Réunion d'hiver SMC 2023

Montréal, 1 - 4 decembre 2023


Analyse géométrique
Org: Siyuan Lu (McMaster University) et Xiangwen Zhang (University of California, Irvine)

MIN CHEN, McGill University
Alexandrov-Fenchel type inequalities for hypersurfaces in the sphere  [PDF]

The Alexandrov-Fenchel inequalities in the Euclidean space are inequalities involving quermassintegrals of different orders and are classical topics in convex and differential geometry. Brendle-Guan-Li proposed a conjecture on the corresponding inequalities for quermassintegrals in the sphere. In this talk, we introduce some new progress on this Conjecture.

A Struwe-Type Decomposition for Weighted $p$-Laplace equations of the Caffarelli-Kohn-Nirenberg Type  [PDF]

In this talk, we establish a Struwe-type decomposition result for a class of critical $p$-Laplace equations of the Caffarelli-Kohn-Nirenberg type in smoothly bounded domains $\Omega \subset \mathbb{R}^n$ for $n \ge 3$. More precisely, we investigate the relative compactness of Palais-Smale sequences associated to the critical elliptic problem \begin{align*} \begin{cases} -\operatorname{div}\left( \left\vert\nabla u\right\vert^{p-2} \nabla u \left\vert x\right\vert^{-ap} \right) = \left\vert u\right\vert^{q-2}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 $q := np/(n-p(1+a-b))$ 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 Harvey-Lawson $T^{m-1}$ 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 3-ball with nonnegative Ricci-curvature contains at least 3 embedded free-boundary minimal 2-disks for any generic metric, and at least 2 solutions even without genericity assumption. Our approach combines ideas from mean curvature flow, min-max theory and degree theory. We also establish the existence of smooth free-boundary mean-convex foliations. This is joint work with Dan Ketover.

FANG HONG, McGill University
Sharpened Minkowski Inequality in Cartan-Hadamard 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 non-positive curvature via harmonic mean curvature flow.

CHAO-MING 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 J-equation, 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.

Strominger system and complex geometry  [PDF]

The Strominger system is the set of fundamental equations of supersymmetric heterotic string theory over a Calabi-Yau threefold. It combines the Yang-Mills 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 $(k-1)$-convex, star-shaped domain $\Omega$. Our proof relies on the solvability of the degenerate k-Hessian 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.

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