Large conformal metrics with prescribed Gaussian and geodesic curvatures

TL;DR

This paper constructs a family of conformal metrics on the unit disk with prescribed Gaussian and geodesic curvatures that blow up at a boundary point, advancing understanding of boundary curvature problems in differential geometry.

Contribution

It introduces a method to produce conformal metrics with prescribed curvatures that exhibit boundary blow-up behavior under generic conditions.

Findings

Constructed conformal metrics with boundary blow-up

Curvatures converge to prescribed functions

Provides a new approach to boundary curvature problems

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. \begin{equation*}\begin{cases}-\Delta u=2K(z)e^u&\hbox{in}\;\mathbb{D}^2,\\ \partial_\nu u+2=2h(z)e^\frac u2&\hbox{on}\;\partial\mathbb{D}^2,\end{cases} \end{equation*} where $K,h$ are the prescribed curvatures. We construct a family of conformal metrics with curvatures $K_\varepsilon,h_\varepsilon$ converging to $K,h$ respectively as $\varepsilon$ goes to $0$, which blows up at one boundary point under some generic assumptions.