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Is there a way to characterize non compact surfaces with constant mean and gaussian curvature. I know that if $K=0=H$ then the surface is a plane. How can I know about the others?

Just to add, for compact surfaces with constant positive curvature I know Liebmann's theorem as well. But i want to do it for non compact surfaces.

Any help????

Assuming $K$ and $H$ constant. If $k_1$ and $k_2$ are principal curvature then

$k_1=H+\sqrt{H^2-K}$ and $k_2=H-\sqrt{H^2-K}$, for compact surfaces I only have $k_1=k_2$ and $k_1>k_2$ is not possible. What can i say about them in this case?

Ali
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  • For any constant $H > 0$, an open subset of the sphere of radius $\frac{1}{H}$ is such a surface, with $K = H^2$. – Travis Willse Apr 04 '15 at 10:30
  • What do I get from this? Can we have negative gaussian curvature? with mean curvature being constant in this case? – Ali Apr 04 '15 at 12:01
  • Do you want mean and Gauss curvature to be the same, or both just constant? – A. Thomas Yerger Apr 04 '15 at 16:42
  • both just constant. I saw it somewhere that the possible surfaces (compact or not compact) are plane, sphere or right circular cylinder but i am not sure. Obviously if it is compact, it will be a sphere be Liebmann (locally) – Ali Apr 04 '15 at 16:44
  • I seem to be completely stuck. I wanted to do it generally, started with compact surfaces and found them to be a part of sphere. Now for non-compact surfaces, I just don't know to use that both are constant – Ali Apr 04 '15 at 16:48
  • @AlfredYerger any idea? – Ali Apr 04 '15 at 16:50
  • If H is positive, K can be negative. Consider a surface with principla curvatures -1 and 5. The mean curvature will be positive but Gauss negative. – A. Thomas Yerger Apr 04 '15 at 16:55
  • how does that help me? – Ali Apr 04 '15 at 17:02

1 Answers1

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It is a nice exercise that the only surfaces with both principal curvatures constant must be (pieces of) a plane, a sphere, or a right circular cylinder. (As a warm-up, prove that a surface with one constant principal curvature and no umbilic points must be a tube around a regular curve.)

Ted Shifrin
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