Principal Investigator: Michael Wolf
Michael Wolf proposes to continue his studies of the applications
of Teichmuller theory to the theory of complete minimal surfaces
in space and of the applications of new techniques in harmonic
maps to singular spaces to problems in Teichmuller theory,
discrete groups and (smooth) harmonic maps theory. In
particular, he proposes to extend his present methods for finding
complete minimal surfaces to periodic minimal surfaces, then
proving their embeddedness via a combination of
Teichmuller-theoretic and minimal surface methods, and then
finally studying limits of families of these surfaces as their
genus increases. He also proposes to classify the minimal maps of
surfaces into a particular building associated to convex
projective structures on Riemann surfaces (in order to identify a
geometrically natural compactification of the space of discrete
faithful representations of a surface group into the Lie group of
convex projective transformations). Finally, he proposes to use
harmonic maps to singular spaces (both with mild singularities
and more serious singularities, like real trees) to study bending
measure coordinates on the Bers slice of Quasifuchsian space, and
harmonic maps between hyperbolic spaces.
The fundamental questions underlying all of this are, "What are
the possible shapes of surfaces we might encounter?", and "What
shapes arise if we require the surfaces to be efficient users of
material, in some sense?" There are, of course, many ways of
interpreting the words "shape" and "efficient", and different
applications of the proposed research would most likely involve
different interpretations of those words. This research project
is an attempt to advance our understanding of the possibilities
for the shapes of soap films (which efficiently use material --
some applications of this theory by others to material science is
in its embryonic stage) and for three other types of shapes. We
also study the possible ways of transforming one surface into
another in an energetically efficient way. It turns out that all
these different problems are deeply interrelated, so advances in
one area often lead to advances in others.
