## Geometric Analysis Seminar

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

## FALL 2017

Wednesday, August 2, 13:30-14:30, Burnside Hall 920
Joshua Ching (University of Sydney)
Title: Singular solutions to nonlinear elliptic equations with gradient dependency
Abstract: Let $N \geq 2$ be the dimension. Let $\Omega \subseteq \mathbb{R}^N$ be a domain containing the origin. We consider non-negative $C^1(\Omega \setminus \{ 0\})$ solutions to the following elliptic equation: ${\rm div} (|x|^{\sigma} |\nabla u|^{p-2}\nabla u)=|x|^{-\tau} u^q |\nabla u|^m$ in $\Omega \setminus \{ 0 \}$, where we impose appropriate conditions on the parameters $m,p,q,\sigma,\tau,N$. We study these solutions from several perspectives including existence, uniqueness, radial symmetry, regularity and asymptotic behaviour. In the model case where $p=2$ and $\sigma=\tau=0$, we impose the conditions $q>0$, $m+q>1$ and $0<m<2$. Here, we provide a sharp classification result of the asymptotic behaviour of these solutions near the origin and infinity. We also provide corresponding existence results in which we emphasise the more difficult case of $m \in (0,1)$ where new phenomena arise. A key step in these proofs is to obtain gradient estimates. Using a technique of Bernstein's and some other ideas, we find a new gradient estimate that is independent of the domain and is applicable in a more general setting than the model case. Via these gradient estimates, we will show a Liouville-type result that extends a theorem of Farina and Serrin (2011). Time permitting, we will also look at further applications of this gradient estimate. In this talk, we present results from Ching and Cîrstea (2015, Analysis & PDE), results from my PhD thesis as well as ongoing research.
Wednesday, August 2, 14:45-15:45, Burnside Hall 920
Laurent Moonens (University of Paris-Sud)
Title: Continuous solutions for divergence-type equations associated to elliptic systems of complex vector fields
Abstract: In this talk, we shall discuss a characterization, obtained with T.H. Picon, of all the distributions $F \in \calD’(\Omega)$ for which one can locally solve by a \emph{continuous} vector field $v$ the divergence-type equation  $$L_{1}^{*}v_{1}+...+L_{n}^{*}v_{n}=F$$ where  $\left\{L_{1},\dots,L_{n}\right\}$ is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on $\Omega \subset \R^{N}$. In case where $(L_1,\dots, L_n)$ is the usual gradient field on $\R^N$, we recover a classical result for the divergence equation, obtained previously by T. De Pauw and W.F. Pfeffer.
Wednesday, August 9, 13:30-14:30, Burnside Hall 920
Florica Cîrstea (University of Sydney)
Title: Nonlinear elliptic equations with isolated singularities
Abstract: In this talk, I will review recent developments on isolated singularities for various classes of nonlinear elliptic equations, which could include Hardy-Sobolev type potentials. In particular, we shall look at fully classifying the behaviour of all positive solutions in different contexts that underline the interaction of the elliptic operator and the nonlinear part of the equation. We also provide sharp results on the existence of solutions with singularities, besides optimal conditions for the removability of all singularities. I will discuss results obtained with various collaborators including T.-Y. Chang (University of Sydney) and F. Robert (University of Lorraine).
Wednesday, August 16, 13:30-14:30, Burnside Hall 920
Chao Xia (Xiamen University)
Title: Uniqueness of stable capillary hypersurfaces in a ball
Abstract:
Capillary hypersurfaces in a  ball $B$ is minimal or CMC hypersurfaces whose boundary intersects $\partial B$ at a constant angle. They are critical points of  some energy functional under volume preserving variation. The study of stability of capillary hypersurfaces in $B$ was initiated by Ros-Vergasta and Ros-Souam in 90's. An open problem is whether any immersed stable capillary hypersurfaces in a  ball in space forms are totally umbilical. In this talk, we will give a complete affirmative answer. We remark that the related uniqueness result for closed hypersurfaces is due to Barbosa-Do Carmo-Eschenburg. The talk is based on a joint work with Guofang Wang.
Wednesday, August 23, 13:30-14:30, Burnside Hall 920
Xinan Ma (University of Science and Technology of China)
Title: The Neumann problem of special Lagrangian equations with supercritical phase
Abstract: In this talk, we establish the global $C^2$ estimates of the Neumann problem of special Lagrangian equations with supercritical phase and the existence theorem by the method of continuity, we also mention the complex version. This is the joint work with Chen chuanqiang and Wei wei.
Wednesday, September 20, 13:30-14:30, Burnside Hall 920
Pengfei Guan (McGill University)
Title: Gauss curvature flows and Minkowski type problems
Abstract: We discuss a class of isotropic flows by power of Gauss curvature of convex hypersurfaces. For each flow, there is an entropy associated to it, and it is monotone decreasing. For this entropy, there is an unique entropy point. The flow preserves the enclosed volume. The main question is to control the entropy point. This was done for standard flows in joint works with Lei Ni, and Ben Andrews and Lei Ni. For isotropic flows, under appropriate assumptions, one prove that the entropy point will keep as origin. From there, one may deduce regularity and convergence. The self-similar solutions are the solutions to corresponding Minkowski type problem. Similar results were also obtained by Bryan-Ivaki-Scheuer via inverse type flows.
Wednesday, September 27, 13:30-14:30, Burnside Hall 920
Gantumur Tsogtgerel (McGill University)
Title: Elliptic estimates for operators with rough coefficients
Abstract:
We will discuss a possible approach to establish elliptic estimates for operators with barely continuous coefficients in a Sobolev-Besov and Triebel-Lizorkin scale. The result would obviously be not new but the proposed approach is relatively elementary and therefore of interest. Please be warned that this is an ongoing project, and the talk is going to be more of a discussion than a polished lecture.
Wednesday, October 4, 13:30-14:30, Burnside Hall 920
Jerome Vetois (McGill University)
Title: Blowing-up solutions for critical elliptic equations on a closed manifold
Abstract:
In this talk, we will look at the question of existence of blowing-up solutions for smooth perturbations of positive scalar curvature-type equations on a closed manifold. From a result of Druet, we know that in dimensions different from 3 and 6, a necessary condition for the existence of blowing-up solutions is that the limit equation agrees with the Yamabe equation at least at one blow-up point. I will present new existence results in situations where the limit equation is different from the Yamabe equation away from the blow-up point. I will also discuss the special role played by the dimension 6. This is a joint work with Frederic Robert.
Wednesday, October 11, 13:30-14:30, Burnside Hall 920
Shaya Shakerian (University of British Columbia)
Title: Borderline Variational problems for fractional Hardy-Schrödinger operators
Abstract:
In this talk, we investigate the existence of ground state solutions associated to the fractional Hardy-Schrödinger operator on Euclidean space and its bounded domains. In the process, we extend several results known about the classical Laplacian to the non-local operators described by its fractional powers. Our analysis show that the most important parameter in the problems we consider is the intensity of the corresponding Hardy potential. The maximal threshold for such an intensity is the best constant in the fractional Hardy inequality, which is computable in terms of the dimension and the fractional exponent of the Laplacian. However, the analysis of corresponding non-linear equations in borderline Sobolev-critical regimes give rise to another threshold for the allowable intensity. Solutions exist for all positive linear perturbations of the equation, if the intensity is below this new threshold. However, once the intensity is beyond it, we had to introduce a notion of Hardy-Schrödinger Mass associated to the domain under study and the linear perturbation. We then show that ground state solutions exist when such a mass is positive. We then study the effect of non-linear perturbations, where we show that the existence of ground state solutions for large intensities, is determined by a subtle combination of the mass (i.e., the geometry of the domain) and the size of the nonlinearity of the perturbations.
Wednesday, October 18, 13:30-14:30, Burnside Hall 920
Shaodong Wang (McGill University)
Title: A compactness theorem for Yamabe problem on manifolds with boundary
Abstract:
In this talk, I will present a compactness result on Yamabe problems on manifolds with boundary. This is from a paper by Zheng-chao Han and Yanyan Li.
Wednesday, October 25, 13:30-14:30, Burnside Hall 920
Guohuan Qiu (McGill University)
Title: One-dimensional convex integration
Abstract:
Convex integration theory was introduced by M.Gromov in his thesis dissertation in 1969 which is a powerful tool for solving differential relations. An important application of the convex integration theory is that it can recover the Nash-Kuiper result on C^{1} isometric embeddings. I will briefly mention the history of h-principle and rigidly theorem for isometric embedding. Then some details about one-dimensional convex integration.
Friday, November 10, 13:30-14:30, Burnside Hall 920
Daniel Pollack (University of Washington)
Title: On the geometry and topology of initial data sets with horizons
Abstract:
One of the central and most fascinating objects which arise in general relativity are black holes. From a mathematical point of view this is closely related to questions of "singularities" and "horizons" which arise in the study of the Einstein equations. We will present a number of results which relate the presence (or absence) of horizons to the topology and geometry of the "exterior region" of an initial data set for Einstein equations. Time permitting we will also connect these results with previous work of Galloway and Schoen on the topology of the black holes themselves. This is joint work with Lars Andersson, Mattias Dahl, Michael Eichmair and Greg Galloway.
Wednesday, November 15, 13:30-14:30, Burnside Hall 920
Fengrui Yang (McGill University)
Title: The Dirichlet Problem for Monge-Ampere equation.
Abstract: In this talk, I will present the main ideas and techniques of the classical paper, ' The Dirichlet Problem for nonlinear Second-Order Elliptic Equations I, Monge-Ampere Equation'. Firstly, I will give a brief introduction of the history of estimating third-order derivatives of Monge-Ampere equation, and then focus on the proof of boundary C2 and C2,a estimates.
Wednesday, November 22, 13:30-14:30, Burnside Hall 920
Saikat Mazumdar (University of British Columbia)
Title: Blow-up analysis for a critical elliptic equation with vanishing singularity
Abstract:
In this talk, we will examine the asymptotic behavior of a sequence of ground state solutions of the Hardy-Sobolev equations as the singularity vanishes in the limit.  If this sequence is uniformly bounded in L-infinity,  we obtain a minimizing solution of the stationary Schrödinger equation with critical growth. In case the sequence blows up, we obtain C0 control on the blow up sequence, and we localize the point of singularity and derive precise blow up rates.
Wednesday, November 29, 13:30-14:30, Burnside Hall 920
Edward Chernysh (McGill University)
Title: Weakly Monotone Decreasing Solutions to an Elliptic Schrödinger System
Abstract:
In this talk we study positive super-solutions to an elliptic Schrödinger system in R^n for n\geq3. We give conditions guaranteeing the non-existence of positive solutions and introduce weakly monotone decreasing functions. We establish lower-bounds on the decay rates of positive solutions and obtain upper-bounds when these are weakly monotone decreasing.
Wednesday, December 6, 13:30-14:30, Burnside Hall 920
Rohit Jain (McGill University)
Wednesday, January 31, 13:30-14:30, Burnside Hall 920
Bruno Premoselli (Université Libre de Bruxelles)

Wednesday, February 14, 13:30-14:30, Burnside Hall 920
Luca Capogna (Worcester Polytechnic Institute)

Friday, February 16, 13:30-14:30, Burnside Hall 920
Loredana Lanzani (Syracuse)

Wednesday, February 21, 13:30-14:30, Burnside Hall 920

## WINTER 2017

Wednesday, January 25, 13:30-14:30, Burnside Hall 920
Gantumur Tsogtgerel (McGill University)
Title: A prescribed scalar-mean curvature problem
Abstract: In this talk, we will be concerned with a problem of prescribing scalar curvature and boundary mean curvature of a compact manifold with boundary. This is an ongoing work motivated by the study of the Einstein constraint equations on compact manifolds with boundary, and builds on the results of Rauzy and of Dilts-Maxwell.

Wednesday, February 1st, 13:30-14:30, Burnside Hall 920
Title: Harnack estimates for curvature flows
Abstract: I will discuss Harnack estimates for curvature flows in the Riemannian and Lorentzian manifolds of constant curvature and that "duality" allows us to obtain a certain type of inequalities, "pseudo"-Harnack inequalities.

Wednesday, February 8, 13:30-14:30, Burnside Hall 920
Jerome Vetois (McGill University)
Title: Decay estimates and symmetry of solutions to elliptic systems in R^n
Abstract: In this talk, we will look at a class of coupled nonlinear Schrödinger equations in R^n. I will discuss a notion of finite energy solutions for these systems and I will present some recent qualitative results on these solutions.

Wednesday, February 22, 13:30-14:30, Burnside Hall 920
Guohuan Qiu (McGill University)
Title: Rigidity of closed self-similar solution to the Gauss curvature flow
Abstract: In the seminar, I will present Choi and Daskalopoulos's recent [arXiv:1609.05487v1] rigidity result about Gauss curvature flow. They proved that a convex closed solution to the Gauss curvature flow in R^n becomes a round sphere after rescaling.

Wednesday, March 8, 13:30-14:30, Burnside Hall 920
Siyuan Lu (McGill University)
Title: Minimal hypersurface and boundary behavior of compact manifolds with nonnegative scalar curvature
Abstract: In the study of boundary behavior of compact Riemannian manifolds with nonnegative scalar curvature, a fundamental result of Shi-Tam states that, if a compact manifold has nonnegative scalar curvature and its boundary is isometric to a strictly convex hypersurface in the Euclidean space, then the total mean curvature of the boundary of the manifold is no greater than the total mean curvature of the corresponding Euclidean hypersurface. In this talk, we give a supplement to Shi-Tam's result by considering manifolds whose boundary includes the outermost minimal hypersurface of the manifold. Precisely speaking, given a compact manifold \Omega with nonnegative scalar curvature, suppose its boundary consists of two parts, \Sigma_h and \Sigma_o, where \Sigma_h is the union of all closed minimal hypersurfaces in \Omega and \Sigma_o is isometric to a suitable 2-convex hypersurface \Sigma in a Schwarzschild manifold of positive mass m, we establish an inequality relating m, the area of \Sigma_h, and two weighted total mean curvatures of \Sigma_o and $\Sigma, respectively. This is a joint work with Pengzi Miao from Miami. Wednesday, March 16, 2:00pm-3:3:00pm, Burnside 1234 Yuanwei Qi (University of Central Florida) Title: Traveling Wave of Gray-Scott model: Existence, Multiplicity and Stability. Abstract: In this talk, I shall present some recent works I have done with my collaborators in rigorously proofing the existence of traveling wave solution to the Gray-Scott model, which is one of the most important models in Turing type of pattern formation after the experiments in early 1990s to validate his theory. We shall also discuss some interesting features of traveling wave solutions. This is a joint work with Xinfu Chen. Wednesday, March 22, 13:30-14:30, Burnside Hall 920 Rohit Jain (McGill University) Title: Regularity estimates for Semi-permeable membrane Flow Abstract: We study a boundary value problem modeling flow through the semi-permeable boundary$\Gamma$with finite thickness$\lambda$and an applied fluid pressure$\phi(x)$. We study optimal regularity estimates for the solution as well as asymptotic estimates as$\lambda \to 0\$.

Wednesday, March 29, 13:30-14:30, Burnside Hall 920
Kyeongsu Choi (Columbia University)
Title: Free boundary problems in the Gauss curvature flow
Abstract: We will discuss the optimal C^{1,1/(n-1)} regularity of the Gauss curvature flow with flat sides, and the C^{\infty} regularity of the flat sides. Moreover, we will study connections between the free boundary problems, the classification to the self-shrinkers, and the prescribed curvature measure equations.

Wednesday, April 5, 13:30-14:30, Burnside Hall 920
Shaodong Wang (McGill University)
Title: Infinitely many solutions for cubic Schrödinger equation in dimension 4
Abstract: In this talk, I will present some recent results in the existence of blow-up solutions to a cubic Schrödinger equation on the standard sphere in dimension four. This is a joint work with Jerome Vetois.

Friday, April 7, 13:30-14:30, Burnside Hall 920
Xinliang An (University of Toronto)
Title: On Gravitational Collapse in General Relativity
Abstract: In the process of gravitational collapse, singularities may form, which are either covered by trapped surfaces (black holes) or visible to faraway observers (naked singularities). In this talk, I will present four results with regard to gravitational collapse for Einstein vacuum equation. The first is a simplified approach to Christodoulouâ€™s monumental result which showed that trapped surfaces can form dynamically by the focusing of gravitational waves from past null infinity. We extend the methods of Klainerman-Rodnianski, who gave a simplified proof of this result in a finite region. The second result extends the theorem of Christodoulou by allowing for weaker initial data but still guaranteeing that a trapped surface forms in the causal domain. In particular, we show that a trapped surface can form dynamically from initial data which is merely large in a scale-invariant way. The second result is obtained jointly with Jonathan Luk. The third result answered the following questions: Can a black holeâ€™â€™ emerge from a point? Can we find the boundary (apparent horizon) of a black holeâ€™â€™ region? The fourth result extends Christodoulouâ€™s famous example on formation of naked singularity for Einstein-scalar field system under spherical symmetry. With numerical and analytic tools, we generalize Christodoulouâ€™s result and construct an example of naked singularity formation for Einstein vacuum equation in higher dimension. The fourth result is obtained jointly with Xuefeng Zhang.

Wednesday, April 19, 13:30-14:30, Burnside Hall 920
Ben Weinkove (Northwestern University)
Title: The Monge-Ampere equation, almost complex manifolds and geodesics
Abstract: I will discuss an existence theorem for the Monge-Ampere equation in the setting of almost complex manifolds. I will describe how techniques for studying this equation can be used to prove a regularity result for geodesics in the space of Kahler metrics. This is joint work with Jianchun Chu and Valentino Tosatti.

Wednesday, April 26, 13:30-14:30, Burnside Hall 920
Chen-Yun Lin (University of Toronto)
Title: An embedding theorem: differential analysis behind massive data analysis
Abstract: High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis. In this talk, I will present a theoretical analysis of the effectiveness of the VDM. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.

## FALL 2016

Wednesday, September 21, 13:30-14:30, Burnside Hall 920
Pengfei Guan (McGill University)
Title: A volume preserving flow and the isoperimetric problem in warped product spaces with general base
Abstract: A flow was introduced in a previous work to handle the isoperimetric problem in sapce forms. We propose to study a similar normalized hypersurface flow in the more general ambient setting of warped product spaces with general base. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the isoperimetric problem in warped product spaces is solved for such domains. This is a join work with Junfang Li and Mu-Tao Wang.

Wednesday, September 28, 13:30-14:30, Burnside Hall 920
Dylan Cant (McGill University)
Title: A Curvature flow and application to an isoperimetric inequality
Abstract: Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under weak assumption on the warp product of surface). This curvature flow preserves the area enclosed by the curve, and this fact is used to prove a general isoperimetric inequality applicable to radial graphs in warped product surfaces under weak assumption on the warp potential.

Wednesday, October 5, 13:30-14:30, Burnside Hall 920
Rohit Jain (McGill University)
Title: Geometric Methods in Obstacle-Type Free Boundary Problems I
Abstract: Obstacle-type free boundary problems naturally appear as mathematical models in science and engineering with some particular motivations arising from contact problems in elasticity, options pricing in financial mathematics, and phenomenological models in superconductor physics. The first talk will focus on geometric methods that have been used to study regularity estimates in Obstacle-Type Free Boundary Problems. The regularity theory for obstacle-type problems (and other type of free boundary problems as well) was much inspired by the regularity theory for minimal surfaces. We will discuss the basic existence, uniqueness and regularity questions in the classical obstacle problem. We will point out generalizations and current problems of interest in this field of research. In the second talk we will focus on an obstacle-type problem arising in stochastic impulse control theory that appeared first as a model for cash management and portfolio optimization under transaction costs. Here the underlying theory for the obstacle problem has to be suitably modified to consider obstacle problems with an implicit and nonlocal obstacle. Regularity estimates will be presented and natural directions for future research discussed.

Wednesday, October 12, 13:30-14:30, Burnside Hall 920
Rohit Jain (McGill University)
Title: Geometric Methods in Obstacle-Type Free Boundary Problems II
Abstract: We will continue studying Geometric Methods in Obstacle-Type Free Boundary Problems. In the second talk we will focus on an obstacle-type problem arising in stochastic impulse control theory that appeared first as a model for cash management and portfolio optimization under transaction costs. Here the underlying theory for the obstacle problem has to be suitably modified to consider obstacle problems with an implicit and nonlocal obstacle. Regularity estimates for the solution and the free boundary will be presented.

Wednesday, October 19, 13:30-14:30, Burnside Hall 920
Guohuan Qiu (McGill University)
Title: Hessian estimate for the Sigma-2 Equation in dimension Three (After Michah Warren and Yu Yuan)
Abstract: Heinz derived a Hessian bound for the two dimensional Monge-Ampere equation by using Uniformization Theorem. Sigma-2=1 in three dimension can be viewed as a equation of a special lagranian graph in C^3. Which is also a three dimensional minimal surface in R^6. Michah Warren and Yu Yuan used this observation and Michael-Simon's sobolev inequalities on generalized submanifolds of R^n to prove a priori interior Hessian estimates for Sigma_2 =1 in three dimension. We will go through their proof in this seminar.

Wednesday, November 2, 13:30-14:30, Burnside Hall 920
Siyuan Lu (McGill University)
Title: Isoperimetric inequality in warped product manifold.
Abstract: We consider isoperimetric inequality in warped product manifold. We discuss two results by Montiel and Bray-Morgan. The paper by Montiel shows that under natural assumption of the warped function, a star shaped constant mean curvature hypersurface must be a coordinate slice. The paper by Bray-Morgan shows that under stronger assumption of the warped function, isoperimetric domain must be a coordinate slice.

Thursday, November 10, 14:30-15:30, Burnside Hall 920
Tatiana Toro (University of Washington)
Title: Almost minimizers with free boundary
Abstract: In recent work with G. David, and ongoing work with G. David and M. Engelstein, we study almost minimizer for functionals which yield a free boundary, as in the work of Alt-Caffarelli and Alt-Caffarelli-Friedman. The almost minimizing property can be understood as the defining characteristic of a minimizer in a problem which explicitly takes noise into account. In this talk we will discuss regularity results for these almost minimizers and as well as the structure of the corresponding free boundary. A key ingredient in the study of the 2-phase problem is the existence of almost monotone quantities.

Wednesday, November 16, 13:30-14:30, Burnside Hall 920
Siyuan Lu (McGill University)
Title: Isoperimetric inequality in warped product manifold II.
Abstract: We will continue to discuss the isoperimetric inequality in warped product manifold. We'll focus on Bray-Morgan's result using comparison to obtain the isoperimetric inequality without the assumption of starshapedness.

Wednesday, December 14, 13:30-14:30, Burnside Hall 920
Pengzi Miao (University of Miami)
Title: Boundary effect of scalar curvature
Abstract: Manifolds with nonnegative scalar curvature arise naturally as maximal slices of physical spacetimes in general relativity. When the manifold is noncompact, there are the Riemannian positive mass theorem and Penrose inequality which give global results on how scalar curvature affects the manifold geometry near infinity. When the manifold is compact, it models bounded domains in such spacetime slices and how the scalar curvature affects its boundary geometry is tied to the quasi-local mass problem. In this talk, I will survey known results on boundary behavior of compact manifolds with nonnegative scalar curvature, and if time permits, I will discuss related open questions.

Previous Talks