Prescription de courbure des feuilles des laminations: retour sur un th\'eor\`eme de Candel

TL;DR

This paper revisits and generalizes Candel's theorem, showing that for compact hyperbolic laminations, negative functions can be realized as curvature functions of unique laminated metrics, with implications for elliptic PDE solutions.

Contribution

It extends Candel's theorem by proving the existence and uniqueness of laminated metrics with prescribed curvature functions in hyperbolic laminations.

Findings

Every negative smooth function is the curvature of a unique laminated metric.

The result is interpreted as a continuity property of elliptic PDE solutions in the Cheeger-Gromov topology.

The theorem applies to compact laminations by hyperbolic surfaces.

Abstract

In the present paper, we revisit a famous theorem by Candel that we generalize by proving that given a compact lamination by hyperbolic surfaces, every negative function smooth inside the leaves and transversally continuous is the curvature function of a unique laminated metric in the corresponding conformal class. We give an interpretation of this result as a continuity result about the solutions of some elliptic PDEs in the so called Cheeger-Gromov topology on the space of complete pointed riemannian manifolds.