# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### There are finitely many surgeries in Perelman's Ricci flow

Although the Ricci flow with surgery has been used by Perelman to solve the Poincaré and Geometrization Conjectures, some of its basic properties are still unknown. For example it has been an open question whether the surgeries eventually stop to occur (i.e. whether there are finitely many surgeries) and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$. In this talk I will show that the number of surgeries is indeed finite and that the curvature is globally bounded by $C t^{-1}$ for large $t$.

##### Optimal transport and regularity of c-convex potentials

The question of regularity in optimal transport and related equations of Monge Ampere type has seen a lot of activity in the past few decades. Starting from the usual quadratic cost in R^n and now ranging

arbitrary costs in Riemannian manifolds (and the related reflector antenna problems). In this talk, we will give an impressionistic description of Caffarelli's regularity theory for the Monge Ampere equation in

Euclidean space, which strongly uses the affine invariance of the equation. We will see when and how such a theory can be pushed to general costs,. The new observation is that in general regularity

arises not so much from affine invariance, but rather from two opposite inequalities for the Mahler volume of c-convex sets (a kind of generalized Blaschke-Santaló inequalities). The validity of such

##### PDEs of Monge-Ampere type

A considerable amount of research activity in recent years has been devoted to the study of nonlinear partial differential equations of Monge-Ampere type (MATEs) in connection with their applications to conformal geometry, optimal transportation and geometric optics. In this talk we will discuss the underlying structural condition found by Ma, Wang and myself and present a selection of recent results motivated by

the applications.

##### Witten spinors on nonspin manifolds

Unlike a 3-dimensional manifold, a higher dimensional manifold need not be spin. On an oriented Riemannian manifold the obstruction to having a spin structure is given by the second Stiefel-Whitney class. I will show that even when this obstruction does not vanish, it is still possible to define a notion of singular spin structure and associated singular Dirac operator. Then, modeling on Witten's proof of the Positive Mass Theorem, I will define the notion of Witten spinor on an asymptotically flat nonspin manifold, show their existence and describe their properties.

##### Critical metrics on connected sums of Einstein four-manifolds

THIS IS A JOINT SEMINAR WITH DIFFERENTIAL GEOMETRY & GEOMETRIC ANALYSIS and JOINT PRINCETON-RUTGERS GEOMETRIC PDEs. PLEASE NOTE DIFFERENT LOCATION. 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 CP^2, and the product metric on S^2 x S^2. 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.** **

##### Yamabe flow, its singularity profiles and ancient solutions

THIS IS A JOINT SEMINAR WITH DIFFERENTIAL GEOMETRY & GEOMETRIC ANALYSIS and JOINT PRINCETON-RUTGERS GEOMETRIC PDEs. PLESE NOTE DIFFERENT LOCATION AND TIME. We will discuss conformally flat complete Yamabe flow and show that in some cases we can give the precise description of singularity profiles close to the extinction time of the solution. We will also talk about a construction

of new compact ancient solutions to the Yamabe flow. This is a joint work with Daskalopoulos, King and Manuel del Pino.** **

##### Ricci flow on quasiprojective manifolds

PLEASE NOTE SPECIAL DAY (WEDNESDAY). The talk is about Ricci flow on noncompact Kaehler manifolds. I'll discuss four types of spatial asymptotics: cuspidal, cylindrical, bulging and conical. The results are about the preservation of the asymptotics, long-time existence, parabolic blowdown limits and the role of the Kaehler-Ricci flow on the divisor. This is joint work with Zhou Zhang

##### Isotropic curvature, macroscopic dimension Filling radius of contractible loops and fundamental group

I will discuss the proof of a conjecture of Gromov's to the effect that manifolds with uniformly positive isotropic curvature (and bounded geometry) are macroscopically 1-dimensional on the scale of the isotropic curvature. One of the main techniques involved is modeled on Donaldson's version of H\"ormander technique to produce (almost) holomorphic sections. We use this to produce sections of the restriction of the complexified tangent bundle of M to a stable embedded minimal disk which destabilize the disk if the distance grows too much. As a consequence we prove that compact manifolds with positive isotropic

curvature have virtually free fundamental groups.

##### Geometric view of conformal PDEs

**THIS SEMINAR HAS BEEN MOVED FROM OCTOBER 25 TO OCTOBER 18. PLEASE NOTE SPECIAL TIME.** In this talk we develop a global correspondence between immersed horospherically convex hypersurfaces $\phi: {\rm M}^n\to H^{n+1}$ and complete conformal metrics $e^{2\rho}g_{S^n}$ on domains $\Omega$ in the boundary $S^n$ at infinity of $H^{n+1}$ such that $\rho$ is the horospherical support function and that $\partial_\infty\phi({\rm M}^n) = \partial\Omega$. For instance, we are able to obtain an explicit correspondence between Obata's Theorem (for conformal metrics) and Alexandrov Theorem (for hypersurfaces). It time permits, we obtain Bernstein and Delaunay theorems for a properly immersed, horospherically convex hypersurface in $H^{n+1}$.

##### Singular perturbation of minimal surfaces

(w/ N. Kapouleas and N.M. M\o ller) I discuss recent work in which we use singular perturbation techniques to show that the space of complete embedded minimal surfaces with four ends and genus $k$ ($\mathcal{M}(k,4)$) is non-empty and non-compact for large $k$.

##### Mean curvature flow of mean convex hypersurfaces

In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow. In the present talk, we explain a new treatment of the theory of mean convex (and k-convex) flows. This includes: (1) an estimate for derivatives of curvatures, (2) a convexity estimate, (3) a cylindrical estimate, (4) a global convergence theorem, (5) a structure theorem for ancient solutions, and (6) a partial regularity theorem. Our new proofs are both more elementary and substantially shorter than the original arguments. Our estimates are local and universal.

##### Singularities of the L^2 curvature flow

The L2 norm of the Riemannian curvature tensor is a natural energy to associate to a Riemannian manifold, especially in dimension 4. A natural path for understanding the structure of this functional and its

minimizers is via its gradient flow, the "L2 flow." This is a quasi-linear fourth order parabolic equation for a Riemannian metric, which one might hope shares behavior in common with the Yang-Mills flow.

We verify this idea by exhibiting structural results for finite time singularities of this flow resembling results on Yang-Mills flow. We also exhibit a new short-time existence statement for the flow exhibiting a

##### Morse index and uniqueness results for free boundary problems

##### Connected sum construction of constant Q-curvature manifolds in higher dimensions

**This is a special talk in addition to the 3:00 pm talk on the same date. **In geometric analysis, gluing constructions are well-known methods to create new solutions to nonlinear PDEs from existing ones. For a compact Riemannian manifold (M, g) of dimension n at least 6 with constant Q-curvature and satisfying a nondegeneracy condition, we show that one can construct many other examples of constant Q-curvature manifolds by a gluing construction. In particular, we prove the existence of solutions of a fourth-order PDE, which implies the existence of a smooth metric with constant Q-curvature on the connected sum. In this talk, I

will begin with denitions of Q-curvature and some background, and then give an overview of the gluing procedure.

##### Uniformity of harmonic map heat flow at infinite time

We will discuss an energy convexity along the harmonic map heat flow with small initial energy and fixed boundary data on the unit 2-disk. In particular, this shows that such weak harmonic map heat flow converges uniformly in time strongly in the W^{1,2}-topology, as time goes to infinity, to the unique limiting harmonic map.

##### H\"older estimates for fully nonlinear parabolic integro-differential equations

**Please note special day, time and location. **We will revisit H\"older estimates for some non local problems we worked recently with G. D\'avila. They arise in stochastic optimal control

driven by purely jump processes. Each one of them can be considered as an extension of the Krylov-Safanov regularity theory for fully non linear second order equations. We will start by motivating these equations, then we will see how the proofs work for simpler models and finally discuss how those ideas can be adapted to the nonlocal setting. ** **

##### On the singularities of the Szego projections on CR manifolds

In this talk, I will report the first part of my paper(Projections in several complex variables, Mem. Soc. Math. France, 2010, 131 p.). In this work, we completely study the heat equation method of Menikoff-Sjostrand and apply it to the Kohn Laplacian defined on a compact orientable connected CR manifold. We then get the full asymptotic expansion of the Szego projection for (0, q) forms when the Levi form is non-degenerate. This generalizes a result of Boutet de Monvel and Sjostrand for (0,0) forms.

##### Differential forms in Heisenberg groups and div-curl systems

In this talk we present a result proved in collaboration with Annalisa Baldi. We prove a family of inequalities for differential forms in Heisenberg groups (Rumin’s complex), that are the natural counterpart of a class of div-curl inequalities in de Rham’s complex proved by Lanzani & Stein, Bourgain & Brezis.

##### The log zoo

The compact four-manifolds that admit a Kahler metric with positive Ricci curvature have been classified in the 19th century: they come in 10 families. In analogy with conical Riemann surfaces (e.g., football, teardrop) and hyperbolic 3-folds with a cone singularity along a link appearing in Thurston's program, one may consider 4-folds with a Kahler metric having "edge singularities", namely admitting a 2-dimensional cone singularity transverse to an immersed minimal surface, a `complex edge'. What are all the pairs (4-fold, immersed surface) that admit a Kahler metric with positive Ricci curvature away from the edge? In joint work with I. Cheltsov we classify all such pairs under some assumptions.

##### A proof of the Alexanderov's Uniqueness Theorem for Convex Surfaces in R3

**PLEASE NOTE SPECIAL DAY (WEDNESDAY). CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT. **A classical uniqueness theorem of Alexandrov says that: if M and M_0 are two closed strictly convex C^2 surface in R^3 and satisfy f(a,b) = f(c,d), at points of M, M_0 with parallel normals, for some C^1 function f(x,y) with \partial_{x}f\partial_{y}f>0, then M is equal to M_0 up to a translation. We will talk about a new PDE proof for this theorem by using the maximal principle and weak uniqueness continuation theorem of Bers-Nirenberg. More generally, we prove a version of this theorem with the minimal regularity assumption: the spherical hessians of the supporting functions for the corresponding convex bodies as Radon measures are nonsingular. This is a joint work with P. Guan and Z. Wang.** **