黎曼曲面间$p$ -调和映射的Radó-Kneser-Choquet定理
The Radó-Kneser-Choquet theorem for $p$-harmonic mappings between Riemannian surfaces
摘要：
在平面上，Radó-Kneser-Choquet定理指出，从单位圆盘到被凸曲线包围的Jordan域的调和映射是一个自同态，只要边界映射是同胚。证明了黎曼曲面间$p$ -调和映射的Radó-Kneser-Choquet内射性准则.在证明单射性准则时，我们用辅助映射逼近解一致椭圆方程组的$p$ -调和映射。我们证明了每个辅助映射有一个积极的雅可比矩阵的同伦论证。对于与雅可比矩阵相关的某个次调和表达式，借助最小值原理，我们使映射在同伦中一直保持单射。
英文摘要：
In the planar setting the Radó-Kneser-Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Radó-Kneser-Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injecticity criterion we approximate the $p$-harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expression that is related to the Jacobian.
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