1) Let C:(-1,1)->S^2 (the unit sphere in R^3). Then let S be the
surface
{t C(s) : 0 < t < infinity, -1 < s < 1}. I wish to determine certain
properties of S such as the coefficients of its first two fundamental
forms. Is t * (parametrization of S^2), i.e., sigma(u,v) = (t
cos(u)cos(v),t cos(u)sin(v), t sin(u)), a valid parametrization? It
seems like sigma should be a function of t as well, but then one could
not compute E, F, and G in the usual
manner. Also, any computations at this point reduce to the case of a
sphere of radius t. Please do at least one of the following: a)
Confirm that the parametrization I gave is correct and respond to the
concerns I give about it. OR b) Give a correct parametrization and
explain why it is the correct one if it is non-obvious. OR c) explain
how to compute the fundamental form coefficients without an actual
written-down
parametrization.
2) Suppose S is a surface whose Gauss map is a diffeomorphism. What
possible values can H(S) (the mean curvature) take?
One solution, which I must confess I do not fully comprehend and
applies seemingly only to the special case of a conformal
diffeomorphism (although perhaps the proof can be modified as most
diffeomorphisms seem to be conformal), is as follows: "G* has the same
Euclidean coordinates as -S, where G is the Gauss map, S is the shape
operator, and f* is the derivative map of f. Let e1, e2 be a
principal frame (i.e., unit vectors in the principal directions) at a
point of S. Then <G*(e1),G*(e2)> (dot product) = 0. Thus G is
conformal iff ||G*(e1)||^2 = ||G*e(2)||^2 > 0 at every point. Hence
the prinipal curvatures satisfy k1^2 = k2^2 > 0 and therefore S is
either a minimal surface or part of a sphere."
Please either give considerably more detail to that proof and explain
why it is sound when ignoring the conformal hypothesis, or ignore that
solution and
provide a new one.
I greatly appreciate anyone working on this question. For this one
and the previous one, if you can only do one part, I'd be happy to
break up the question into distinct parts so as to be able to
recognize your contribution monetarily. Finally, time is
unfortunately somewhat limited here; responses by Tuesday night would
be most appreciated, and by Wednesday noon rather necessary.
Thanks again for all your help,
Dubois |
