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CATEGORIES:Partial Differential Equations seminar
SUMMARY:Global existence and convergence of smooth solutio
ns to Yang-Mills gradient flow over compact four-m
anifolds - Paul Feehan
DTSTART;TZID=Europe/London:20140616T150000
DTEND;TZID=Europe/London:20140616T160000
UID:TALK52197AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/52197
DESCRIPTION:Given a compact Lie group and a principal bundle o
ver a closed Riemannian manifold\, the quotient sp
ace of connections\, modulo the action of the grou
p of gauge transformations\, has fundamental signi
ficance for algebraic geometry\, low-dimensional t
opology\, the classification of smooth four-dimens
ional manifolds\, and high-energy physics.\n\nThe
quotient space of connections is equipped with the
Yang-Mills energy functional and Atiyah and Bott
(1983) had proposed that its gradient flow with re
spect to the natural Riemannian metric on the quot
ient space should prove to be an important tool fo
r understanding the topology of the quotient space
via an infinite-dimensional Morse theory. The cri
tical points of the energy functional are gauge-eq
uivalence classes of Yang-Mills connections. Howev
er\, thus far\, smooth solutions to the Yang-Mills
gradient flow have only been known to exist for a
ll time and converge to critical points\, as time
tends to infinity\, in relatively few cases\, incl
uding (1) when the base manifold has dimension two
or three (Rade\, 1991 and 1992\, in dimension two
and three\; G. Daskalopoulos\, 1989 and 1992\, in
dimension two)\, (2) when the base manifold is a
complex algebraic surface (Donaldson\, 1985)\, and
(3) in the presence of rotational symmetry in dim
ension four (Schlatter\, Struwe\, and Tahvildar-Za
deh\, 1998). Global existence of solutions with up
to finitely many point singularities (caused by t
he ``bubbling'' phenomenon) was proved independent
ly by Struwe (1994) and Kozono\, Maeda\, and Naito
(1995). However\, the question of global existenc
e of smooth solutions over general compact\, Riema
nnian\, four-dimensional base manifolds has thus f
ar remained unresolved.\n\nIn this talk we shall d
escribe our proof of the following result: Given a
compact Lie group and a smooth initial connection
on a principal bundle over a compact\, Riemannian
\, four-dimensional manifold\, there is a unique\,
smooth solution to the Yang-Mills gradient flow w
hich exists for all time and converges to a smooth
Yang-Mills connection on the given principal bund
le as time tends to infinity.
LOCATION:CMS\, MR13
CONTACT:Prof. ClĂ©ment Mouhot
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