有界紧致流形上规定量曲率和中曲率的三分性定理
Trichotomy Theorem for Prescribed Scalar and Mean Curvatures on Compact Manifolds with Boundaries
摘要：
在这篇文章中，我们给出了规定的标量和中曲率函数的度量逐点共形或共形等价的黎曼度量，配备在一个紧流形上的边界，与尺寸至少$ 3 $。根据共形拉普拉斯算子第一特征值的符号对结果进行了分类。这导致了一个“三分定理”的标量和中曲率函数，这是一个完整的扩展“三分定理”给出的卡兹丹和华纳。在欧拉示性数为负的条件下，我们还讨论了边界为点态共形或共形等价于原度量的紧致Riemann曲面上的Gauss曲率和测地曲率问题.其关键步骤是构造一种处理边界条件上零阶非线性项的单调迭代格式。
英文摘要：
In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with dimensions at least $ 3 $. The results are classified by the sign of the first eigenvalue of the conformal Laplacian. This leads to a "Trichotomy Theorem" in terms of both scalar and mean curvature functions, which is a full extension of the "Trichotomy Theorem" given by Kazdan and Warner. We also discuss prescribing Gauss and geodesic curvature problems on compact Riemann surfaces with boundary for metrics either pointwise conformal or conformally equivalent to the original metric, provided that the Euler characteristic is negative. The key step is a general version of monotone iteration scheme which handle the zeroth order nonlinear term on the boundary conditions.
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