An Equivalence of Scalar Curvatures on Hermitian Manifolds

作者：Dabkowski Michael G^*; Lock Michael T

来源：Journal of Geometric Analysis, 2017, 27(1): 239-270.

DOI：10.1007/s12220-016-9680-4

• Kahler manifolds
• Chern scalar curvature
• Differential geometry

For a Kahler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kahler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu-Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kahler. However, we prove that a certain class of non-compact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.

• 出版日期2017-1