## 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 C**î**rstea **(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)

**Wednesday, October 4,****
13:30-14:30, Burnside Hall 920**

**Jerome Vetois**** **(McGill University)

**Wednesday, October 11,****
13:30-14:30, Burnside Hall 920**

**Shaya Shakerian**** **(University of British Columbia)

**Wednesday, October 18,****
13:30-14:30, Burnside Hall 920**

**Shaodong Wang**** **(McGill University)

**Wednesday, October 25,****
13:30-14:30, Burnside Hall 920**

**Guohuan Qiu**** **(McGill University)

**Friday, November 10,****
13:30-14:30, Burnside Hall 920**

**Daniel Pollack**** **(University of Washington)

**Wednesday, November 15,****
13:30-14:30, Burnside Hall 920**

**Fengrui Yang **(McGill University)

**Wednesday, November 22,****
13:30-14:30, Burnside Hall 920**

**Saikat Mazumdar**** **(University of British Columbia)

**Wednesday, November 29,****
13:30-14:30, Burnside Hall 920**

**Rohit Jain**** **(McGill University)

**Wednesday, December 6,****
13:30-14:30, Burnside Hall 920**

**Vladmir Sicca**** **(McGill University)

## 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**

**Mohammad Najafi Ivaki** (Concordia University)

**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**