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SymplecticEinstein Report Summary

Project ID: 646649
Funded under: H2020-EU.1.1.

Periodic Reporting for period 1 - SymplecticEinstein (The symplectic geometry of anti-self-dual Einstein metrics)

Reporting period: 2015-09-01 to 2017-02-28

Summary of the context and overall objectives of the project

"The project "SymplecticEinstein" is founded on a new formulation of Einstein’s equations in dimension 4. This new approach reveals a surprising link between four-dimensional Einstein manifolds and six-dimensional symplectic geometry. The project will exploit this interplay in both directions: using Riemannian geometry to prove results about symplectic manifolds and using symplectic geometry to prove results about Reimannian manifolds.

Solutions to Einstein's field equations are possible models for the universe. Finding solutions is as difficult as it is important. The aim of this project is to exploit the hidden symplectic geometry of these equations which I recently discovered together with co-authors. This confluence of two different geometries makes many new potential techniques available to either side. The overall objective is to both find new solutions to Einstein's equations and better understand the solutions we already have. Moreover, I will use techniques from the study of Einstein metrics to explore the symplectic manifolds which arise this way."

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

"Work in the first period has focused on the following questions.

1. Construction of new solutions of Einstein's equations. To date there are four known classes of examples of compact 4-dimensional solutions: locally homogeneous spaces (with lots of local symmetries), Kähler-Eintsein surfaces (a construction which again relies on symmetries, this time internal ones), Dehn fillings (a method of producing new Einstein metrics from hyperbolic geometry) and "isolated" examples of which there are just two (the Page metric and the Chen-LeBrun-Webber metric). Using the techniques outlined in the project proposal, in joint work with Bruno Premoselli, I have succeeded in finding a new class of examples. They are again built using hyperbolic geometry, and can be compared to those coming from Dehn fillings in terms of difficulty and importance. Moreover they are the first compact Einstein metrics with negative sectional curvatures which are not locally homogeneous. This work has recently been completed, and will soon be published.

Another question I have attacked here is the study of hyperkähler metrics on a 4-manifold with boundary. It is a classical question to ask how solutions to partial differential equations can be described in terms of their boundary values. In joint work with Jason Lotay and Michael Singer, I resolved this for hyperkähler metrics, which are special solutions to Einstein's equations.

2. Geometric flows to classify symplectic structures. Another main theme of the project is the classification of certain symplectic structures. I have investigated an important flow (coming from 7-dimensional G2 geometry) which attempts to deform a 4-dimensional "hypersymplectic" structure in to one giving a solution of Einstein's equations. In joint work with Chengjian Yao I have shown that this flow exists for as long as the scalar curvature remains bounded. This goes beyond what is known for the 4-dimensional Ricci flow, a famous geometric flow which has had huge success in 3-dimensional geometry. This work has recently been completed and will soon be published.

3. Singularity formation. In general, families of solutions to Einstein's equations can form singularities. I have been able to that a large class of singularities cannot occur when the metrics are constructed using the symplectic formalism on which this project is based. This is important for two reasons: firstly it shows that these Einstein metrics have a very different behaviour from, say, hyperkähler ones. Secondly it is important for constructing new examples: one takes a metric which is almost a solution to the equations and tries to deform it so it becomes a solution. The arguments I have developed here show that as one carries out this deformation, the singularities which can arise are of a very restricted nature.

4. Symplectic geometry of asymptotically hyperbolic spaces. Another aim of the project is to use symplectic geometry (and in particular ideas from symplectic field theory) to study asymptotically hyperbolic 4-manifolds. A first step has been taken, in joint work with the theoretical physicists Yannick Herfray, Kirill Krasnov and Carlos Scarinci. We have worked out how to epxand the symplectic structure in powers of a boundary defining function. This gives precise information how the symplectic geometry degenerates as you run out towards infinity in space. Subsequently, in solo work, I have computed the asymptotic behaviour of holomorphic curves - they limit to conformal geodesics. This is an important part of understanding how to adapt the ideas of symplectic field theory to this setting."

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

In all of the above areas, the progress has been substantially beyond the state of the art:

1. Examples of compact Einstein 4-manifolds are extremely rare. The ones I find in my joint work with Premoselli are the first to be found in the last 10-15 years. Moreover they are the first Einstein metrics which are negatively curved, but not locally homogeneous. This answers a long standing question in the field.

2. My joint work with Yao on the G2 flow for hypersymplectic structures gives a new extension result which is the first time results for the G2 flow are stronger than those available for Ricci flow.

3. My work on singularity formation, ruling out for example isolated orbifold singularities in families of anti-self-dual Einstein metrics, is a substantial advance. It will be significant for the construction of new examples of such metrics, via continuity methods or similar approaches.

4. My investigation of asymptotically hyperbolic 4-manifolds has led to a new geometric interpretation of conformal geodesics: they are the boundaries of J-holomorphic curves in twistor space which are asymptotically cylindrical. This opens the way to using ideas from symplectic field theory to study these curves and their projections which are minimal surfaces in the 4-manifold.

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