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Abstract:
Properly embedded constant mean curvature (CMC) surfaces,
which model complete noncompact soap bubbles and equilibrium fluid
droplets, have a particular asymptotic behavior. This leads to a pair
of natural questions: How well do the asymptotic data determine the
surface? Can one describe the moduli space of all CMC surfaces with a
given (finite) topology, using these asymptotic data as natural
parameters? The key to answering these questions, at least in part,
is the nondegeneracy of the linearized mean curvature (or second
variation of area, or Jacobi) operator. We'll report on recent joint
work with K. Grosse-Brauckmann, N. Korevaar, J. Ratzkin and
J. Sullivan, where we have found good estimates on the nullity of the
Jacobi operator; though we have shown the operator is nondegenerate
for a large class of surfaces, many interesting open problems remain.
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