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arXiv · 2006.12900

Large conformal metrics with prescribed Gaussian and geodesic curvatures

Luca Battaglia·Maria Medina·Angela Pistoia
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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.

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Luca Battaglia, Maria Medina, Angela Pistoia. 2020-06-23. Large conformal metrics with prescribed Gaussian and geodesic curvatures. https://arxiv.org/abs/2006.12900

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