School of Mathematics and Statistics - Theses

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A length preserving geometric heat flow for curves

Pihan, Denis M. ( 1998)

This thesis is concerned with the formulation and analysis of a length preserving geometric heat flow for curves, which is the steepest descent flow for the thread problem. The thread problem is the classical problem of minimising the area functional of a surface, subject to the constraint of keeping part of the boundary fixed, while the remainder has some prescribed length. We derive the gradient flow for the thread problem, for the case that the thread has co-dimension one, establish its short-time existence, examine appropriate maximum principles and discuss its properties. Concentrating on the flow of curves in the plane, we establish that closed, convex, embedded curves converge exponentially to a circle, which solves the thread problem and the corresponding isoperimetric problem.