Geometric Analysis
Org:
Siyuan Lu (McMaster University) and
Xiangwen Zhang (University of California, Irvine)
[
PDF]
 MIN CHEN, McGill University
AlexandrovFenchel type inequalities for hypersurfaces in the sphere [PDF]

The AlexandrovFenchel inequalities in the Euclidean space are inequalities involving quermassintegrals of different orders and are classical topics in convex and differential geometry. BrendleGuanLi proposed a conjecture on the corresponding inequalities for quermassintegrals in the sphere. In this talk, we introduce some new progress on this Conjecture.
 EDWARD CHERNYSH, McGill
A StruweType Decomposition for Weighted $p$Laplace equations of the CaffarelliKohnNirenberg Type [PDF]

In this talk, we establish a Struwetype decomposition result for a class of critical $p$Laplace equations of the CaffarelliKohnNirenberg type in smoothly bounded domains
$\Omega \subset \mathbb{R}^n$ for $n \ge 3$. More precisely, we investigate the relative compactness of PalaisSmale sequences associated to the critical elliptic problem
\begin{align*}
\begin{cases}
\operatorname{div}\left( \left\vert\nabla u\right\vert^{p2} \nabla u \left\vert x\right\vert^{ap} \right) = \left\vert u\right\vert^{q2}u\left\vert x \right\vert^{bq} & \text{in } \Omega,\ u = 0 & \text{on } \partial \Omega.
\end{cases}
\end{align*}
Here, $1 < p < n$ and $q := np/(np(1+ab))$ under suitable conditions for $a,b$. In doing so, we highlight crucial differences between the weighted setting and the pioneering work of Michael Struwe in the unweighted model $p=2$ case.
 TRISTAN COLLINS, University of Toronto
Uniqueness of Cylindrical Tangent Cones to some Special Lagrangians [PDF]

I will explain a proof of the following result: if an exact special Lagrangian $N \subset \mathbb{C}^n$ has a multiplicity one, cylindrical tangent cone of the form $\mathbb{R}^k\times C$ where $C$ is a special Lagrangian cone with smooth, connected link, then this tangent cone is unique provided $C$ satisfies an integrability condition. This applies, for example, to the HarveyLawson $T^{m1}$ cones for $m\ne 8,9$. This is joint work with Y. Li.
 ROBERT HASLHOFER, University of Toronto
Free boundary minimal disks in convex balls [PDF]

We prove that every strictly convex 3ball with nonnegative Riccicurvature contains at least 3 embedded freeboundary minimal 2disks for any generic metric, and at least 2 solutions even without genericity assumption. Our approach combines ideas from mean curvature flow, minmax theory and degree theory. We also establish the existence of smooth freeboundary meanconvex foliations. This is joint work with Dan Ketover.
 FANG HONG, McGill University
Sharpened Minkowski Inequality in CartanHadamard Spaces [PDF]

Minkowski inequality describes the relationship between total mean curvature of a surface and its area. Extension of Minkowski inequality to hyperbolic space and finding the sharp inequality have been a long standing problem. We will discuss a recent paper by M. Ghomi and J. Spruck and sharper inequality we get based on their proof, in which we generalized Minkowski inequality to general spaces with nonpositive curvature via harmonic mean curvature flow.
 CHAOMING LIN, Ohio State University
On the solvability of general inverse $\sigma_k$ equations [PDF]

In this talk, first, I will introduce general inverse $\sigma_k$ equations in Kähler geometry. Some classical examples are the complex Monge–Ampère equation, the Jequation, the complex Hessian equation, and the deformed Hermitian–Yang–Mills equation. Second, by introducing some new real algebraic geometry techniques, we can consider more complicated general inverse $\sigma_k$ equations. Last, analytically, we study the solvability of these complicated general inverse $\sigma_k$ equations.
 SEBASTIEN PICARD, UBC
Strominger system and complex geometry [PDF]

The Strominger system is the set of fundamental equations of supersymmetric heterotic string theory over a CalabiYau threefold. It combines the YangMills equation on a vector bundle with a constraint equation on the Riemannian curvature tensor. We will survey current developments on these equations.
 LING XIAO, University of Connecticut
Generalized Minkowski inequality via degenerate Hessian equations on exterior domains [PDF]

In this talk, we will talk about the proof of a sharp generalized Minkowski inequality that holds for any smooth, strictly $(k1)$convex, starshaped domain $\Omega$. Our proof relies on the solvability of the degenerate kHessian equation on the exterior domain $R^n\setminus\Omega.$
 ZIHUI ZHAO, Johns Hopkins University
Unique continuation and the singular set of harmonic functions [PDF]

Unique continuation property is a fundamental property for harmonic functions, as well as a large class of elliptic and parabolic PDEs. It says that if a harmonic function vanishes at a point to infinite order, it must vanish everywhere. In the same spirit, we are interested in quantitative unique continuation problems, where we use the local growth rate of a harmonic function to deduce some global estimates, such as estimating the size of its singular or critical set set. In this talk, I will talk about some recent results together with C. Kenig on boundary unique continuation.