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 injectivity 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.

中文翻译：

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黎曼曲面之间的$ p $调和映射的Radó–Kneser–Choquet定理

在平面环境中，Radó–Kneser–Choquet定理指出，假设边界映射是同胚的，则从单位圆盘到由凸曲线定界的Jordan域上的谐波映射是微分同构的。我们证明了黎曼曲面之间的$ p $谐波映射的Radó–Kneser–Choquet的内射性准则。

在我们的内射性标准证明中，我们用求解均匀椭圆系统的辅助映射来逼近$ p $-调和图。我们通过同伦论证证明每个辅助映射都有一个正雅可比矩阵。我们借助最小原理，针对与雅可比行列式相关的某些次谐波表达，通过同构性将地图始终保持内射性。