CMC Tori
A classical field in
geometry is the study of curves and surfaces in 3-space. Smooth surfaces are
basically determined by their curvature behavior. This is measured by two
functions the Gaußian curvature and the mean curvature.
Fix a point on a surface and a direction at this point. The change of the
normal in this direction gives the directional curvature. By averaging this
directional curvature over all directions one gets the mean curvature at
this point.
A longstanding question is: How do surfaces with constant mean curvature (cmc)
look like?
If the mean curvature of a surface is identically zero one usually speaks of
minimal surfaces and the study of these surfaces is a field for itself. If the
mean curvature is constant but not zero the surfaces are called cmc-surfaces.
Trivial examples for compact cmc-surfaces are round spheres. These examples
were of course already known in the last century, but until the seventies of
this century is was an open question whether these are the only compact
cmc-surfaces.
Besides spheres the next simplest class of compact surfaces are tori (doughnut
shaped surfaces). 1986 Wente and other mathematicians were able to prove that
cmc-tori must exist and made the first pictures for special cases. 1992 Bobenko
succeeded in giving formulas which describe all cmc-tori in terms of theta
functions. Still this is only a theoretical result and it was part of my own work to develop a theory and tools on
computers in order to visualize these surfaces.
Examples:
Pictures: genus 2 , genus 3, genus 4, genus 5
Video(quicktime 72,2 MB) Video(avi 66,7 MB)
Genus 2
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Genus 3
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Genus 4
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Genus 5
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