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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 *K*_{0}.
The inequality is

were *L* is the length of the curve and *w*(

*x*)

is the winding
number of 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
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