Conformal Metrics on the Disk with Prescribed Negative Gaussian Curvature and Boundary Geodesic Curvature

TL;DR

This paper investigates the existence of conformal metrics on the disk with prescribed negative Gaussian and boundary geodesic curvatures, addressing bubbling phenomena and establishing compactness results through variational methods.

Contribution

It provides new existence results for negative curvature prescriptions on the disk, analyzing blow-up behavior and compactness under bounded Morse index.

Findings

Existence of conformal metrics with prescribed negative Gaussian and boundary curvatures.

Analysis of bubbling phenomena and blow-up solutions.

Compactness results for solutions with bounded Morse index.

Abstract

We study the problem of prescribing the Gaussian curvature on the disk and the geodesic curvature on its boundary via a conformal change of the metric. In this paper the case of negative Gaussian curvature is treated, a regime for which the bubbling behavior of approximate solutions is not so well understood. This is due to the possible appearance of blow-up solutions with diverging length and area. We give an existence result under assumptions on the curvatures which are somewhat natural, in view of some obstructions inherent to the problem. Our strategy is variational and relies on the study of certain families of approximated problems. By performing a refined blow-up analysis for solutions with bounded Morse index, we conclude compactness.