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CATEGORIES:Differential Geometry and Topology Seminar
SUMMARY:Hypersymplectic structures on 4-manifolds and the
G2 Laplacian flow - Joel Fine\, ULB
DTSTART;TZID=Europe/London:20171018T160000
DTEND;TZID=Europe/London:20171018T170000
UID:TALK72461AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/72461
DESCRIPTION:A hypersymplectic structure on a 4-manifold is a t
riple w_1\, w_2\, w_3 of symplectic forms such tha
t any non-zero linear combination of these forms i
s again symplectic. The prototypical example is a
the triple of Kähler forms of a hyperkähler metric
. Donaldson has conjectured that up to isotopy\, t
his is the only example. More precisely\, Donaldso
n conjectures that on a compact 4-manifold\, any h
ypersymplectic triple is isotopic through cohomolo
gous hypersymplectic triples to a hyperkähler trip
le. This is a special case of a famous folklore co
njecture: a compact symplectic 4-manifold with c_1
=0 and b_+=3 admits a compatible integrable comple
x structure making it hyperkähler. I will describe
an approach to Donaldson’s conjecture which goes
via G2 geometry. It gives a natural flow of hypers
ymplectic structures which tries to deform a given
triple into a hyperkähler one. It can be thought
of as an analogue of Ricci flow adapted to this co
ntext. I will then explain joint work with Chengji
an Yao\, which shows that the hypersymplectic flow
exists as long as the scalar curvature of the ass
ociated G2 metrics remains bounded. It is intrigui
ng that this is a stronger existence result than w
hat is currently known for Ricci flow. I will not
assume any prior knowledge of Ricci flow\, or G2 g
eometry. \n
LOCATION:MR13
CONTACT:Ivan Smith
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