Conformal metrics of the disk with prescribed Gaussian and geodesic curvatures

TL;DR

This paper investigates the existence of conformal metrics on the disk with prescribed Gaussian and geodesic curvatures, extending classical problems and providing existence results using degree theory.

Contribution

It introduces new existence criteria for conformal metrics with specified curvatures on the disk, involving harmonic extensions and degree computations.

Findings

Existence results depend on conditions involving Gaussian and geodesic curvatures.

The role of harmonic extension H is crucial in the existence criteria.

Degree theory is used to establish the existence of solutions.

Abstract

This paper is concerned with the existence of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures. Being more specific, given nonnegative smooth functions $K: \overline{\mathbb{D}} \to \mathbb{R}$ and $h: \partial \mathbb{D} \to \mathbb{R}$, we consider the problem of finding a conformal metric realizing $K$ and $h$ as Gaussian and geodesic curvatures, respectively. This is the natural analogue of the classical Nirenberg problem posed on the disk. As we shall see, both curvatures play a role in the existence of solutions. Indeed we are able to give existence results under conditions that involve $K$ and $H$, where $H$ denotes the harmonic extension of $h$. The proof is based on the computation of the Leray-Schauder degree in a compact setting.