Conformal metrics with prescribed Gaussian and Geodesic curvatures
Identificadores
URI: https://hdl.handle.net/10481/80126Metadatos
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Société Mathematique de France
Materia
Prescribed curvature problem Conformal metric Variational methods Blow-up analysis
Fecha
2019-01-27Referencia bibliográfica
Published version: López, R., Malchiodi, A. & Ruiz, D. (2022). Conformal metrics with prescribed gaussian and geodesic curvatures. Annales Scientifiques de l'École Normale Supérieure, 5, t. 55. DOI : [10.24033/asens.2516]
Patrocinador
Spanish Government MTM2015-68210-P PGC2018-096422-B-100; project Geometric Variational Problems and Finanziamento a supporto della ricerca di base from Scuola Normale Superiore; Ministry of Education, Universities and Research (MIUR) FQM-116 J. Andalucia 2015KB9WPT001Resumen
We consider the problem of prescribing the Gaussian and
the geodesic curvatures of a compact surface with boundary by a conformal
deformation of the metric. We derive some existence results using a
variational approach, either by minimization of the Euler-Lagrange energy
or via min-max methods. One of the main tools in our approach is
a blow-up analysis of solutions, which in the present setting can have diverging
volume. To our knowledge, this is the first time in which such an
aspect is treated. Key ingredients in our arguments are: a blow-up analysis
around a sequence of points different from local maxima; the use of
holomorphic domain-variations; and Morse-index estimates.