Smooth approximation of Lipschitz projections

TL;DR

This paper demonstrates that Lipschitz projection-valued functions on connected closed Riemannian manifolds can be uniformly approximated by smooth projections with nearly the same Lipschitz constant, addressing a question posed by Rieffel.

Contribution

It provides a method to approximate Lipschitz projections smoothly while preserving Lipschitz bounds, solving an open problem in the field.

Findings

Any Lipschitz projection can be approximated by smooth projections with similar Lipschitz constants.

The approximation is uniform on connected closed Riemannian manifolds.

Answers a previously open question by Rieffel.

Abstract

We show that any Lipschitz projection-valued function p on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions q with Lipschitz constant close to that of p. This answers a question of Rieffel.