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Prescribing curvatures in the disk via conformal changes of the metric: the case of negative Gaussian curvature

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

TL;DR

This work studies prescribing Gaussian curvature on the disk via conformal deformations in the negative-curvature regime under symmetry constraints. It casts the problem variationally through the energy $\mathcal{I}$ and employs Struwe's monotonicity trick alongside Morse-index estimates to obtain a $G$-symmetric solution via mountain-pass geometry, despite the lack of compactness. A thorough blow-up analysis ties concentration phenomena to boundary Liouville-type limits on $\mathbb{R}^2_{+}$, yielding compactness under symmetry and distinguishing the $G=O(2)$ case from $G=\langle g_k\rangle$; a nonexistence result further demonstrates the sharpness of the assumptions. Overall, the paper advances the understanding of prescribing curvature on disks with negative Gaussian curvature by combining variational methods, blow-up analysis, and symmetry considerations.

Abstract

This paper deals with the question of prescribing the Gaussian curvature on a disk and the geodesic curvature of its boundary by means of a conformal deformation of the metric. We restrict ourselves to a symmetric setting in which the Gaussian curvature is negative, and we are able to give general existence results. Our approach is variational, and solutions will be searched as critical points of an associated functional. The proofs use a perturbation argument via the monotonicity trick of Struwe, together with a blow-up analysis and Morse index estimates. We also give a nonexistence result that shows that, to some extent, the assumptions required for existence are necessary.

Prescribing curvatures in the disk via conformal changes of the metric: the case of negative Gaussian curvature

TL;DR

This work studies prescribing Gaussian curvature on the disk via conformal deformations in the negative-curvature regime under symmetry constraints. It casts the problem variationally through the energy and employs Struwe's monotonicity trick alongside Morse-index estimates to obtain a -symmetric solution via mountain-pass geometry, despite the lack of compactness. A thorough blow-up analysis ties concentration phenomena to boundary Liouville-type limits on , yielding compactness under symmetry and distinguishing the case from ; a nonexistence result further demonstrates the sharpness of the assumptions. Overall, the paper advances the understanding of prescribing curvature on disks with negative Gaussian curvature by combining variational methods, blow-up analysis, and symmetry considerations.

Abstract

This paper deals with the question of prescribing the Gaussian curvature on a disk and the geodesic curvature of its boundary by means of a conformal deformation of the metric. We restrict ourselves to a symmetric setting in which the Gaussian curvature is negative, and we are able to give general existence results. Our approach is variational, and solutions will be searched as critical points of an associated functional. The proofs use a perturbation argument via the monotonicity trick of Struwe, together with a blow-up analysis and Morse index estimates. We also give a nonexistence result that shows that, to some extent, the assumptions required for existence are necessary.
Paper Structure (5 sections, 12 theorems, 133 equations, 4 figures)

This paper contains 5 sections, 12 theorems, 133 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and approximate solutions
  3. Blow-up analysis and proof of Theorem \ref{['thm: existence']}
  4. Proof of Theorem \ref{['thm: nonexistence']}
  5. Tests functions

Key Result

Theorem 1.1

Under hypotheses (eq: hypothesis_1), (eq: hypothesis_2), (eq: hypothesis_3) and (eq: hypothesis_symmetry), the problem (eq: Problem 1) admits a $G$-symmetric solution.

Figures (4)

  • Figure 1: Construction in $\mathbb{H}$.
  • Figure 2: $A_1=\mathbb{D}\cap B_{q_1}(2r)$.
  • Figure 3: $A^+$ and $A_1$.
  • Figure 4: $A_1=\partial\mathbb{D}\cap B_{p_1}(r)$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Proposition \ref{['prop: Mountain-pass']}
  • Proposition 2.4
  • proof
  • ...and 14 more