Shubham Dwivedi, Pure Mathematics, University of Waterloo
"Minimal Varieties in Riemannian Manifolds - Part II"
The
goal
is
to
go
through
the
paper
"Minimal
Varieties
in
Riemannian
manifolds"
by
Jim
Simons.
We
will
start
with
a
brief
overview
of
the
historical
development
in
the
theory
of
minimal
surfaces
and
how
Simons'
paper
established
a
very
important
case
for
the
so
called
Bernstein's
Problem.
We
will
then
start
with
a
basic
introduction
to
the
theory
of
connections
on
a
Riemannian
Manifold,
followed
by
Riemannian
Submanifold
theory.
In
particular,
we
will
derive
various
formulas
relating
the
connections
and
Riemann
Curvature
Tensor
of
a
submanifold
and
it's
ambient
manifold.
We
will
define
the
second
fundamental
form
of
a
submanifold
and
will
use
this
to
define
the
Mean
Curvature.
We
will
then
define
variations
and
start
with
minimal
varieties.
If
time
permits,
we
will
derive
the
minimal
surface
equation
and
give
many
examples
of
minimal
submanifolds.