Flows
The 1-dimensional model: curves moving by curvature become round.
Concept
The one-dimensional model, and the animation on the home page. Move a plane curve along its normal with speed equal to its curvature:
Gage–Hamilton (1986) showed convex curves shrink to round points. Grayson (1987) showed every embedded curve becomes convex first. The pattern is the same one you see in Ricci flow: the flow smooths, rounds out, and then becomes extinct.
On arXiv
We study the spectral stability of Dirichlet eigenvalues on an embedded annulus whose boundary evolves by curve shortening flow while the ambient surface evolves under the two dimensional Ricci flow using variational formulas, Rellich–type identities, and harmonic capacity methods, we relate eigenvalue variations to geometric deficit and modulus. We establish quantitative bounds comparing the spectrum of the evolving annulus with that of a flat cylinder of equal modulus. As a consequence, we obtain geometric stability and a spectral gap estimate controlled by the deficit functional.