Critical Points of Gaussian Curvature

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2 total messages Started by Daniel A. Asimov Tue, 22 Dec 1992 00:49

#139

Author: Daniel A. Asimov
Date: Tue, 22 Dec 1992 00:49

38 lines
1874 bytes

The Four-Vertex Theorem states that the curvature function on a smooth 
simple closed curve in the plane has at least 4 critical points.

Now consider a closed hypersurface M^n smoothly embedded in R^(n+1). 

Let "Gaussian curvature" K denote the determinant of the Jacobian of the
Gauss map G: M -> S^n (obtained by setting G(x) =  the outward unit normal 
to M at x).

QUESTION:
	Is there some generalization of the Four-Vertex Theorem for the 
	Gaussian curvature of such hypersurfaces in R^(n+1) ?

	Intuitively, if M were a non-degenerate ellipsoid, it would apparently
	have 2n critical points of its Gaussian curvature K: M -> R.  So, is 
	there evidence for or against the possibility that

		The Gaussian curvature K: M -> R of an 
		arbitrary smooth hypersurface 
		M^n in R^(n+1) must have at least 2n critical points.
	?

	Possible cases to consider:	a)	M is convex 
					b)	M is topologically S^n 
					c)	M is arbitrary.


Dan Asimov
Mail Stop T045-1
NASA Ames Research Center
Moffett Field, CA 94035-1000

asimov@nas.nasa.gov
(415) 604-4799

P.S.  To avoid questions of differentiability I have assumed that everything
in sight is smooth, although a finite number of derivatives would almost
certainly suffice.

Re: Critical Points of Gaussian Curvature

#140

Author: Paul Burchard
Date: Tue, 22 Dec 1992 09:47

37 lines
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In article <1992Dec22.004910.20943@nas.nasa.gov> asimov@nas.nasa.gov (Daniel A.  
Asimov) writes:
> QUESTION:
> 	Is there some generalization of the Four-Vertex Theorem for the 
> 	Gaussian curvature of such hypersurfaces in R^(n+1) ?
> 
> 	Intuitively, if M were a non-degenerate ellipsoid, it would apparently
> 	have 2n critical points of its Gaussian curvature K: M -> R.  So, is 
> 	there evidence for or against the possibility that
> 
> 		The Gaussian curvature K: M -> R of an 
> 		arbitrary smooth hypersurface 
> 		M^n in R^(n+1) must have at least 2n critical points.
> 	?
> 
> 	Possible cases to consider:	a)	M is convex 
> 					b)	M is topologically S^n 
> 					c)	M is arbitrary.
> 
[The ellipsoid would have 2(n+1) critical points...I am responding to that
conjecture here.] 

I proposed this same problem (in the case of topological spheres) to the crew
here at the Geometry Center.  Ken Brakke found a very neat and simple
counterexample, namely a BANANA!  This has only four critical points.  By
slightly "banana-izing" a sphere, you can also get convex counterexamples
with just four critical points.

Still, I would expect that the minimum number of critical points for the
curvature will always be greater than the trivial minimum of 2.

P.S. Ken won $10 for his ingenuity...
--
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Paul Burchard	<burchard@geom.umn.edu>
``I'm still learning how to count backwards from infinity...''
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