Metrics of constant curvature on a Riemann surface with two corners on the boundary

Jürgen Jost; Guofang Wang; Chunqin Zhou

doi:10.1016/j.anihpc.2007.11.001

JournalsaihpcVol. 26, No. 2pp. 437–456

Metrics of constant curvature on a Riemann surface with two corners on the boundary

Jürgen Jost
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04013, Germany
Guofang Wang
University of Magdeburg, Faculty of Mathematics, P.O. Box 4120, D-39016 Magdeburg, Germany
Chunqin Zhou
Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China

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Abstract

In this paper we classify all solutions of

⎩ ⎨ ⎧ - Δ u = ∣ x ∣^{2 α} e^{u}, \frac{\partial u}{\partial t} = c_{1} ∣ x ∣^{α} e^{\frac{u}{2}}, \frac{\partial u}{\partial t} = c_{2} ∣ x ∣^{α} e^{\frac{u}{2}}, in R_{+}^{2}, on \partial R_{+}^{2} \cap {s > 0}, on \partial R_{+}^{2} \cap {s < 0}

with the finite energy condition

R_{+}^{2} \int ∣ x ∣^{2 α} e^{u} d x < \infty, \partial R_{+}^{2} \int ∣ x ∣^{α} e^{\frac{u}{2}} d s < \infty.

Here $c_{1}$ , $c_{2}$ are constants and $α > - 1$ .

Cite this article

Jürgen Jost, Guofang Wang, Chunqin Zhou, Metrics of constant curvature on a Riemann surface with two corners on the boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 2, pp. 437–456

DOI 10.1016/J.ANIHPC.2007.11.001