Seminar on low-dimensional mathematics
Monday, June 19, 2006

"The Full Converse to the Four Vertex Theorem"
H.Gluck (UPenn)

In this talk I will describe Bjorn Dahlberg's proof of the full converse to the
Four Vertex Theorem: any continuous real-valued function on the circle which
is either constant or has at least two local maxima and two local minima
is the curvature function of a simple closed curve in the plane.

The necessity of this condition was proved by Syamadas Mukhopadhyaya in 1909 for strictly
positive curvature, and by Adolf Kneser in 1912 in the general case.

I proved the sufficiency for strictly positive curvature in 1971, as a special case of a
theorem about the existence of n-spheres in n+1-space with preassigned strictly positive
Gaussian curvature.  Dahlberg proved the sufficiency in general in 1997, but died in
January 1998.  His paper was recovered by his students after his death, edited, and only
appeared last year.

In the talk I will review this background, and then focus on the new ideas introduced by
Dahlberg.

This talk represents joint work with Dennis DeTurck, Dan Pomerleano and Shea Vick.

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http://www.pdmi.ras.ru/~lowdimma