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

Rafael López-Soriano, Francisco J. Reyes-Sánchez, David Ruiz

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.

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

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.
Paper Structure (10 sections, 20 theorems, 159 equations, 1 figure)

This paper contains 10 sections, 20 theorems, 159 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the Main Results and Preliminaries
  3. The Perturbation Framework and Variational Settings
  4. Analytic tools
  5. Existence of solutions of the perturbed problems
  6. Mountain-pass geometry for $\boldsymbol{\varepsilon>0}$
  7. Higher dimensional linking geometry for $\boldsymbol{\varepsilon<0}$
  8. Existence of a Critical Value
  9. Blow-up analysis
  10. Proof of the Main Theorems

Key Result

Theorem 1.1

Assume eq:assumption_H, and suppose that $(a)$ and $(b^-)$ or $(b^+)$ are satisfied, where: Then problem eq:Problem admits at least one solution.

Figures (1)

  • Figure 1: The path $\gamma=\gamma_1+\gamma_2+\gamma_3$ for $p\in\mathbb{S}^1$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3: Lebedev-Milin Inequality, OsgoodPhillipsSarnak1988
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 26 more