi-regular case.

Gang Liu   (Berkeley). Title: Gromov-Haudorff convergence of Kähler manifolds and the finite generation conjecture

Abstract: We study the uniformization conjecture of Yau by using the Gromov-Haudorff convergence. As a consequence, we confirm Yau's finite generation conjecture. More precisely, on a complete noncompact Kähler manifold with nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated. We prove if M is a complete noncompact Kähler manifold with nonnegative bisectional curvature and maximal volume growth, then it is biholomorphic to an affine algebraic variety. We also confirm a conjecture of Ni on the existence of polynomial growth holomorphic functions on complete Kähler manifolds with nonnegative bisectional curvature.

John Lott   (Berkeley). Title: Ricci measure for some singular Riemannian metrics

Abstract: We define the Ricci curvature, as a measure, for certain singular torsion-free connections on the tangent bundle of a manifold. The definition uses an integral formula and vector-valued half-densities. We give relevant examples in which the Ricci measure can be computed. In the time dependent setting, we give a weak notion of a Ricci flow solution on a manifold.

Yanir Rubinstein   (Maryland). Title: The degenerate special Lagrangian equation

Abstract: In joint work with J. Solomon, we introduce the degenerate special Lagrangian equation and develop the basic analytic tools to construct and study its solutions. This equation governs geodesics in the space of positive Lagrangians. It has intriguing relations to uniqueness, existence and stability notions related to special Lagrangians, Lagrangian mean curvature flow, the topology of Hamiltonian isotopy classes, and the Arnold conjecture.

Natasa Sesum   (Rutgers). Title: Ancient solutions in curvature flows

Abstract: We will discuss ancient solutions in the context of the mean curvature flow, the Ricci flow and the Yamabe flow. In the first part of the talk we will mention some previous results on the classification of ancient solutions in the Ricci flow and the existence of infinitely many ancient solutions in the Yamabe flow. In the second part of the talk we will mention the most recent result about the unique asymptotics of non-collapsed ancient solutions to the mean curvature flow which is a joint work with Daskalopoulos and Angenent.

Christina Sormani   (CUNY GC and Lehman College). Title: Convergence of manifolds with boundary

Abstract: I will present joint work with my doctoral student, Raquel Perales, on the convergence of manifolds with boundary that have nonnegative Ricci curvature. At first I assume no regularity conditions on the boundary and describe the glued limits of such sequences (which may exist even when the sequence has no Gromov-Hausdorff limit). Then I will present new solo work of Perales concerning sequences of such manifolds with assumptions on the mean curvature of the boundary and other hypotheses.

Song Sun   (Stony Brook). Title: Gromov-Hausdorff limits of Kähler-Einstein manifolds

Abstract: A sequence of Riemannian manifolds with bounded Ricci curvature and non-collapsing volume has Gromov-Haudorff limits, which are in general only metric spaces. For projective manifolds endowed with Kähler metrics, we will show these limits are naturally projective varieties, and discuss further connections with algebraic geometry of the limits and their singularities. This talk is based on joint work with Simon Donaldson.

Gábor Székelyhidi   (Notre Dame). Title: Kähler-Einstein metrics along the smooth continuity method

Abstract: I will discuss an equivariant version of the Yau-Tian-Donaldson conjecture on the existence of Kähler-Einstein metrics, strengthening the result of Chen-Donaldson-Sun. This potentially gives new examples of Kähler-Einstein manifolds, and it can also be applied to the existence problem for Kähler-Ricci solitons. It is joint work with Ved Datar.

Jeff Viaclovsky   (Wisconsin). Title: Critical metrics on connected sums of Einstein four-manifolds

Abstract: I will discuss a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on CP², and the product metric on S²xS². Using these metrics in various gluing configurations, critical metrics are found on connected sums for a specific Riemannian functional, which depends on the global geometry of the factors. This is joint work with Matt Gursky. If time permits, I will also discuss some recent results regarding the asymptotic expansion of the Kuranishi map.

Bing Wang   (Wisconsin). Title: The Kähler-Ricci flow on Fano manifolds

Abstract: Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we prove the Hamilton-Tian conjecture and the partial C⁰ conjecture of Tian. In this talk, we will describe the framework of the solution, explain where are the technical difficulties and show how to overcome them. This is a joint work with X.X. Chen.

Mu-Tao Wang   (Columbia). Title: The spacetime Ricci curvature in Riemannian geometry

Abstract: Several fundamental results in Riemannian geometry such as the comparison theorem and the second variation formula rely on the sign of the Ricci curvature. However, many model spaces such as the Schwarzschild manifold (uniqueness of black holes) and the Bryant soliton (uniqueness of Ricci solitons) are not Einstein. On the other hand, the Schwarzschild manifold can be embedded as a time slice of a spacetime that has zero spacetime Ricci curvature. Such a spacetime point of view gives natural interpretations of previous results, helps in deriving brand new estimates and theorems, and inspires unexpected conjectures. I shall discuss several such examples in my talk.

Guofang Wei   (UCSB). Title: Analysis and geometry on manifolds with integral Ricci curvature lower bounds

Abstract: Recently Tian and Zhang showed that L⁴ norm of Ricci curvature is uniformly bounded under Kähler-Ricci flow. We will give a survey on some tools about analysis and geometry on manifolds with integral Ricci curvature bounds: Laplacian comparison, volume comparison, Sobolev inequality, Poincare inequality, maximal principle, gradient estimate.

Xiaokui Yang   (Northwestern). Title: The Kähler-Ricci flow, Ricci flat metrics and collapsing limits

Abstract: We discuss the Kähler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kähler metrics. This strengthens previous work of Song-Tian and others. We will also talk about some analogous results for degenerations of Ricci-flat Kähler metrics. This is joint work with Valentino Tosatti and Ben Weinkove.

Ruobing Zhang   (Princeton). Title: Quantitative nilpotent structure and epsilon-regularity on collapsed manifolds with Ricci curvature bounds

Abstract: In this talk we discuss the ε-regularity theorems for Einstein manifolds and more generally manifolds with just bounded Ricci curvature, in the collapsed setting. A key tool in the regularity theory of noncollapsed Einstein manifolds is the following: If a bigger geodesic ball on an Einstein manifold is sufficiently Gromov-Hausdorff-close to a ball on the Euclidean space of the same dimension, then in fact the curvature on a smaller ball is uniformly bounded. No such results are known in the collapsed setting, and in fact it is easy to see without more such results are false. It turns out that the failure of such an estimate is related to topology. Our main theorem is that for the above setting in the collapsed context, either the curvature is bounded, or the local nilpotent rank drops. There are generalizations of this result to bounded Ricci curvature and even just lower Ricci curvature. This is a joint work with Aaron Naber.