John Loftin, Rutgers University Newark
“Minimal Lagrangian Surfaces in CH2”
An
immersed
minimal
Lagrangian
surface
in
the
complex
hyperbolic
plane
CH2
inherits
a
conformal
structure
from
the
induced
metric.
Moreover,
the
components
of
the
second
fundamental
form
can
be
written
in
terms
of
a
cubic
differential
U
which
is
holomorphic
with
respect
to
the
conformal
structure.
The
integrability
condition
for
the
conformal
factor
of
the
metric
is
an
elliptic
semilinear
equation
which
is
very
similar
to
the
Uhlenbeck’s
equation
for
minimal
surfaces
in
RH3.
Solving
this
equation
on
a
hyperbolic
Riemann
surface
induces
a
prepresentation
of
the
fundamental
group
into
SU(2,1)
with
0
Toledo
invariant.
When
the
developed
minimal
Lagrangian
surface
is
close
enough
to
a
Lagrangian
plane
in
CH2,
we
may
call
the
induced
representation
”almost
real-Fuchsian.”
We
investigate
the
geometry
of
these
representations
and
develop
a
dictionary
between
the
theories
of
minimal
surfaces
in
RH3
and
minimal
Lagrangian
surfaces
in
CH2.
This
is
joint
work
with
Ian
McIntosh
and
with
Zeno
Huang
and
Marcello
Lucia.