Evolution of curves on a surface driven by the geodesic curvature and external force

Abstract

We study a flow of closed curves on a given graph surface driven by the geodesic curvature and external force. Using vertical projection of surface curves to the plane we show how the geodesic curvature-driven flow can be reduced to a solution of a fully nonlinear system of parabolic differential equations. We show that the flow of surface curves is gradient-like, i.e. there exists a Lyapunov functional nonincreasing along trajectories. Special attention is placed on the analysis of closed stationary surface curves. We present sufficient conditions for their dynamic stability. Several computational examples of evolution of surface curves driven by the geodesic curvature and external force on various surfaces are presented in this article. We also discuss a link between the geodesic flow and the edge detection problem arising from the image segmentation theory.

Keywords:

• Geodesic curvature
• External force
• Flow of surface curves
• Linearized stability
• Lyapunov functional
• Closed geodesic curve

Keywords:

• AMS Classifications: 35K65
• 35B35
• 35K55
• 53C44

Acknowledgements

This work was supported by VEGA grants 1/0313/03, 1/0259/03 and APVT-20-040902 grant. The authors are also thankful to the Stefan Banach International Mathematical Center – Center of Excellence, Institute of Mathematics PAN in Warsaw and ICM, Warsaw University, where a substantial part of the article was finalized and numerical experiments were completed.