JHU-UMD Complex Geometry Seminar Archives for Academic Year 2019

Compactness of Kahler-Einstein manifolds of negative scalar curvature

When: Tue, October 1, 2019 - 4:30pm
Where: JHU - Maryland 201
Speaker: Jian Song (Rutgers) - https://sites.google.com/site/jiansongrutgers/home
Abstract: Abstract: Let K(n,V) be the set of n-dimensional Kahler-Einstein manifolds (X, g) satisfying Ric(g)=-g with volume bounded above by V. We show that any sequence (X_i, g_i) in K(n,V) converge, after passing to a subsequence, in pointed Gromov-Hausdorff toplogy, to a finite union of complete Kahler-Einstein metric spaces. The limiting metric space is biholomorphic to an n-dimensional semi-log canonical model with its non log terminal locus removed. Our result is a high dimensional generalization for the compactness of constant curvature metrics on Riemann surfaces of high genus. We will also give some applications to the Weil-Petersson metric on the moduli space of canonically polarized manifolds.

On global invariants of CR manifolds

When: Tue, October 8, 2019 - 4:30pm
Where: Kirwan Hall 3206
Speaker: Jeffrey Case (Penn State) - http://www.personal.psu.edu/jqc5026/
Abstract: A secondary global invariant of a CR manifold is the integral of a scalar pseudohermitian invariant which is independent of the choice of pseudo-Einstein contact form. All such invariants are biholomorphic invariants of domains C^n. One example is the Burns—Epstein invariant in C^2, which gives a nice characterization of the ball. In this talk I will describe two important families of secondary global invariants. The first, the total Q-prime curvatures, give a nice analytic interpretation of the Burns—Epstein invariant. The second, the total I-prime curvatures, show that the theory of secondary global invariants is much richer than the analogous theory of global conformal invariants, and in particular disproves a conjecture of Hirachi.

Ricci flow and contractibility of spaces of metrics

When: Tue, November 12, 2019 - 4:30pm
Where: JHU Maryland 201
Speaker: Bruce Kleiner (NYU) - https://math.nyu.edu/people/profiles/KLEINER_Bruce.html
Abstract: 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.

On the regularity of Ricci flows coming out of metric spaces

When: Tue, March 3, 2020 - 4:30pm
Where: JHU Ames 234
Speaker: Felix Schulze (University of Warwick) - https://www.felixschulze.eu
Abstract: We consider smooth, not necessarily complete, Ricci flows, (M,g(t))_{t \in (0,T)} with Ric(g(t))\geq-1 and |Rm(g(t))|\leq c/t for all t\in(0,T) coming out of metric spaces (M,d_0) in the sense that (M,d(g(t)),x_0)->(M,d_0,x_0) as t->0 in the pointed Gromov-Hausdorff sense. In the case that B_{g(t)}(x_0,1)\Subset M for all t \in (0,T) and d_0 is generated by a smooth Riemannian metric in distance coordinates, we show using Ricci-harmonic map heat flow, that there is a corresponding smooth solution \tilde{g}(t)_{t\in (0,T)} to the \delta-Ricci-DeTurck flow on an Euclidean ball B_r(p_0)\subset R^n, which can be extended to a smooth solution defined for t\in [0,T). We further show, that this implies that the original solution g can be extended to a smooth solution on B_{d_0}(x_0,r/2) for t \in [0,T), in view of the method of Hamilton. This is joint work with Alix Deruelle and Miles Simon.