Amanda Petcu, Department of Pure Mathematics, University of Waterloo
"Partial progress on a conjecture of Donaldson by Fine and Yao"
Given
a
compact
hypersymplectic
manifold
$X^4$,
Donaldson
conjectured
that
the
hypersymplectic
structure
can
be
deformed
through
cohomologous
hypersymplectic
structures
to
a
hyperkähler
structure.
Fine
and
Yao
consider
a
manifold
with
closed
$G_2$-structure
that
is
set
up
as
$\mathbb{T}^3
\times
X^4$.
They
examine
the
$G_2$-Laplacian
flow
under
in
this
setting
and
give
a
flow
of
hypersymplectic
structures
which
evolve
according
to
the
equation
\[\partial_t
\underline{\omega}
=
d(Q
d^*(Q^{-1}
\underline{\omega}))\]
where
$\underline{\omega}$
is
the
triple
that
gives
the
hypersymplectic
structure
and
$Q$
is
a
$3
\times
3$
symmetric
matrix
that
relates
the
symplectic
forms
$\omega_i$
to
one
another.
Lotay-Wei
have
established
long
time
existence
of
the
$G_2$-Laplacian
flow
provided
the
velocity
of
the
flow
remains
bounded.
Fine—Yao
use
this
extension
theorem
in
their
setup
and
manage
to
improve
it
by
proving
long
time
existence
of
the
hypersymplectic
flow
provided
the
torsion
tensor
$T$
remains
bounded.
Furthermore,
one
can
relate
the
scalar
curvature
and
torsion
tensor
of
manifold
with
closed
$G_2$-structure
and
thus
they
conclude
long
time
existence
for
the
hypersymplectic
flow
provided
the
scalar
curvature
remains
bounded.
In
this
talk
we
will
go
over
some
details
from
this
paper
by
Fine-Yao.
MC 5403