Michael Gage - Remarks on B. Süssmann's proof of the Banchoff-Pohl inequality

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Michael Gage - Remarks on B. Süssmann's proof of the Banchoff-Pohl inequality

MICHAEL GAGE, University of Rochester, Rochester, New York 14627, USA

Remarks on B. Süssmann's proof of the Banchoff-Pohl inequality

This is an expository talk describing Bernd Suessmann's use of the curve shortening flow to prove the Banchoff-Pohl isoperimetric inequality for non-simple closed curves on simply connected surfaces with Gauss curvature bounded above by a non-positive constant K₀. The inequality is

$\begin{displaymath}L^2 - 4\pi\int_M w(x)^2\, dA(x)+K_0\Bigl(\int_M \vert w(x)\vert\,dA(x)\Bigr)^2 \ge 0 \end{displaymath}$

were L is the length of the curve $\gamma$ and w(x) is the winding number of $\gamma$ about the point x. The idea of the proof is to show that the left hand side cannot be increased under the curve shortening flow. This is sufficient because the curve shortening flow deforms an arbitrary closed curve to a ``circular'' point (the curve may temporarily develop cusps during process) and the left hand side is non-negative for curves near these points.

The inequalities Süssmann derives in order to prove that this quantity decreases under the curve shortening flow are interesting and probably more powerful than the final result.

Next: Miroslav Lovric - Multivariate Up: 1) Differential Geometry and Global Previous: Ailana Fraser - On