论文标题
在带有Dirichlet边界条件的Riemannian度量标准的共形变形下，规定的标量曲率问题
Prescribed Scalar Curvature Problem under Conformal Deformation of A Riemannian Metric with Dirichlet Boundary Condition
论文作者
论文摘要
在本文中，我们首先表明，对于所有紧凑的riemannian流形，具有非空的平稳边界和尺寸至少3个，存在一个与原始度量标准的指标，指标，持续的标态曲率在内部持续的标态曲率，并且在边界上与边界上的边界相关的范围至少是一系列cur的一系列cur，在边界上均具有量表。侧面的保形变形。这些类型的结果既是类比，也是对不同类型的歧管的卡兹丹和华纳的“三分法定理”的扩展。这些问题的关键步骤是获得具有Dirichlet边界条件的Yamabe方程的正平滑解。
In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the interior, and constant scalar curvature on the boundary by considering the boundary as a manifold of its own with dimension at least 2. We then show a series of prescribed scalar curvature results in the interior and on the boundary, with pointwise conformal deformation. These type of results is both an analogy and an extension of Kazdan and Warner's "Trichotomy Theorem" on a different type of manifolds. The key step of these problems is to obtain a positive, smooth solution of a Yamabe equation with Dirichlet boundary conditions.
