Schedule for: 24w5260 - Conformal and CR Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Meet and Greet at BIRS Lounge (Professional Development Centre, 2nd floor)

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this talk I will report my recent research on applications of potential theory in conformal geometry. I will review curvature equations in conformal geometry. Motivated from the work of Huber on the classification of open surfaces and the work of Schoen-Yau on locally conformally flat manifolds, we want to study singular behavior of solutions to curvature equations under certain curvature conditions in conformal geometry. Our approach is potential theoretic. We will demonstrate we can understand the singular behavior of the potentials outside “thin" sets. Consequently, we are able to derive geometric and topological consequences from the Hausdorff dimensions of singularities based on positivities of curvature.

Branson’s Q-curvature in conformal geometry has been one of the main objects of study for more than 20 years. Using the analogy between conformal and CR geometries, we can also define CR Q-curvature. However, it is less interesting because its integral always vanishes, so you cannot obtain a global CR invariant. It later turns out that the vanishing of CR Q-curvature allows us to define Q-prime curvature, whose integral provides a non-trivial global invariant, including the Burns-Epstein invariant in 3 dimensions. In this introductory talk, I plan to explain the origin of CR Q-curvature from the perspective of complex analysis of strictly pseudoconvex domains and its connection with Q-prime curvature. I also plan to discuss generalizations of Q-prime curvature that provide further global CR invariants, based on work by Taiji Marugame and Yuya Takeuchi.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will first review the history of interior C^2 estimate for fully nonlinear equations. Notably, very few equations were known to have such properties. In the second part, I will discuss my recent work on interior C^2 estimate for Hessian quotient equation. Such equation has deep connections with the Monge-Ampere equation, Hessian equation and special Lagrangian equation. I will then discuss the main idea behind the proof.

In 1974, Loewner and Nirenberg established that any smooth bounded Euclidean domain admits a conformally flat metric which is complete in the interior and has constant negative scalar curvature. Generalisations to compact manifolds with boundary, asymptotic expansions of solutions and other related problems have since received significant attention from many authors (Aviles, McOwen, Mazzeo etc.) In this talk I will discuss recent work with Luc Nguyen on the k-Loewner--Nirenberg problem, in which one replaces the scalar curvature with the k-curvature of a Riemannian manifold. From the PDE perspective, this is equivalent to solving a fully nonlinear, non-uniformly elliptic equation with infinite boundary data. These equations exhibit some interesting regularity properties and unexpected existence phenomena, which will be the focus of the talk.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Poincare-Einstein (PE) manifolds such as the Poincare ball model are complete Einstein manifolds with a well-defined conformal boundary. There is a deep connection between the conformal geometry of the boundary of a PE manifold and the Riemannian geometry of its interior. A first step in studying the moduli space of PE manifolds is to develop a good understanding of its global invariants. In even dimensions, renormalized curvature integrals give many such invariants. In this talk, I will discuss a general procedure for computing renormalized curvature integrals on PE manifolds that is independent of Alexakis' classification. In particular, this explains the connection between the Gauss-Bonnet-type formulas of Albin and Chang-Qing-Yang for the renormalized volume, and explicitly identifies the scalar conformal invariant in the latter formula. Our procedure also produces similar formulas for compact Einstein manifolds. If time permits, I will mention more examples and applications. This talk is based on joint works with Jeffrey Case, Ayush Khaitan, Aaron Tyrrell, and Wei Yuan.

We obtain a formula for the renormalized energy of a solution to the Yang-Mills equations on a Poincare-Einstein 6-manifold. The method generalizes an older proof for renormalized volumes by Chang-Qing-Yang to a much more general setting.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

It is known that the total sigma_2 curvature of a positive-scalar curvature metric conformal to the round one on the sphere, normalized by its volume, is uniquely (up to Mobius transformations) minimized by the round metric. We show that if a metric almost minimizes, then it is almost round (up to Mobius transformations). We obtain optimal exponents for two different notions of closeness. This is a stability result for an optimization problem whose Euler-Lagrange equation is fully nonlinear.

We consider the classical problem of prescribing the scalar and boundary mean curvatures of a Riemannian manifold of dimension $n \geq 3$ via conformal deformations of the metric. We deal in particular with negative scalar curvature, which to our knowledge is the least treated case in the literature. We employ a variational approach to prove new existence results, especially in three dimensions. One of the principal issues for this problem is to obtain compactness properties, due to the fact that intricate bubbling phenomena may occur. In higher dimensions, using a Ljapunov-Schmidt procedure, we obtain new existence results on the unit ball when the prescribed functions are close to constants.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will discuss my joint work with Zhizhang Wang on fully nonlinear flows of noncompact spacelike hypersurfaces in Minkowski space. In particular, we prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show that after rescaling the flow converges to a self-expander.

I'll present joint work with Rayssa Caju and Almir Silva Santos describing the local structure of the moduli space of complete, conformally flat, constant Q-curvature metrics on a finitely punctured sphere. We show this moduli space has a local structure as a real-analytic variety, compute its formal dimension and show that it is smooth when the linearized Q-curvature operator is injective on the right function space. We also show that a natural parameter space has a natural symplectic structure, and that near the nondegenerate points the moduli space has the structure of a Lagrangian submanifold.

Ricci solitons are generalizations of Einstein metrics that can be viewed as fixed points of the Ricci flow.  Expanding Ricci solitons asymptotic to cones may be useful in resolving conical singularities of the Ricci flow which appear beginning in dimension four.  I will discuss joint work with Richard Bamler in which we develop a degree theory for four-dimensional asymptotically conical expanding Ricci solitons, which in particular implies the existence of expanders asymptotic to a large class of cones.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Given an embedded closed n-dimensional submanifold Sigma in the closed Riemannian (n+k)-manifold M, where k < n + 2, we define extrinsic global conformal invariants of Sigma by renormalizing the volume associated to the unique singular Yamabe metric with singular set Sigma. In the case n is odd, the renormalized volume is an absolute conformal invariant, while if n is even, there is a conformally invariant energy term given by the integral of a local Riemannian invariant. In particular, the volume gives a global conformal invariant of a knot embedding in the three-sphere. This is work with my student, Sri Rama Chandra Kushtagi.

In this talk we will discuss minimal surfaces in asymptotically hyperbolic spaces and a corresponding "renormalized" area that is conformally invariant. Inspired by work in the compact setting, we show that knowledge of the renormalized area on a relatively small subset of minimal surfaces determines the asymptotic expansion of the metric, including the conformal infinity. As a further application, we show that renormalized area can determine the conformal structure of the boundary of a hyperbolic 3-manifold.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Conformal geodesics are special curves in conformal manifolds. Fine and Herfray characterized them using holography — specifically, through the asymptotic Dirichlet problem for proper minimal surfaces in Poincaré-Einstein spaces. The speaker has observed that this characterization can be reformulated and computationally simplified using proper harmonic maps from the hyperbolic plane. I want to extend this to CR geometry, a more complicated setting. As the first step toward this, in this talk, I focus on the case of the standard CR sphere, which is the boundary at infinity of the complex hyperbolic space. I will present how we can naturally obtain along this line a system of ODEs of (partially) fourth order whose solutions are “horizontal circles” in the sphere. One of the ideas we need to tackle the CR case is based on a paper by Donnelly in 1994.

I'll discuss the boundary behavior of geodesics in conformally compact but not necessarily asymptotically hyperbolic manifolds and in an analogous class of complete K\"ahler metrics on strongly pseudoconvex domains. This is joint work with Achinta Nandi.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We establish the $C^4$-compactness for the solution set of the $Q$-curvature problem on a smooth compact Riemannian manifold for $5 \leq n \leq 24$, under the validity of the positive mass theorem. A counterexample for $n \geq 25$ has been constructed by Wei and Zhao in 2013. Compactness up to dimension 9 (under some conditions) has been proved by YY Li and Xiong in 2019. We execute the Khuri--Marques--Schoen program for $Q$-curvature. Our key observation is that the linearized problem can be reduced to an overdetermined linear system, which admits a nontrivial solution thanks to an unexpected algebraic structure of the Paneitz operator. (Joint work with Liuwei Gong and S. Kim.)

We deal with the problem of characterizing rotationally symmetric solutions to $$\begin{equation} \label{cb} \begin{cases} \Delta u = -n , & \text{in $\Omega$} , \\ u = 0 , & \text{on $\partial\Omega$} \end{cases} \end{equation}$$ when considering ring-shaped domains in the $n$-dimensional Euclidean space ($n \geq 3$). Through a suitable conformal reformulation of the problem, we derive key gradient estimates, which are crucial to the proof of our main comparison result. In particular, if we assume that the set of maximum points of the solutions has positive $\mathcal{H}^{n-1}$-measure, we are able to characterize ring-shaped model solutions to the corresponding \emph{partially} overdetermined problem. Our analysis also provides some sufficient conditions for solutions to problem~$\eqref{cb}$ to be rotationally symmetric. This talk is based on joint work with V. Agostiniani, S. Borghini and L. Mazzieri.

The CR Paneitz operator, a CR invariant fourth-order linear differential operator, plays a crucial role in three-dimensional CR geometry. It is deeply connected to global embeddability, the CR positive mass theorem, and the logarithmic singularity of the Szegő kernel. In this talk, I will discuss recent progress on the spectrum of the CR Paneitz operator. Specifically, I will focus on differences in its nature depending on whether it is embeddable or not.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The $k$-Yamabe problem is a fully non-linear extension of the classical Yamabe problem that seeks for metrics of constant $k$-curvature. In this talk I will discuss this equation from the point of view of geometric flows and provide existence and classification results on soliton solutions for the $k$-Yamabe flow. This is joint work with Maria Fernanda Espinal

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

The purpose of the talk is two-fold. First we revisit our previous work on the conformal quotient equation and prove that the $\sigma_2$ Yamabe problem can be solved in the class of conformal metrics of positive scalar curvature, which answers a question of Jeffrey Case. Then we propose a conjecture on a new type of optimal Sobolev inequality involving $Q$-curvatures and provide evidence by proving the first case. This is a joint project with Yuxin Ge.