Conformal metrics with prescribed Gaussian and geodesic curvatures

Abstract

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.

Gauge Theory Formulation of Hyperbolic Gravity

“…Some aspects of the problem have been considered in the mathematical literature. References that we found particularly useful include [18][19][20][21] and references therein. A typical approach is to start from a metric g 0 with geodesic boundaries and to look for a metric g = e 2σ g 0 in the same conformal class having the required extrinsic curvature k i on the i th boundary component C i .…”

“…A very useful recent reference to understand the metrics for which certain boundaries have k > 1 is [20]. The basic idea, first put forward by Rosenberg in [19] (see also [23]) is the following.…”

Large conformal metrics with prescribed Gaussian and geodesic curvatures

“…The case of nonconstant curvatures was addressed for the first time by Cherrier in [7], where the existence of a solution to (2) is proved provided the curvatures are not too big in a reasonable geometric sense. Recently, Lopez-Soriano, Malchiodi and Ruiz in [20] considered surfaces with negative Euler characteristic and negative Gaussian curvature, studied the problem via a variational point of view and obtained solutions to (2) by minimization and min-max techniques.…”

Concentration solutions to singularly prescribed Gaussian and geodesic curvatures problem

“…When K and κ are not constants, Cherrier [8] proved the existence of a solution for not big curvatures. Recently, López-Soriano, Malchiodi and Ruiz [20] considered the negative Gaussian curvature case, i.e K < 0, and they derived some existence results using a variational approach. The higher-dimensional analogue of this question is to prescribe scalar curvature of a manifold and mean curvature of the boundary.…”