Schedule for: 20w5077 - New Directions in Geometric Flows (Cancelled)

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will discuss the Kahler Ricci flow on quasi-projective varieties. Analytically I will discuss a complex parabolic Monge Ampere equation on a compact complex manifold, in the presence of both singular and degenerate terms in the equation. These singularities and degeneracies will correspond to an associated projective variety with log canonical singularities. Connections will be made to earlier work on the flow for rough or degenerate initial data, the conical Kahler Ricci flow, and the flow of cusp like singularities.

I will present recent work (joint with Yong Wei, Changwei Xiong and Yitao Lei) on volume-preserving flows of convex Euclidean hypersurfaces by powers of elementary symmetric functions of curvatures, with a volume constraint. We use the machinery of curvature measures to prove asymptotic convergence to a sphere (or to a Wulff shape in anisotropic settings), and derive a delicate curvature estimate to prove smooth convergence.

We show that the mean curvature flow of generic closed surfaces in R^3 avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in R^4 is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons. This is joint work with Otis Chodosh, Kyeongsu Choi and Christos Mantoulidis.

We present a notion of super Ricci flows on finite weighted graphs using a dynamic version of geodesic convexity of the relative entropies with respect to discrete optimal transport metrics. We study the heat flow on varying finite weighted graph structures and give a characterization of super Ricci flows via gradient- and transport estimates for the heat flow.

In this talk, I will outline the proof of the high dimensional mean convex neighborhood theorem for mean curvature flow. I will focus on explaining why, in contrast with the two-dimensional case, smoothness does not follow easily from the asymptotic analysis. I will then outline how the proof can be closed via a variant of the moving plane method, through which smoothness is a conclusion rather than an assumption. This is based on a joint work with Kyeongsu Choi, Robert Haslhofer and Brian White

In this talk, I will discuss Hamilton-Tian's conjecture for KR flow on Fano manifolds. We will give a proof of the conjecture by using the local regularity of complex Monge-Ampere equation. Our proof is based on a recent result of Liu-Szekelyhidi for Tian's partial $C^0$-estimate for polarized Kaehler metrics with Ricci curvature bounded below.

We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from $C^0$ initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from $C^0$ initial data.

The Ricci flow of rotationally symmetric metrics has been a source of interesting dynamics for the flow that include the formation of slow blow-up degenerate neckpinch singularities and the forward continuation of the flow through neckpinch singularities. A natural next source of examples is then the Ricci flow of doubly-warped product metrics. This structure allows for a potentially larger collection of singularity models compared to the rotationally symmetric case. Indeed, formal matched asymptotic expansions suggest a non-generic set of initial metrics on a closed manifold form finite-time, type II singularities modeled on a Ricci-flat cone at parabolic scales. I will outline the formal matched asymptotics of this singularity formation and the topological argument used to prove existence of Ricci flow solutions with these asymptotics. Finally, we will discuss applications of these solutions to questions regarding the possible rates of singularity formation and the blow-up of scalar curvature.

A classical result in Monge-Ampere equation states that the paraboloids are the only convex entire solutions to $\det D^2 u = 1$. In this talk, we discuss our result on the generalization of this classical result to the case when n=2 and the equation’s right-hand side is $(1+|x|^2)^{\beta}$. This corresponds to the classification of 2-surfaces which are translators of powers of Gauss curvature flow. Our proof is based on a spectral analysis around a soliton, B. Andrews’ classification of shrinking solitons to powers of curve shortening flow, and Daskalopoulos-Savin’s result on the behavior of solutions to M-A equation with homogeneous right-hand sides. This is a joint work with Kyeongsu Choi and Soojung Kim.

We establish interior regularity for convex viscosity solutions of the special Lagrangian equation. Our result states that all such solutions are real analytic in the interior of the domain. This is joint work with Ravi Shankar and Yu Yuan.

Given a Lorentzian spacetime with a foliation by spatial hypersurfaces, the vacuum Einstein equations reduce to a flow. When the hypersurfaces have constant mean curvature, this is called the Einstein flow. I'll discuss two aspects of the flow. The first is about the initial behavior of a flow coming out of a crushing singularity. The second is about the long-time behavior of a flow in an expanding universe.

We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an n-dimensional projective manifold X with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time in a fixed domain containing a fibre of X over its canonical model Xcan. This assumption on the Ricci curvature always holds when the Kodaira dimension of X is n, n−1 or when the general fibre of X over its canonical model is a complex torus. In particular, the normalized Kahler-Ricci flow converges in Gromov-Hausdorff topolopy to its canonical model when X has Kodaira dimension 1 with KX being semi-ample and the general fibre of X over its canonical model being a complex torus. We also prove the Gromov-Hausdorff limit of collapsing Ricci-flat Kahler metrics on a holomorphically fibred Calabi-Yau manifold is unique and is homeomorphic to the metric completion of the corresponding twisted Kahler-Einstein metric on the regular part of its base.

It is known that given a cone there may be more than one self-expanders asymptotic to the cone. In this talk, we discuss some global features of space of asymptotically conical self-expanders. This is joint with J. Bernstein.

Non-collapsing plays a fundamental role in the analysis of mean curvature flow. In this talk, I will discuss how Brian White's measure theoretic approach can be generalized to yield the non-collapsing for mean curvature flow with free boundary, provided the barrier is mean convex. This is joint work with N. Edelen, R. Haslhofer and J. Zhu.

In the lecture I will discuss recent joint work with Richard Bamler, which uses Ricci flow through singularities to construct deformations of spaces of metrics on 3-manifolds. We show that the space of metrics with positive scalar on any 3-manifold is either contractible or empty; this extends earlier work by Fernando Marques, which proved path-connectedness. We also show that for any spherical space form M, the space of metrics with constant sectional curvature is contractible. This completes the proof of the Generalized Smale Conjecture, and gives a new proof of the original Smale Conjecture for $S^3$.

In order to analyze the heat flow with Neumann boundary conditions on (not necessarily convex) domains of metric measure spaces, we introduce synthetic Ricci bounds which also take care of the curvature of the boundary. In terms of these distributional valued Ricci bounds we prove associated gradient estimates for the heat flow with Neumann boundary conditions on domains of metric measure spaces which improve upon previous results — both in the case of non-convex domains and in the case of convex domains.

We define a Perelman like functional for any Asymptotically Locally Euclidean metric. Such an energy has been introduced by Haslhofer in the setting of Asymptotically Flat metrics with non-negative scalar curvature. Our main result is a Lojasiewicz inequality for this energy. Applications to dynamical stability and rigidity of integrable and stable ALE Ricci flat metric with non-negative scalar curvature will be given. This is joint work with Tristan Ozuch.

The Gibbons-Hawking ansatz is a powerful method for constructing a large family of hyperkaehler 4-manifolds (which are thus Ricci-flat), which appears in a variety of contexts in mathematics and theoretical physics. I will describe recent progress in understanding the theory of minimal surfaces and mean curvature flow in these 4-manifolds. In particular, I will explain a proof of a version of the Thomas-Yau Conjecture in Lagrangian mean curvature flow in this setting. This is joint work with G. Oliveira.

We extend the concept of singular Ricci flow to 3 dimensional complete noncompact manifold with possibly unbounded curvature. As an application, we show the existence of a smooth Ricci flow starting from a complete 3 dimensional manifold with non-negative Ricci curvature. This partially resolves a conjecture by Topping.

The lecture describes joint work with Simon Brendle on the evolution of hypersurfaces in a Riemannian manifold that have the sum of any two principal curvatures positive everywhere. The speed is give by the harmonic mean of all pairwise sums of principal curvatures, leading to a fully non-linear evolution system. Surprisingly it turns out that this flow has somewhat better properties than mean curvature flow in an ambient manifold, allowing for a complete classification of singularities and a surgery algorithm that eventually leads to a classification of all two-convex hypersurfaces in a natural class of Riemannian manifolds.