Gap phenomena for a class of fourth-order geometric differential operators on surfaces with boundary

TL;DR

This paper proves a gap phenomenon for immersed surfaces satisfying certain fourth-order geometric PDEs with boundary conditions, showing small curvature implies the surface is a union of spheres, planes, or both.

Contribution

It establishes new gap results for a broad class of fourth-order geometric operators on surfaces with boundary, including Willmore and biharmonic surfaces, under various boundary conditions.

Findings

Small L^2-norm of curvature implies the surface is totally umbilic.

Under flat boundary conditions, only planes are possible.

Results hold without symmetry assumptions, for surfaces of any genus or boundary shape.

Abstract

In this paper we establish a gap phenomenon for immersed surfaces with arbitrary codimension, topology and boundaries that satisfy one of a family of systems of fourth-order anisotropic geometric partial differential equations. Examples include Willmore surfaces, stationary solitons for the surface diffusion flow, and biharmonic immersed surfaces in the sense of Chen. On the boundary we enforce either umbilic or flat boundary conditions: that the tracefree second fundamental form and its derivative or the full second fundamental form and its derivative vanish. For the umbilic boundary condition we prove that any surface with small L^2-norm of the tracefree second fundamental form or full second fundamental form must be totally umbilic; that is, a union of pieces of round spheres and flat planes. We prove that the stricter smallness condition allows consideration for a broader range of differential operators. For the flat boundary condition we prove the same result with weaker hypotheses, allowing more general operators, and a stronger conclusion: only pieces of planes are allowed. The method used relies only on the smallness assumption and thus holds without requiring the imposition of additional symmetries. The result holds in the class of surfaces with any genus and irrespective of the number or shape of the boundaries.