You are currently browsing the monthly archive for October 2010.
A wonderful meditation community in San Luis Obispo, White Heron Sangha. Reminds me a lot of Megan’s Rainbow Falls Sangha in Irvine.
True Love: A Practice for Awakening the Heart, another great little book by Tich Nhat Hanh. Highly recommended.
Albums and movies that have been in my play list the past month.
Albums: Blood on the Tracks by Bob Dylan, Layla and Other Assorted Love Songs by Derek and the Dominos, Rumors by Fleetwood Mac, Sea Change by Beck, Wildflowers by Tom Petty.
Movies: (500) Days of Summer, Annie Hall, Eternal Sunshine of the Spotless Mind, Forgetting Sarah Marshall.
He spoke last night at the the Cohen Center. Unfortunately, I wasn’t able to attend (math has been eating up my life recently).
Earlier this month I gave a lecture at the differential geometry seminar on the relation between minimal surfaces and the area functional. This is a summary of the lecture.
Definition. A regular parametrized surface is called minimal if its mean curvature is zero everywhere.
We shall try to understand why the word “minimal” is used for such surfaces. One of the original motivations for the development of the theory was the study of soap films formed when dipping closed wires into soapy water. These films tend to form surfaces of least area. In 1760 Joseph Lagrange recognized the connection between surfaces of least area and minimal surfaces and proposed the problem of showing the existence of minimal surfaces with a given boundary. This is now known as Plateau’s problem, named after the Belgian physicist Joseph Plateau for his experiments with soap films.
Formally, Plateau’s problem can be stated as follows. Given a curve , find a minimal surface
having
as its boundary. As we shall see, least-area surfaces are minimal. Thus, another version of Plateau’s problem is to find a least-area surface having
as its boundary.
What is a necessary condition that have least-area among all surfaces with boundary
? The answer may be found in a simplified version of the calculus of variations as follows.
Suppose that is a surface of least-area with boundary
. Consider the nearby surfaces which look like slightly deformed versions of
,
Here,
is a function on the domain of
which has the effect, when multiplied by a small
and added to
, of moving points of
a small bit and leaving
fixed. That is,
on
where
is the boundary of the domain of
and
. Let us parametrize
by
where
,
. Recall that the surface area of
is given by
Now, taking the derivative with respect to , which passes inside the integral, we obtain
We assumed that was a minimum so
. Therefore, setting
in the equation above, we get
Now, let and
Computing ,
, and applying Green’s theorem we then get
since on
. Of course the first integral is zero as well, so we end up with
Since this is true for all such , by the fundamental lemma of the calculus of variations, we must have
But this shows that that mean curvature is identically zero! Therefore, we have shown the following necessary condition for a surface to be area minimizing: if is area minimizing then
is minimal.
I particularly like the previous result as it has a very nice application of Green’s theorem. Unfortunately, the previous result is somewhat limiting since it only considers if our surface is parametrized as a graph. By dropping this restriction we can arrive at a slightly more general result.
We define the normal variation of a surface in
to be a family of surfaces
representing how
changes when pulled in a normal direction. Let
denote the area of
. We show that the mean curvature of
vanishes if, and only if, the first derivative of
vanishes at
.
Let be a regular parametrized surface and choose a bounded domain
. Suppose that
is differentiable and
, where
is the union of the domain
with its boundary
. Let
denote a unit vector field such that
is perpendicular to
for all
; that is,
.
Definition. The normal variation of , determined by
is the map
given by
for
and
. For each fixed
, the map
given by
is a parametrized surface.
We denote by ,
, and
the coefficients of the first fundamental form of
Then the area
of
is
Lemma. We have where
denotes the mean curvature of
.
Theorem. Let be a regular parametrized surface and let
be a bounded domain in
. Then
is minimal on
if, and only if,
for all such
and all normal variations of
.
Proof. If is minimal then
is identically zero and so
for any
. Conversely, assume that
for any
, but that there exists some
for which
. We choose
such that
and
is identically zero outside a small neighborhood of
. But then,
for the variation determined by this
. This contradiction shows that
. Since
is arbitrary,
is minimal.
It should be pointed out that we have said nothing about the second derivative of at
, so that a minimal surface, although a critical point of
, may not actually be a minimum. This is a question concerning the stability of minimal surfaces and might possibly be discussed in a later post.
A cool commercial I saw the other day. It’s like a public service announcement for math. I wonder what job openings IBM has?
I’m not sure if my first post deserves some sort of special treatment that will set it apart from the posts that are to come later or if I should just write something. I suppose by writing the previous sentence I have chosen the former.