## Geometric Analysis Seminar

Organizers: Pengfei Guan and Jerome Vetois. Seminars are usually held on Wednesdays, 13:30-14:30, in Burnside Hall Room 920.

## 2015--2016

Wednesday, September 9, 13:30-14:30, Burnside 920
M. Moller (Princeton and ICTP, Trieste)
Gluing of Solutions to Nonlinear PDEs of Mean Curvature Type

Wednesday, September 16, 13:30-14:30, Burnside 920
Guohuan Qiu (McGill and University of Science and Technology of China)
Title: The Neumann problem for Hessian equations.
Abstract: In this seminar, we consider the existence of the Neumann problem for Hessian equations. We solved this problem which is also a conjecture raised by N.S. Trudinger in 1986. First, we will briefly review the history of Neumann problem and the standard steps about solving this nonlinear problems using continuity method. Then we will discuss main obstacles to solving Neumann problems of fully nonlinear problems before. Last I will talk about details of our method to overcome these difficulties which mainly lies in C^2 estimates. (This is joint work with my advisor Xinan Ma in USTC.)

Wednesday, September 23, 13:30-14:30, Burnside 920
Chao Xia (McGill University)
Title: Inverse anisotropic curvature flow and Minkowski's inequality.
Abstract: In this talk, I will first discuss the inverse anisotropic mean curvature flow from a star-shaped hypersurface and show such flow exists for long time and converges to a rescale Wulff shape. Next I will discuss general inverse anisotropic curvature flows from convex hypersurfaces. As an application,we prove Minkowski's inequality for mixed volumes.

Wednesday, September 30, 13:30-14:30, Burnside 920
Jerome Vetois (McGill University)
Title: Blow-up phenomena for linear perturbations of the Yamabe equation.
Abstract: We will consider the question of the stability of solutions to the Yamabe equation under the effect of linear perturbations. We will see that the linear potential of the Yamabe equation plays a critical role for this question in dimensions four and above. I will present results which show the existence of blow-up phenomena for this equation, looking at different explosion profiles (one-peak solutions, multi-peak solutions, isolated and non-isolated explosion points).

Monday, October 5, 13:30-14:30, Burnside 920 (Joint seminar in Analysis and Geometric Analysis)
Biao Ou (University of Toledo)
Title: An equality for the geodesic curvature of certain curves on a two-dimensional Riemann surface.
Abstract: We prove an equality for the geodesic curvature function of certain closed curves in a local domain of a two-dimensional Riemannian surface. We address its connection to the local Gauss-Bonnet theorem. We also show that the equality leads to a four-vertex theorem for simple and closed curves on a two-dimensional Riemannian surface with a constant Gauss curvature.

Wednesday, October 28, 13:30-14:30, Burnside 920
Pengfei Guan (McGill)
Title: Flow by powers of the Gauss curvature.
Abstract: We discuss a joint work with Ben Andrews and Lei Ni on the Gauss curvature flow by powers. We prove that convex hypersurfaces in ${\mathbb R}^{n+1}$ contracting under the flow by any power $\alpha>\frac{1}{n+2}$ of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the initial body we prove that the limit is the round sphere.

Wednesday, November 11, 13:30-14:30, Burnside 920
Shaodong Wang (McGill)
Title: Infinitely many solutions for the Schrodinger equations in R^n with critical growth
Abstract

Wednesday, November 18, 13:30-14:30, Burnside 920
Shaodong Wang (McGill)
Title: Infinitely many solutions for the Schrodinger equations in R^n with critical growth (part II)
Abstract

Wednesday, Nov. 25, 13:30-14:30, Burnside 920
Teng Fei (MIT)
Title: Some new solutions to the Strominger system.
Abstract: The Strominger system is a system of PDEs derived by Strominger in his study of compactification of heterotic strings with torsion. It can be thought of as a generalization of Ricci-flat metrics on non-Kahler Calabi-Yau 3-folds. We present some new solutions to the Strominger system on a class of noncompact Calabi-Yau 3-folds constructed by twistor technique. These manifolds include the resolved conifold Tot(O(-1,-1)->P^1) as a special case.

Wednesday, December 2, 13:30-14:30, Burnside 920
Pierre-Damien Thizy (University of Cergy-Pontoise)
Title: Schrodinger-Poisson systems in closed manifolds.
Abstract: The Schrodinger-Poisson system we investigate in this talk arises when we look for standing waves solutions of the full (time dependent) Schrodinger-Maxwell system in the electrostatic case. After a short introduction, we will give recent stability/instability results of the set of the solutions to this system with respect to small perturbations of the "phase" (temporal frequency). In the process of this talk, we will make the connection of these results with some existence, nonexistence, uniqueness and multiplicity results concerning the positive solutions to this system.

Wednesday, December 9, 13:30-14:30, Burnside 920
Junfang Li (University of Alabama)
Title: An integral formula with geometric applications in Riemannian and Pseudo-Riemannian manifolds.
Abstract: In this talk, we will present a recent joint work with Chao Xia. We first prove a general integral formula for bounded domains in Riemannian manifolds. This formula includes Reilly's integral formula and the recent work of Qiu-Xia as special cases. In the second part of the talk, we will apply this formula to prove 1) Heitz-Karcher type inequalities, 2) Minkowski inequality, 3) two almost Schur type of Theorems. All these geometric inequalities hold for the so-called substatic Riemannian manifolds which consists of a large family Riemannian manifolds including all the space forms. We note that Heitze-Karcher inequality naturally leads to an Alexandrov rigity theorem for substatic warped product spaces. Thus we recovered S. Brendle's recent work by a completely different approach. The results in this talk are focused on Riemannian manifolds, however it has deep roots from Pseudo-Riemannian spaces.

Wednesday, January 27, 13:30-14:30, Burnside 920
Siyuan LU (McGill)
Title: Interior C^2 estimate of Monge-Ampere equation in 2-dimension.
Abstract: Heinz's work and related development.

Wednesday, February 3, 13:30-14:30, Burnside 920
Guohuan Qiu (University of Science and Technology of China & McGill)
Title: On degenerate case of prescribed curvature measure problems.
Abstract: We discuss a problem that curvature measure of radial compact C^1,1 Hypersurface is prescribed by a given nonnegative smooth function f on sphere. And we mainly focus on the regularity issue about this problem. This is well studied problems when the given function is positive. But even in prescribed mean curvature problem, no regularity results can permit f to be zero anywhere to our knowledge. While its nature to consider the case when f can touch zero somewhere on sphere, and it will cause degeneracy in the fully nonlinear problems. In this talk, We will talk about our recent development on this topic.

Wednesday, February 10, 13:30-14:30, Burnside 920
Siyuan LU (McGill)
Title: Interior C^2 estimate of Monge-Ampere equation in 2-dimension II.
Abstract: Continuation of the discussion of Heinz's work and related development.

Wednesday, February 17, 13:30-14:30, Burnside 920
Frederic Robert (Universite de Lorraine, France)
Title: Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing nonlinearity
Abstract: We consider a positive solution to a nonlinear elliptic equation on a punctured ball. The linear part is the classical Laplacian. When the nonlinear part is positive and critical, this is similar to the classical problem studied by Caffarelli-Gidas-Spruck. When the nonlinear part is negative and a pure power, the problem is associated to a natural convex functional and the singularities are completely understood. In the present work, we mix the two nonlinearities. We show the existence of several potential behaviors. Two of them are natural extensions of the case of constant-sign nonlinearity. Two other behaviors are arising from the interaction of the two nonlinearity. In this talk, I will describe all the possible behaviors and I will show how the methods of apriori analysis in nonlinear elliptic problems are helping understanding this problem. This is joint work with Florica Cirstea (Sydney)

Wednesday, March 9, 13:30-14:30, Burnside 920
Siyuan Lu (McGill)
Title: Curvature estimates for embedded surfaces into Riemannian manifolds
Abstract: We discuss the a priori mean curvature estimate for embedded surfaces in Riemannian manifold. By a modified Heinz's interior C^2 estimate and a Shi-Tam inequality, we prove that the mean curvature of strictly convex surface is bounded. This bound is independent of the structure of the ambient space and the location of the surface, which is main contribution of the work.

Wednesday, April 6, 13:30-14:30, Burnside 920
Saikat Mazumdar (Universite de Lorraine, France)
Title: HIGHER ORDER ELLIPTIC PROBLEMS WITH CRITICAL SOBOLEV GROWTH ON A COMPACT RIEMANNIAN MANIFOLD: BEST CONSTANTS AND EXISTENCE
Abstract: We investigate the existence of solutions to a nonlinear elliptic problem involving the critical Sobolev exponent for a Polyharmomic operator on a Riemannian manifold M. We first show that the best constant of the Sobolev embedding on a manifold can be chosen as close as one wants to the Euclidean one, and as a consequence derive the existence of minimizers when the energy functional goes below a quantified threshold. Next, higher energy solutions are obtained by Coron’s topological method, provided that the minimizing solution does not exist and the manifold satisfies a certain topological assumption. To perform the topological argument, we obtain a decomposition of Palais-Smale sequences as a sum of bubbles and adapt Lions’s concentration-compactness lemma.

Wednesday, April 13, 13:30-14:30, Burnside 920
Dima Jakobson (McGill University)
Title: Zero and negative eigenvalues of the Yamabe operator.
Abstract. This is joint work with Rod Gover, Asma Hassannezhad and Michael Levitin. After recalling old results (joint with Canzani, Gover and Ponge) about conformal invariants that arise from nodal sets of eigenfunctions lying in a kernel of a conformally covariant operator on a compact manifold, we show that for the conformal Laplacian, zero is _not_ an eigenvalue for generic Riemannian metrics. We proved previously that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any compact manifold of dimension $n \geq 3$; we show that if the number of negative eigenvalues increases for a sequence of metrics, then that sequence cannot satisfy certain natural pre-compactness assumptions, and hence cannot have "convergent subsequences." If time permits, we shall discuss related results for operators on graphs.

Wednesday, May 4, 13:30-14:30, Burnside 920
Title: Ancient solutions to geometric flows
Abstract: We will discuss ancient solutions to geometric flows such as the Mean Curvature flow, Ricci flow and Yamabe flow. We will discuss the classification of ancient or eternal solutions as well as the construction of ancient solutions from the gluing is solitons.

Thursday, July 7, 13:30-14:30, Burnside Hall 920
Yuxin Ge, (university of Toulouse)
Title: On the comformally compact Einstein manifolds
Abstract: In this talk, I will discuss 4-dimensional conformally compact Einstein manifolds and in particular the compactness result of these manifolds and some relations to the conformal invariants at infinity.

Wednesday, July 27, 13:30-14:30, Burnside Hall 920
Pierre Damien Thizy (University of Cergy-Pontoise)
Title: Blow-up analysis for the Moser-Trudinger equation in dimension 2
Abstract: We will first introduce the Moser-Trudinger elliptic equation with critical exponential non-linearity and give some variational motivations to study it. Then, we will give the main results in the literature concerning this equation. At last we will give our result obtained with Olivier Druet about the blow-up analysis for this equation. These results answer questions asked by Adimurthi-Struwe, Druet, Martinazzi-Malchiodi, and Del Pino-Musso-Ruf.