gradient descent evolution of surfaces

Hi all,

I'm trying to understand someone's PhD thesis on the topic of
variational surface evolution and its application in computer vision,
and I'm having trouble working out how he evaluates some particular
types of expressions involving the gradient.

This is the notation I will use here:

M : a closed m-dimensional manifold in R^(m+1)
x : point in R^(m+1), and sometimes only x in M
Sm : m-dimensional unit sphere in R^(m+1)
n : unit normal vector to M at an x. (as a function, denotes the Gauss
map: n : M -> Sm)
f (or g) : M -> R
f~: diff'ble extension of f, R^(m+1)->R, such that everywhere on the
manifold, f~ = f
<v1, v2> : dot product of v1 and v2
grad(f) : gradient of f : R^(m+1) -> R
grad_Sm(f) : gradient of f along the m-surface, in R^(m+1)
(Sm represents the m-dimensional unit-sphere, and is used because the
"Gauss map" on the m-surface M is
a function n : M -> Sm)
div(v) : divergence of v

I think it'll be easier if I specify the concerned references
directly, with the hope that someone with the time, patience and
knowledge can take a brief look at them and help clarify things.

For an overview of the author's research, please take a look at the
following presentation:
http://www.uib.no/People/nmaxt/oslo-talk/Solem_Oslo-2005.pdf

Of particular interest to my problem is the gradient projection given
on page 10 of this presentation, and the gradient descent evolution of
the surface on page 15.
I will reproduce these two equations below (with a slight difference
from the original document), referring to them as the gradient
projection eq. and the steepest descent eq. respectively:

grad_Sm(f(x)) = grad(f~) - <grad(f~), n>n
grad_M(E) = div(grad_Sm(g) + (g)n)

My first equation uses n instead of x as done by the author in the
presentation, because in the presentation he has written the equation
for the specific case when M is the unit sphere. I simply wanted to
emphasise that M could be any surface, and n is the unit normal at the
point under consideration. Also, to remove any potential confusion
(one that I experienced initially), for the general case, the S^m in
LaTeX graphic is being generated. Reload this page in a moment. is
merely used to indicate of the fact that the 'Gauss map' on any closed
manifold is given by the map n : M -> S^m

Now, referring to the author's PhD thesis:
http://homeweb.mah.se/~tsjeso/publications/Solem-thesis-2006.pdf

My main problem is how the author applies these equations to some
specific examples of energy functionals, on page 40 and 41 of the
thesis.

Referring first to the (simpler) example on page 41 (section 3.7),
with
g = -1/2*<v,n>.

The way the author applies the gradient projection eq. seems to imply
that
grad_Sm(v*n) = v - <v,n>n
or to narrow it down even further
grad(v*n) = v
(with the assumption - I suppose - that the vector field 'v' is
defined throughout the space, and that the gradient of the vector
field 'v' is a function), and maybe it's just some simple property
from vector calculus of the gradient operator that I haven't been able
to apply, but I don't get it.
Ditto with the more complex examples on page 40 of the thesis, eqs.
3.27 and 3.25.

I don't have a lot of mathematical knowledge, so forgive me if I made
a slip-up somewhere in my understanding of the problem and the
author's solution (and in which case I would welcome any corrections)

I'd be grateful for any help...
Thanks!

PS I posted the same question on the following forum website, if you
care to look at prettier version with the math stuff formatted with
Latex:
http://www.physicsforums.com/showthread.php?t=168125

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