On stationary solutions to the non-vacuum Einstein field equations

  • Bing-Long ChenEmail author


We derive a local curvature estimate for four-dimensional stationary solutions to the Einstein equations coupled with electro-magnetic fields or scalar fields. In particular, it implies that any such stationary geodesically complete solution with vanishing Poynting vector and proper coupling constants is flat. We also generalize the results in static case to higher dimensions.

Mathematics Subject Classification

Primary 53c50 Secondary 83c20 



The author is grateful to Professors S. T. Yau, X. P. Zhu and Dr. J. B. Li for helpful discussions. The work was partially supported by grants NSFC11521101, 11025107.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsSun Yat-sen UniversityGuangzhouPeople’s Republic of China

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