# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Kähler Ricci flow on Del Pezzo surfaces

##### A new example with positive sectional curvature

I will discuss the construction of a new example with positive sectional curvature on a 7-dimensional manifold homeomorphic to the unit tangent bundle of the 4-sphere. The metric is of Kaluza Klein type on an orbifold principle bundle over the 4-sphere and is closely related to the geometry of self dual Einstein and 3- Sasakian metrics.

##### The volume of a differentiable stack

We extend to the setting of Lie groupoids the notion of the cardinality of a finite groupoid (a rational number, equal to the Euler characteristic of the correspondingdiscrete orbifold). Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associatedstack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a naturalline bundle over the stack. Sections of a square root of this line bundle constitute an "intrinsic Hilbert space'' of the stack. The talk will not require prior knowledge of groupoids or stacks.

##### The space of positive scalar curvature metrics on the three-sphere

In this talk we will discuss a proof of the path-connectedness of the space of positive scalar curvature metrics on the three-sphere. The proof uses the Ricci flow with surgery and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundamental.

##### The singular set of $C^{1}$ smooth surfaces in the Heisenberg group

##### Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space

Complete Riemannian manifolds with nonnegative Ricci curvature have been well studied. Riemannian manifolds with a negative lower bound for Ricci curvature are considerably more complicated and less understood. I will first survey some recent results on such manifolds with positive bottom of spectrum. Then I will discuss a rigidity theorem which characterizes hyperbolic manifolds. The proof uses idea from potential theory and Brownian motion on Riemannian manifolds

##### An Exotic Sphere with Positive Sectional Curvature

I'll discuss joint work with Peter Petersen that shows that the Gromoll-Meyer exotic 7-sphere admits a metric of positive sectional curvature. I'll discuss the history of the problem and give a coarse outline of the proof.

##### Harmonic maps between singular spaces

We will discuss regularity questions of harmonic maps from a simplicial complex to metric spaces of non-positive curvature. We will also discuss the relation with rigidity questions of group actions on these spaces.

##### Ends of locally symmetric spaces

We intend to explain joint work with Lizhen Ji and Peter Li on relating the size of the bottom spectrum to the number of ends for locally symmetric spaces.

##### Hyperdiscriminant polytopes, Chow Polytopes, and K-energy asymptotics

Let (X,L) be a polarized algebraic manifold. I have recently proved that the Mabuchi energy of (X,L) is bounded from below along any degeneration if and only if the Hyperdiscriminant polytope contains the Chow polytope (with respect to the various Kodaira embeddings). This completes the analysis initiated by Ding and Tian in their 1992 Inventiones paper "Kahler Einstein metrics and the Generalized Futaki Invariant," and therefore gives the final form to Tian's concept of K-semistability.

##### Compactness Properties of the Space of Genus-$g$ Helicoids

I will discuss a recent application of the work of Colding and Minicozzi of structure of embedded minimal surfaces in $\Real3$ to the study of compactness properties of the space of genus-$g$ helicoids. I will introduce the theory of Colding and Minicozzi and then show how it can be used to show (among other results) that the space of genus-one helicoids is compact (modulo symmetries). (Joint work with C. Breiner)

##### Geometric flows with rough initial data

In a recent joint work with Herbert Koch (University of Bonn) we showed the existence of a global unique and analytic solution for the mean curvature flow (in arbitrary codimensions) and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. In this talk I will explain our construction and, if time permits, I will show how similar constructions can be used to obtain the existence of a global unique and analytic solution of the Ricci-DeTurck flow on euclidean space for bounded initial metrics which are close to the euclidean metric in $L^\infty$ and of the harmonic map flow for initial maps whose image is contained in a small geodesic ball.

##### Ricci flow on ALE spaces

##### Greatest lower bounds on the Ricci curvature of Fano manifolds

On Fano manifolds we study the supremum of the possible t such that there exists a metric in the first Chern class with Ricci curvature bounded below by t. For the projective plane blown up in one point we show that this supremum is 6/7.

##### The Einstein-Weyl Equations, Scattering Maps, and Holomorphic Disks

This talk will show that conformally compact, globally hyperbolic, Lorentzian Einstein-Weyl 3-manifolds are in natural one-to-one correspondence with orientation-reversing diffeomorphisms of the 2-sphere. The proof hinges on a holomorphic-disk analog of Hitchin's mini-twistor correspondence.

##### On the structure of Lagrangian submanifolds

This is a report on a recent joint project with Lars Schaefer. We derive results related to the minimality of Lagrangian submanifolds. In particular, these apply to Lagrangian 3-folds and to Lagrangian submanifolds in twistor spaces over quaternionic Kaehler manifolds. We then use a splitting theorem to give a better description in dimensions four and five.

##### 11-dimensional supergravity and Dirichlet problem for forms on asymptotically hyperbolic spaces

##### A Second Boundary Value Problem for special Lagrangian submanifolds

Given any two uniformly convex regions in Euclidean space, we show that there exists a unique diffeomorphism between them, such that the graph of the diffeomorphism is a special Lagrangian submanifold in the product space. This is joint work with Simon Brendle.

##### Lagrangian Mean Curvature flow for entire Lipschitz graphs

We prove existence of longtime smooth solutions to mean curvature flow of entire Lipschitz Lagrangian graphs. As an application of the estimates for the solution, we establish a Bernstein type result for translating solitons. The results are from joint work with Albert Chau and Weiyong He.

##### Ancient solutions to the curve shortening flow and the Ricci flow in 2 dimensions

I will give a classification of ancient convex closed embedded ancient solutions to the curve shortening flow and the ancient solutions to the Ricci flow on surfaces. This is a joint work with Daskalopoulos and Hamilton.