论文标题
与边界紧凑的歧管上的曲率的存在和障碍物
Existence and obstructions for the curvature on compact manifolds with boundary
论文作者
论文摘要
我们研究了具有边界可以拥有的给定紧凑型歧管的一组曲率函数。首先，我们证明高斯 - 基因定理所需的符号是给定函数作为地理曲率或某些同类等效度量的高斯曲率的必要条件。我们的方法使我们能够解决在偶然的共形案例中无法解决的问题。此外，我们获得了有关连形变形的深度，更精致的信息。我们证明了在共形环境中具有规定曲率的指标的新存在和不存在的结果，这取决于欧拉的特征。
We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the geodesic curvature or the Gaussian curvature of some conformally equivalent metric. Our approach allows us to solve problems that are impossible to solve in the pointwise conformal case. Moreover, we obtain a deep and more delicate information on pointwise conformal deformations. We prove new existence and nonexistence results for metrics with prescribed curvature in the conformal setting, which depend on the Euler characteristic.
