Every conformal class contains a metric of bounded geometry

TL;DR

This paper proves that every conformal class of semi-Riemannian metrics on a manifold contains a metric with uniformly bounded curvature derivatives, and extends this to foliated manifolds and other geometric structures.

Contribution

It introduces a method to find metrics within any conformal class with controlled curvature derivatives and geometric properties, including completeness and injectivity radius.

Findings

Existence of metrics with bounded curvature derivatives in every conformal class.

Extension of results to foliated manifolds with controlled leaf geometry.

Framework for applying similar techniques to other geometric structures.

Abstract

We show that on every manifold, every conformal class of semi-Riemannian metrics contains a metric $g$ such that each $k$-th-order covariant derivative of the Riemann tensor of $g$ has bounded absolute value $a_k$. This result is new also in the Riemannian case, where one can arrange in addition that $g$ is complete with injectivity and convexity radius greater than 1. One can even make the radii rapidly increasing and the functions $a_k$ rapidly decreasing at infinity. We prove generalizations to foliated manifolds, where curvature, second fundamental form and injectivity radius of the leaves can be controlled similarly. Moreover, we explain a general principle that can be used to obtain analogous results for Riemannian manifolds equipped with arbitrary other additional geometric structures instead of foliations.