David Hoffman,
MSRI
``Properly embedded minimal surfaces of finite topology''
April 22, 1999
Not counting the plane, the first minimal surfaces (those
that locally minimize area) were discovered in the 18th Century:
the catenoid by Euler in 1744 and the helicoid by Meusnier in 1776.
These clasical minimal surfaces are topologically the sphere with
a point (two for the catenoid) removed. No new examples of
complete properly embedded minimal surfaces of finite topology
in Euclidean space were found until the mid 1980s. There has been
progress in the last few years in understanding the general nature of
the space of all such surfaces. For instance, it is now known that
if such a surface has more than one topological end, it must have
finite total curvature, and each end must be asymptotic to either
a plane or a catenoid. The discovery in the mid 90s of a ``genus-one
helicoid'' has focused research on trying to understand and classify
one-ended examples. Other than the plane, are they all asymptotic
to the helicoid? Are the plane and the the helicoid the only simply
connected examples? The talk will include a short computer animation.
See the home page of the
Scientific
Graphics Project for some relevant images.