## Periodic Reporting for period 3 - SymplecticEinstein (The symplectic geometry of anti-self-dual Einstein metrics)

Reporting period: 2018-09-01 to 2020-02-29

"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.

"

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 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 appeared as a preprint and is submitted for publication.

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. This work has been published in the Forum of Mathematics, Sigma.

This boundary value problem is a precursor for a more intricate question: to describe anti-self-dual Poincaré-Einstein 4-manifolds in terms of their conformal infinities. In joint work with Rafe Mazzeo and Michael Singer, we have analysed this problem for small perturbations of a given initial anti-self-dual Poincaré-Einstein metric. We prove that the deformation problem is unobstructed giving a smooth moduli space of solutions (modulo gauge which is the identity at infinity). We also characterise the small deformations of the conformal infinity which can be ""filled in"" by anti-self-dual Poincaré-Einstein metrics. This generalises a famous result of Biquard, who dealt with the case of the hyperbolic metric on the interior of the 4-ball. Our work is a key initial step in understanding the moduli space of anti-self-dual Poincaré-Einstein metrics. This work will soon be completely written up and submitted for publication.

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 been accepted for publication in Duke Mathematical Journal.

In an attempt to understand the long-time behaviour of this hypersymplectic flow, I encouraged Chengjian Yao to consider a cohomogeneity one example on the 4-torus, given a PDE in two variables. He attacked this problem in joint work with Hongnian Huang and Yuanqi Wang, proving long-time existence and convergence of the flow. This goes some way to justifying the hope that the hypersymplectic flow is able to solve the classification of hypersymplectic structures. Their work is published in the Journal of the London Mathematical Society.

In joint work with Giovanni Mascellani, we are working to adapt this flow to study definite connections on 4-manifolds. We have devised a new parabolic flow which is the formal analogue of the hypersymplectic flow. We have proved short time existence and are now working on the Shi-type estimates. This work will soon be completely written up and submitted for publication.

3. Singularity formation.

In general, families of solutions to Einstein's equations can form singularities. I have been able to show 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. This work was published in the Transactions of the London Mathematical Society.

These ideas are also central to understanding singularity formation in both the hypersymplectic flow and its analogue for definite connections (as described in part 2).

4. Symplectic geometry and enumerative invariants of asymptotically hyperbolic 4-manifolds.

Another aim of the project is to use symplectic geometry, and in particular ideas from Gromov-Witten 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. This work is published in Classical and Quantum Gravity.

The main goal now is to define enumerative invariants of these manifolds, given by counting special J-holomorphic curves in their twistor spaces. These curves correspond to special conformal harmonic maps, i.e. branched minimal immersions into the 4-manifold. With no additional boundary conditions, these curves come in infinite dimensional families. The challenge is to single out appropriate boundary conditions which will give a Fredholm problem.

In joint work with Manh-Tien Nguyen, we have shown that, whilst the energy of a proper map from a Riemann surface into an AH 4-manifold is infinite, it is possible to ""renormalise"" it so as to obtain a finite quantity, which we call renormalised energy. Critical points of renormalised energy are both harmonic and conformal, but they also satisfy a stronger condition, governing the way in which they approach infinity. One consequence is that the boundary curve must be a conformal geodesic. It remains to be seen if this is sufficient to give a Fredholm theory and can be used to define an invariant ""counting"" the number of such maps."

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 appeared as a preprint and is submitted for publication.

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. This work has been published in the Forum of Mathematics, Sigma.

This boundary value problem is a precursor for a more intricate question: to describe anti-self-dual Poincaré-Einstein 4-manifolds in terms of their conformal infinities. In joint work with Rafe Mazzeo and Michael Singer, we have analysed this problem for small perturbations of a given initial anti-self-dual Poincaré-Einstein metric. We prove that the deformation problem is unobstructed giving a smooth moduli space of solutions (modulo gauge which is the identity at infinity). We also characterise the small deformations of the conformal infinity which can be ""filled in"" by anti-self-dual Poincaré-Einstein metrics. This generalises a famous result of Biquard, who dealt with the case of the hyperbolic metric on the interior of the 4-ball. Our work is a key initial step in understanding the moduli space of anti-self-dual Poincaré-Einstein metrics. This work will soon be completely written up and submitted for publication.

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 been accepted for publication in Duke Mathematical Journal.

In an attempt to understand the long-time behaviour of this hypersymplectic flow, I encouraged Chengjian Yao to consider a cohomogeneity one example on the 4-torus, given a PDE in two variables. He attacked this problem in joint work with Hongnian Huang and Yuanqi Wang, proving long-time existence and convergence of the flow. This goes some way to justifying the hope that the hypersymplectic flow is able to solve the classification of hypersymplectic structures. Their work is published in the Journal of the London Mathematical Society.

In joint work with Giovanni Mascellani, we are working to adapt this flow to study definite connections on 4-manifolds. We have devised a new parabolic flow which is the formal analogue of the hypersymplectic flow. We have proved short time existence and are now working on the Shi-type estimates. This work will soon be completely written up and submitted for publication.

3. Singularity formation.

In general, families of solutions to Einstein's equations can form singularities. I have been able to show 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. This work was published in the Transactions of the London Mathematical Society.

These ideas are also central to understanding singularity formation in both the hypersymplectic flow and its analogue for definite connections (as described in part 2).

4. Symplectic geometry and enumerative invariants of asymptotically hyperbolic 4-manifolds.

Another aim of the project is to use symplectic geometry, and in particular ideas from Gromov-Witten 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. This work is published in Classical and Quantum Gravity.

The main goal now is to define enumerative invariants of these manifolds, given by counting special J-holomorphic curves in their twistor spaces. These curves correspond to special conformal harmonic maps, i.e. branched minimal immersions into the 4-manifold. With no additional boundary conditions, these curves come in infinite dimensional families. The challenge is to single out appropriate boundary conditions which will give a Fredholm problem.

In joint work with Manh-Tien Nguyen, we have shown that, whilst the energy of a proper map from a Riemann surface into an AH 4-manifold is infinite, it is possible to ""renormalise"" it so as to obtain a finite quantity, which we call renormalised energy. Critical points of renormalised energy are both harmonic and conformal, but they also satisfy a stronger condition, governing the way in which they approach infinity. One consequence is that the boundary curve must be a conformal geodesic. It remains to be seen if this is sufficient to give a Fredholm theory and can be used to define an invariant ""counting"" the number of such maps."

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.The work on renormalised energy of maps from a Riemann surface to an asymptotically hyperbolic manifold has led to a new geometric interpretation of conformal geodesics: they are the boundaries of critical maps for the renormalised energy functional. This opens the way to using ideas from minimal surface theory to study these conformal geodesics.

Expected results until the end of the project:

1. Foundational results on a new geometric flow, defined for definite connections. This flow is modelled on the G2 flow for hypersymplectic structures. It aims to deform a given definite connection into one which defines an anti-self-dual Einstein metric. This will hopefully lead to one of the key goals of the project, that the only compact 4-manifolds which admit positive definite connections are the four-sphere and the complex projective plane. At the time of writing, short-time existence and uniqueness of the flow has been proved. The next step is to prove that the flow extends for as long as scalar curavture is bounded (as in my work with Yao on the G2 flow for hypersymplectic structure). Then singularity formation will need to be addressed. (This part of the project is joint work with Giovanni Mascellani.)

2. New examples of compact Einstein 4-manifolds The techniques developed in my work with Premoselli can also be applied to other situations. We treat the case of branched covers of certain hyperbolic 4-manifolds, we will next look at hyperbolic 4-orbifolds, with codimension 2 singularities. We aim to find a smooth Einstein metric on the underlying manifold. We also expect to show that in some cases one can take an Einstein metric with cone singularity and continuously open the cone, keeping the metric Einstein, until one arrives at a smooth Einstein metric. This is the real 4-dimensional analogue of recent celebrated work of Chen-Donaldson-Sun in the Kähler-Einstein setting.

3. In joint work with Bruno Premoselli, we are considering the following problem. Given an asymptotically complex-hyperbolic (ACH) Einstein metric, we aim to find a path of Einstein metrics g(t) which are asymptotically hyperbolic (AH) and which converge as t tends to 0 to the original ACH metric. Moreover, by using the language of definite connections, we aim to show that if the original ACH metric is also anti-self-dual, then the pertubed AH metrics are also anti-self-dual. This will give infinitely many new examples of these metrics. It will also show that the conformal geometry and infinity can degenerate without the interior metric becoming singular.

4. Enumerative invariants of anti-self-dual Poincaré-Einstein 4-manifolds. I hope to define an invariant counting maps which are critical points of renormalised energy. This hinges on a compactness result which in turn relies on the symplectic geometry of twistor space. The choice of boundary conditions will also be very important. Asking that the disc becomes asymptotically totally geodesic seems to give a good theory, in which the boundary curves are actually conformal geodesics. Supposing the invariant can be defined, the next challenge will be to compute it. It would be interesting to find two different anti-self-dual Poincaré-Einstein metrics on the 4-ball with different values of the invariant. This would ultimately prove that the metrics could not be joined by a path of anti-self-dual Poincaré-Einstein metrics.

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.The work on renormalised energy of maps from a Riemann surface to an asymptotically hyperbolic manifold has led to a new geometric interpretation of conformal geodesics: they are the boundaries of critical maps for the renormalised energy functional. This opens the way to using ideas from minimal surface theory to study these conformal geodesics.

Expected results until the end of the project:

1. Foundational results on a new geometric flow, defined for definite connections. This flow is modelled on the G2 flow for hypersymplectic structures. It aims to deform a given definite connection into one which defines an anti-self-dual Einstein metric. This will hopefully lead to one of the key goals of the project, that the only compact 4-manifolds which admit positive definite connections are the four-sphere and the complex projective plane. At the time of writing, short-time existence and uniqueness of the flow has been proved. The next step is to prove that the flow extends for as long as scalar curavture is bounded (as in my work with Yao on the G2 flow for hypersymplectic structure). Then singularity formation will need to be addressed. (This part of the project is joint work with Giovanni Mascellani.)

2. New examples of compact Einstein 4-manifolds The techniques developed in my work with Premoselli can also be applied to other situations. We treat the case of branched covers of certain hyperbolic 4-manifolds, we will next look at hyperbolic 4-orbifolds, with codimension 2 singularities. We aim to find a smooth Einstein metric on the underlying manifold. We also expect to show that in some cases one can take an Einstein metric with cone singularity and continuously open the cone, keeping the metric Einstein, until one arrives at a smooth Einstein metric. This is the real 4-dimensional analogue of recent celebrated work of Chen-Donaldson-Sun in the Kähler-Einstein setting.

3. In joint work with Bruno Premoselli, we are considering the following problem. Given an asymptotically complex-hyperbolic (ACH) Einstein metric, we aim to find a path of Einstein metrics g(t) which are asymptotically hyperbolic (AH) and which converge as t tends to 0 to the original ACH metric. Moreover, by using the language of definite connections, we aim to show that if the original ACH metric is also anti-self-dual, then the pertubed AH metrics are also anti-self-dual. This will give infinitely many new examples of these metrics. It will also show that the conformal geometry and infinity can degenerate without the interior metric becoming singular.

4. Enumerative invariants of anti-self-dual Poincaré-Einstein 4-manifolds. I hope to define an invariant counting maps which are critical points of renormalised energy. This hinges on a compactness result which in turn relies on the symplectic geometry of twistor space. The choice of boundary conditions will also be very important. Asking that the disc becomes asymptotically totally geodesic seems to give a good theory, in which the boundary curves are actually conformal geodesics. Supposing the invariant can be defined, the next challenge will be to compute it. It would be interesting to find two different anti-self-dual Poincaré-Einstein metrics on the 4-ball with different values of the invariant. This would ultimately prove that the metrics could not be joined by a path of anti-self-dual Poincaré-Einstein metrics.