Abstract
For a Kähler 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-Kähler 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 Kähler. 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.
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Acknowledgments
The authors would like to thank Jeff Viaclovsky for his insight into this problem and many useful comments, as well as Ethan Leeman and Matthew Gursky for many valuable discussions. Michael T. Lock was partially supported by NSF Grant DMS-1148490.
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This article is dedicated Konstantin Leyzerovsky on the occasion of his birthday.
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Dabkowski, M.G., Lock, M.T. An Equivalence of Scalar Curvatures on Hermitian Manifolds. J Geom Anal 27, 239–270 (2017). https://doi.org/10.1007/s12220-016-9680-4
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DOI: https://doi.org/10.1007/s12220-016-9680-4