Lifespan theorem for constrained surface diffusion flows

TL;DR

This paper establishes a lifespan theorem for constrained surface diffusion flows of hypersurfaces in three and four dimensions, providing bounds on existence time and curvature concentration, applicable to general initial data.

Contribution

The paper extends lifespan results to constrained surface diffusion flows, generalizing previous work on Willmore flow to broader initial conditions.

Findings

Positive lower bound on smooth solution existence time.

Upper bound on total curvature during the flow.

Applicability to general initial hypersurfaces.

Abstract

We consider closed immersed hypersurfaces in $\R^{3}$ and $\R^4$ evolving by a class of constrained surface diffusion flows. Our result, similar to earlier results for the Willmore flow, gives both a positive lower bound on the time for which a smooth solution exists, and a small upper bound on a power of the total curvature during this time. By phrasing the theorem in terms of the concentration of curvature in the initial surface, our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows.