Weak Counterexamples to $L^2$ Curvature Estimates for Minimizing Surfaces

TL;DR

This paper constructs a sequence of smooth minimizing surfaces in metrics approaching Euclidean space, demonstrating that the $L^2$ norm of their second fundamental form can diverge, challenging existing curvature estimates.

Contribution

It provides explicit counterexamples showing divergence of $L^2$ curvature norms in minimizing surfaces under converging metrics.

Findings

Divergence of $L^2$ curvature norms in constructed surfaces

Counterexamples to previous curvature estimates

Convergence of metrics to Euclidean space

Abstract

We construct a sequence of smooth minimizing surfaces in a sequence of metrics converging to the standard Euclidean metric, so that they have diverging $L^2$ norm of second fundamental form.