# Conformal and CR geometry (12w5072)

Arriving in Banff, Alberta Sunday, July 29 and departing Friday August 3, 2012

## Organizers

Spyros Alexakis (University of Toronto )

(University of Washington)

Kengo Hirachi (University of Tokyo)

Paul Yang (Princeton University)

## Objectives

The goal of this workshop is to bring together researchers from a variety
of backgrounds who find common ground in conformal and CR geometry to
report on recent progress and to stimulate further developments. The
subject remains extremely fertile, with important recent progress opening
up new questions. Areas of focus will include the following:

1. Algebraic questions and asymptotics

2. Index theory related to conformal/CR geometry

3. Geometric analysis in the conformal/CR setting

4. Study of Poincare-Einstein manifolds

Many of the recent developments have their conceptual roots in the
celebrated Atiyah-Patodi-Singer heat kernel proof of the index theorem.
This already brought together in an essential way geometric analysis,
index theory, formal asymptotics, and algebra in the form of classical
invariant theory. Fefferman's program to develop an analogous theory for
the Bergman kernel transferred the ideas to the CR setting and opened up
the subject. The program was later extended to include conformal
geometry.

Much progress has been made in the study of local invariants, for example
in the construction of invariant operators in conformal and CR geometry.
These led to the important discovery by Branson of Q-curvature, motivated
partly by considerations related to index theory and anomalies.
Q-curvature has been intensely studied as an object at the nexus of our
four focus areas; in particular by analysts as a generalization of
scalar curvature in Yamabe-type problems. Very recently, Juhl has
discovered that Q-curvature possesses wholly unexpected stunning recursive
structure. His discoveries center around his introduction of residue
families which are defined in terms of formal asymptotics. These are
clearly a fundamental new tool whose potential has barely begun to be
realized. Another major recent development involving invariants is
Alexakis' resolution of the Deser-Schwimmer conjecture concerning integral
invariants in conformal geometry. This too involved the introduction of
new methods which have not yet been digested, due partly to their
complexity. Important open problems remain, some again related to index
theory. For example, one open question is to understand the structure of
the integrands in the Tian-Yau-Catlin-Zelditch expansion, which can be
viewed as a local version of the Riemann-Roch theorem.

Very general algebraic machinery for constructing and studying invariant
differential operators for parabolic geometries, vast Lie-theoretic
generalizations of conformal and CR geometries, has been developed. In
particular, Cap-Slovak-Soucek constructed Bernstein-Gelfand-Gelfand
sequences for all parabolic geometries; these are sequences of invariant
operators that refine the de Rham complex for each such geometry. One
natural goal in the analysis of these sequences (which sometimes become
complexes) would be to find index theorems and analytic torsions. A new
feature here which complicates the analysis is that the sequences/complexes
contain differential operators of various orders so that the associated
Laplacians are of higher order, and may not be (sub-)elliptic. A first
step in this direction has been carried out by Rumin-Seshadri; they defined
and studied the analytic torsion for a BGG complex for 3-dimensional CR manifolds.

New objects such as Q-curvature mentioned above which were discovered
through formal considerations have become the focus of analytic work.
Another example is the $sigma_k$-Yamabe problem, a generalization of the
Yamabe problem introduced by Viaclovsky, which has spawned vast activity
and progress in the analysis of fully nonlinear partial differential
equations. The study of Bach-flat metrics in dimension 4 has required the
development of new analytic tools for fourth order operators. A natural
problem is to make a similar study in higher even dimensions, replacing the
Bach tensor by the ambient obstruction tensor--another local invariant
discovered through formal geometric considerations. In CR geometry, the
analysis of invariant operators lies at the interface of several complex
variables, harmonic analysis and nonlinear partial differential equations.
Recent work in 3 dimensional CR geometry of Cheng-Malchiodi-Yang on a
positive mass theorem and of Chanillo-Chiu-Yang on the global imbedding
problem involves positivity of the CR version of the Paneitz operator in
fundamental ways. In another direction, there has recently been important
work establishing CR analogues of Sobolev inequalities which exhibit
conformal invariance and which are fundamental in many analytic questions.
Substantial open problems remain in this direction as well.

Poincare-Einstein metrics were introduced as a real version of the
Kahler-Einstein metrics of Cheng-Yau; they are asymptotically real
hyperbolic. There has been work studying existence, uniqueness and
regularity issues for these metrics, but the existence and uniqueness
issues are not settled and much work remains to be done. The study of
Poincare-Einstein metrics has been greatly stimulated by its interaction
with the AdS/CFT correspondence in physics. In particular, the notions of
renormalized volume and area introduced in the physics literature are now
important objects of study by geometric analysts. Renormalized volume
plays a role in formulae of Chern-Gauss-Bonnet type on these spaces,
realizing another connection with index theory. A question with direct
bearing on the physical motivation behind Poincare-Einstein metrics is the
problem of finding the extrema of renormalized volume and area, for
Poincare-Einstein metrics and for complete minimal surfaces inside
them. For minimal surfaces, the renormalized area has been shown by
Alexakis-Mazzeo to be essentially equivalent to the Willmore energy of the
surface. In light of this, the extremization problem can be viewed as a
natural analogue of the Willmore problem for complete surfaces with a free
boundary. This line of inquiry thus relates our subject with geometric
variational problems (such as the well-studied harmonic map problem), in a
new and understudied setting. The theory of asymptotically complex
hyperbolic Einstein manifolds, whose boundaries at infinity are CR
manifolds, is less developed than the real case and should be a fruitful arena for future study.

Our proposed list of participants includes experts in all of these areas.
We intend that this workshop will continue the pattern of rich interaction
which has characterized this field. We also hope to encourage younger
researchers and give them the opportunity to learn from the perspectives of
their more senior colleagues.