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SymplecticEinstein SIGNED

The symplectic geometry of anti-self-dual Einstein metrics

Total Cost €


EC-Contrib. €






Project "SymplecticEinstein" data sheet

The following table provides information about the project.


Organization address
postcode: 1050

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Belgium [BE]
 Project website
 Total cost 1˙162˙880 €
 EC max contribution 1˙162˙880 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2014-CoG
 Funding Scheme ERC-COG
 Starting year 2015
 Duration (year-month-day) from 2015-09-01   to  2021-08-31


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 
1    UNIVERSITE LIBRE DE BRUXELLES BE (BRUXELLES) coordinator 1˙162˙880.00


 Project objective

This project is founded on a new formulation of Einstein's equations in dimension 4, which I developed together with my co-authors. This new approach reveals a surprising link between four-dimensional Einstein manifolds and six-dimensional symplectic geometry. My 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.

Our new idea is to rewrite Einstein's equations using the language of gauge theory. The fundamental objects are no longer Riemannian metrics, but instead certain connections over a 4-manifold M. A connection A defines a metric g_A via its curvature, analogous to the relationship between the electromagnetic potential and field in Maxwell's theory. The total volume of (M,g_A) is an action S(A) for the theory, whose critical points give Einstein metrics. At the same time, the connection A also determines a symplectic structure omega_A on an associated 6-manifold Z which fibres over M.

My project has two main goals. The first is to classify the symplectic manifolds which arise this way. Classification of general symplectic 6-manifolds is beyond current techniques of symplectic geometry, making my aims here very ambitious. My second goal is to provide an existence theory both for anti-self-dual Poincaré--Einstein metrics and for minimal surfaces in such manifolds. Again, my aims here go decisively beyond the state of the art. In all of these situations, a fundamental problem is the formation of singularities in degenerating families. What makes new progress possible is the fresh input coming from the symplectic manifold Z. I will combine this with techniques from Riemannian geometry and gauge theory to control the singularities which can occur.


year authors and title journal last update
List of publications.
2017 Joel Fine and Chengjian Yao
Hypersymplectic 4-manifolds, the G2-Laplacian flow and extension assuming bounded scalar curvature.
published pages: , ISSN: , DOI:
Accepted for publication in Duke Mathematical Journal 2020-02-25
2017 Joel Fine, Jason Lotay, Michael Singer
The space of hyperkähler metrics on a 4-manifold with boundary
published pages: 50pp, ISSN: 2050-5094, DOI: 10.1017/fms.2017.3
The Forum of Mathematics, Sigma Volume 5, issue 6 2020-02-25
2018 Joel Fine and Bruno Premoselli
Examples of compact Einstein four-manifolds with negative curvature
published pages: , ISSN: , DOI:
Preprint, submitted for publication 2020-02-25
2016 Joel Fine, Yannick Herfray, Kirill Krasnov, Carlos Scarinci
Asymptotically hyperbolic connections
published pages: 185011, ISSN: 0264-9381, DOI: 10.1088/0264-9381/33/18/185011
Classical and Quantum Gravity 33/18 2020-02-25
2018 Hongnian Huang, Yuanqi Wang, Chengjian Yao
Cohomogeneity-one G2-Laplacian flow on the 7-torus
published pages: 349-368, ISSN: 0024-6107, DOI: 10.1112/jlms.12137
Journal of the London Mathematical Society 98/2 2020-02-25
2017 Baptiste Chantraine, Georgios Dimitroglou Rizell, Paolo Ghiggini, Roman Golovko
Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors
published pages: , ISSN: , DOI:
Preprint submitted for publication 2020-02-25
2017 Jean-François Barraud, Agnès Gadbled, Roman Golovko, Hông Vân Lê
Novikov fundamental group
published pages: , ISSN: , DOI:
Preprint, submitted for publication 2020-02-25

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