Foliations of hyperbolic space and minimal surfaces with voids

David Chopp
Northwestern University

I will present two recent geometric computations done using the level set
method. In the first problem (joint with J. Velling), we provide evidence
that hyperbolic space can be foliated by a family of constant mean curvature
disks sharing an ideal boundary. While it has been proven to be possible
when the boundary is star-shaped, we show that the result is probably true
for arbitrary smooth boundaries. In the second problem (joint with M. Torres
and T. Walsh), we compute periodic minimal surfaces in domains which contain
voids. The area of the surface is measured only outside the voids. With the
presence of voids, we show that the periodic surface of least area is not
necessarily flat.


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