arXiv subjects

Francisco J. Reyes-Sánchez

Publications and source records attributed to Francisco J. Reyes-Sánchez.

2 recordsLinked to original sources

arXiv2026

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

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.

math.AP↗

arXiv2024

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

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.

math.DG↗