# A rigidity theorem for ideal surfaces with flat boundary

## TL;DR

This paper proves that ideal surfaces with flat boundary conditions and small curvature norm must be planar, based on a sixth order nonlinear elliptic PDE related to mean curvature extremization.

## Contribution

It establishes a rigidity theorem for surfaces with flat boundary satisfying a complex nonlinear PDE, showing they are necessarily planar under small curvature conditions.

## Key findings

- Surfaces with flat boundary and small curvature are necessarily planar.
- The result applies to surfaces satisfying a sixth order nonlinear elliptic PDE.
- Small $L^2$-norm of the second fundamental form implies planarity.

## Abstract

We consider surfaces with boundary satisfying a sixth order nonlinear elliptic partial differential equation corresponding to extremising the $L^2$-norm of the gradient of the mean curvature. We show that such surfaces with small $L^2$-norm of the second fundamental form and satisfying so-called `flat boundary conditions' are necessarily planar.