C^1-regularity for local graph representations of immersions

TL;DR

This paper proves that immersions with uniform local graph representations become smooth with small C^1-norm if their initial C^0-norm is sufficiently small, ensuring regularity under certain conditions.

Contribution

It establishes C^1-regularity for local graph representations of immersions based on small C^0-norm conditions, advancing understanding of immersion regularity.

Findings

Graph functions become smooth with small C^1-norm

Regularity depends on small initial C^0-norm

Results apply to immersions with uniform local graph representations

Abstract

We consider immersions admitting uniform graph representations over the affine tangent space over a ball of fixed radius r>0. We show that for sufficiently small C^0-norm of the graph functions, each graph function is smooth with small C^1-norm.