Comparison Theorems for 3D Manifolds With Scalar Curvature Bound
Comparison Theorems for 3D Manifolds With Scalar Curvature Bound
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标量曲率有界的 3D 流形的比较定理
DOI：
10.1093/imrn/rnab307
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发表时间：
2021
影响因子：
1
通讯作者：
Wang, Jiaping
中科院分区：
文献类型：
--
作者：
Munteanu, Ovidiu;Wang, Jiaping
对于标量曲率从下有界的三维完全非紧流形，得到了两个明显的比较结果。第一个是格林的功能。当标量曲率非负时，它表明除非流形本身与欧几里得空间等距，否则能级集上的能量衰减率严格小于欧几里得空间的衰减率。当流形的Ricci曲率在无穷远处渐近非负时，结果又转化为格林函数的水平集的锐利面积比较。第二个结果提供了一个标量曲率下界的底谱的明显上界，与Cheng的经典结果相反，后者涉及一个里奇曲率下界。
Two sharp comparison results are derived for 3D complete noncompact manifolds with scalar curvature bounded from below. The 1st one concerns the Green’s function. When the scalar curvature is nonnegative, it states that the rate of decay of an energy quantity over the level set is strictly less than that of the Euclidean space unless the manifold itself is isometric to the Euclidean space. The result is in turn converted into a sharp area comparison for the level set of the Green’s function when in addition the Ricci curvature of the manifold is assumed to be asymptotically nonnegative at infinity. The 2nd result provides a sharp upper bound of the bottom spectrum in terms of the scalar curvature lower bound, in contrast to the classical result of Cheng, which involves a Ricci curvature lower bound.