# Geometric Analysis and General Relativity (16w5054)

Arriving in Banff, Alberta Sunday, July 17 and departing Friday July 22, 2016

## Organizers

(University of Miami)

David Maxwell (University of Alaska Fairbanks)

Richard Schoen (University of California Irvine)

(University of Washington)

## Objectives

This workshop will bring together a wide variety of researchers from around the world whose expertise and interests lie within the realms of geometric analysis and general relativity. Over the past few years there have been remarkable advances on the mathematical problems of general relativity and this workshop will build on this by bringing together the primary participants in this recent progress as well as young researchers, including post-docs and advanced graduate students, working in this area.

Rather than try to equally include the totality of research in mathematical relativity this workshop will focus on work which is primarily of geometric and/or elliptic nature from an analytical point of view. This will increase the likelihood of fostering significant mathematical interactions and collaborations amongst the participants. Below we outline a number of the themes that will be well represented during the workshop.

The subject of marginally outer trapped surfaces (MOTS) and their relationship with solutions of Jang's equation, were initiated in the work of Schoen and Yau in proving the space-time version of the positive mass theorem. Over the last decade, this subject has seen a major resurgence and blossoming in the field. Significant recent mathematical work in this area has been done by Andersson, Eichmair, Galloway, Mars, Metzger, Pollack and Schoen, among others. Some of this work was viewed within the context of quasi-local models for black holes introduced by Hayward and subsequently developed by Ashtekar and Krishnan. MOTS play an important role in the analysis, by Galloway-Schoen and Galloway, of the topology of higher dimensional black holes. More recently the advances in understanding the analysis of MOTS has been applied to questions of topological censorship in the work of Eichmair, Galloway and Pollack.

Global inequalities have played a large role in general relativity, going back at least as far as the proof the positive mass theorem due to Schoen and Yau and more recently via the proofs of the Riemannian Penrose inequality (which is deeply connected to Penrose's cosmic censorship conjecture) by Huisken-Ilmanen and Bray. Attempts to generalize this later work to a Lorentzian space-time setting are topics of current research. The long standing attempt to generalize the notion of mass in relativity to one which is quasi-locally defined has seen new ideas introduced due to the work of Shi and Tam and, more recently, to the extensive work of Wang and Yau. Recognizing the need for low regularity formulations of the fundamental quantities in relativity Huisken has developed formulations of many global inequalities in terms of isoperimetric inequalities which hold in a very rough setting. This perspective has been investigated by Eichmair and Metzger. Fundamental questions of uniqueness and stability of black holes has led to a great deal of important work, by Dain, Chrusciel, Schoen and others, on the angular momentum-mass inequality for axisymmetric black holes.

The analysis of initial data sets and the implications for their spacetime developments has been the subject of a great deal of recent attention. Research in this area has benefited from the introduction of gluing techniques from geometric analysis. Corvino's construction of vacuum spacetimes which are Schwarzschild outside of a compact set, and the extension of this method by Corvino-Schoen and Chru\'sciel-Delay have had a significant impact. The combination of this with conformal gluing methods by Chru\'sciel, Isenberg and Pollack have led to many interesting constructions. Recently, the construction of initial data sets exhibiting gravitational screening by Carlotto and Schoen has made clear that solutions of the constraint equations have a great deal more flexibility than was previously understood. This latter work has led to a number of interesting conjectures.

Aside from gluing, the analysis of the set of solutions of the constraint equations has largely relied on the so-called conformal method. This technique allows for a parameterization of the set of constant-mean curvature solutions of the constraints, but its applicability in the far-from-CMC regime is poorly understood. Lately there have been two major pushes toward understanding the conformal method for arbitrary mean curvatures: the limit equation analysis by Dahl, Gicquaud and Humbert, and select far-from CMC'' existence results due to Holst, Nagy, Tsogtgerel and Maxwell. These results have generated a large amount of recent activity, including results by younger researchers Dilts, Ng\^o and Sakovich. Despite these efforts, basic questions of the existence and uniqueness for the conformal method remain open, and indeed a very recent paper by Gicquaud and Ng\^o shows that our current far-from CMC existence results can be seen as merely another variety of near-CMC theorems. In light of these difficulties, Maxwell has reexamined the conformal method by working with special cases and has used this analysis to developed a related drift'' method that offers new perspectives and the potential of fertile ground for future research.

In 2010, BIRS ran a very successful meeting on Geometric Analysis and General Relativity'' organized by Andersson, Dafermos, Galloway and Pollack. In 2015 there will be numerous meetings to celebrate the 100 year anniversary of general relativity, however most of these will focus primarily on physics and less on mathematics. Notable recent meetings centered on geometric analysis and general relativity include a very active semester long program at MSRI in Autumn 2013 and an upcoming Oberwolfach Workshop on Mathematical Aspects of General Relativity" in July, 2015.

This workshop at BIRS brings together many of the researchers responsible for the recent developments discussed above. This will be an opportunity to report on the most recent advances as well as to indicate those areas in which future progress is both likely and important. The workshop will be conducted with a limited number of talks covering a wide range of topics and will allow for ample time for informal interaction among the participants. A substantial effort will be made to encourage junior researchers to attend and participate.