Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate
Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate
复制标题
DOI：
10.1002/cpa.3160430703
复制
发表时间：
1990-10
影响因子：
3
通讯作者：
José F. Escobar
中科院分区：
文献类型：
--
作者：
José F. Escobar
设（M ', go）为边界维数为n>= 3的紧致黎曼流形。本文研究了在M的边界aM上具有恒定平均曲率的常数标量曲率与g共形相关的度量空间，其中最重要的情况是当M是n维欧氏空间中的球时。在这种情况下，我们完全确定了这个度量空间。事实上，我们证明了如果球B上的度规g ={x E R ': 1x1 51)与具有恒定标量曲率R的欧几里得度规保形相关，并且在aB上具有恒定的平均曲率h，则g具有恒定的截面曲率。此外，在莫比乌斯变换和常数范围内，g是半径为r的测地线球在S ‘ ’上的度规，是半径为r ' ' + l的球面上的诱导度规，r ‘ ’是欧几里德度规或双曲空间W ‘ ’上的度规。这三种情况是根据标量曲率R的符号是正、零还是负来区分的。半径r由平均曲率h决定。
Let (M", go) be a compact Riemannian manifold with boundary of dimension n>= 3. In this paper we study the space of metrics conformally related to g,, of constant scalar curvature, with constant mean curvature on aM, the boundary of M. The most important case to be considered is when M is the ball in n-dimensional Euclidean space. In this case, we identify completely this space of metrics. In fact we prove that if a metric g on the ball B,={x E R": 1x1 5 1) is conformally related to the Euclidean metric with constant scalar curvature R,, and has constant mean curvature h, on aB,, then g has constant sectional curvature. Moreover, up to a Mobius transformation and up to a constant, g is the metric on a geodesic ball of radius r on S", the sphere in R"+ l with the induced metric, R" with the Euclidean metric or else hyperbolic space W". The three cases are distinguished according to whether the sign of the scalar curvature R, is positive, zero, or negative, respectively. The radius r is determined by the mean curvature h,.