Large conformal metrics with prescribed Gaussian and geodesic curvatures

Abstract

We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on the unit disk. This is equivalent to solving the following P.D.E.where K, h are the prescribed curvatures. We construct a family of conformal metrics with curvatures Kε, hε converging to K, h respectively as ε goes to 0, which blows up at one boundary point under some generic assumptions.

Concentration solutions to singularly prescribed Gaussian and geodesic curvatures problem

“…then the blow-up must occur at a critical point of ϕ(ξ) = H(ξ) + H(ξ) 2 + K(ξ), where H is the harmonic extension of κ. When finishing our writing the other day, we find that Battaglia-Medina-Pistoia [3] considered the inverse problem to [16]. They constructed a family of solutions with convergent curvatures K ε → K and κ ε → κ as ε goes to 0, which blows up at one boundary critical point of ϕ(ξ) under some non-degeneracy conditions.…”

“…Note that ε 2 K(x) and εκ(x) stand for the prescribed Gaussian curvature and the prescribed geodesic curvature of the boundary. We mention that the problem (2) is the perturbation case of zero curvatures, which do not be included in [3] and [16]. Note that the universal constant 1 is the original geodesic curvature of the boundary ∂D.…”