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Results in Mathematics

, 74:164 | Cite as

A Variation Problem for Stress–Energy Tensor

  • Yingbo HanEmail author
Article
  • 54 Downloads

Abstract

In this paper, we introduce a functional \(\Phi _S\) related to the stress–energy tensor of a smooth map between two Riemannian manifolds. We derive the first variation formula and the second variation formula of \(\Phi _S\). We also use the stress–energy tensor to obtain some Liouville type results for some special maps. Finally, we obtain that the maps from or into the compact convex hypersurfaces \(M^m\ (m\ge 5)\) of \(R^{m+1}\) which are stable stress–energy stationary maps and satisfy the inequality \(3\lambda _m<\sum _{i=1}^{m-1}\lambda _i\) must be constant.

Keywords

Stress–energy tensor Liouville type results variation formula 

Mathematics Subject Classification

Primary 58E20 58E99 

Notes

Acknowledgements

The author would like to thank the referees whose valuable suggestions make this paper more perfect. This work was written while the author visited Department of Mathematics of the University of Oklahoma in USA. He would like to express his sincere thanks to Professor Shihshu Walter Wei for his help, hospitality and support. This work was also supported by the National Natural Science Foundation of China (Grant No. 11201400, 11971415, 11701494), China Scholarship Council (201508410400), Nanhu Scholars Program for Young Scholars of XYNU, the Universities Young Teachers Program of Henan Province (2016GGJS-096) and Teacher Education Project of XYNU (2019-JSJYYJ-12).

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXinyang Normal UniversityXinyangPeople’s Republic of China

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