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Computational Methods
for
Riemann Surfaces
and
Helicoids with Handles
Ph.D. Thesis (2005) (pdf
9.61 MB)
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The purpose of this thesis is a numerical investigation of helicoids with
handles. Helicoids with handles are complete minimal surfaces of finite
topology, i.e., they have finite genus and finitely many ends. In fact,
helicoids have just one end, which is asymptotic to the simple helicoid.
Among all minimal immersions the embedded examples are the most interesting.
Embedded minimal surface of infinite topology abound. In the 19th century Scherk
discovered his famous families of singly and doubly periodic embedded examples.
Since then, many more examples of infinite topology have been constructed, but
until 1984 the only known embedded examples of finite topology were the plane,
the catenoid (Euler, c. 1744), and the helicoid (Meusnier, 1776).
In 1984 Costa [Cos84] wrote down the
Weierstrass data for his famous minimal torus having one planar and two
catenoidal ends that are all disjoint and separately embedded. The following
year David Hoffman and William Meeks, working with Jim Hoffman, were able to
generate computer pictures of the Costa surface indicating that it is embedded,
which they proved soon afterward [HoffmanMeeks85].
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The family of complete minimal surfaces has an important subclass of surfaces
with finite total curvature given by all surfaces M for which the integral of
the absolute Gauss curvature ∫M|K|
dA is finite. Costa's surface has
finite total curvature, and many more examples of finite total curvature have
been generated since. The minimal surfaces that we are concerned with have
infinite total curvature because the helicoidal end has infinite total
curvature.
In 1993, Hoffman, Karcher, and Wei added a single handle to the helicoid,
creating the first infinite total curvature example of finite positive genus [HWK93].
This helicoid with one handle (He1)
inspired the theory of infinite total curvature minimal surfaces much like
Costa's surface did the finite curvature case. Once again, plots of the surface
strongly suggest that it is embedded, and Weber, Hoffmann, and Wolf recently
proved this by exposing He1 as the limit
of a family of embedded minimal surfaces that are invariant under screw-motion [WHW03]. |
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