# Conformal and CR geometry (12w5072)

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

## Organizers

Spyros Alexakis (University of Toronto )

Robin Graham (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.

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.