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A note on Hang-Wang’s hemisphere rigidity theorem

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Abstract

Let (M, g) be a compact manifold with boundary and \(Ric_g\ge (n-1)g\), Hang and Wang proved that (M, g) is isometric to the standard hemisphere if \(\partial M\) is convex and isometric to \({\mathbb {S}}^{n-1}(1)\). We prove some rigidity theorems when \(\partial M \) is isometric to a product manifold where one factor is the standard sphere.

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References

  1. Brendle, S.: Rigidity phenomena involving scalar curvature. Surv. Differ. Geom. 17, 179–202 (2012)

    Article  MathSciNet  Google Scholar 

  2. Brendle, S., Marques, F.C., Neves, A.: Deformations of the hemisphere that increase scalar curvature. Invent. Math. 185, 175–197 (2011)

    Article  MathSciNet  Google Scholar 

  3. Chen, X., Lai, M., Wang, F.: The Obata equation with Robin boundary condition, arXiv:1901.02206

  4. Ge, J.: Problems related to isoparametric theory, arXiv:1910.12229

  5. Hang, F., Wang, X.: Rigidity theorems for compact manifolds with boundary and positive Ricci curvature. J. Geom. Anal. 19(3), 628–642 (2009)

    Article  MathSciNet  Google Scholar 

  6. Miao, P., Wang, X.: Boundary effect of Ricci curvature. J. Differ. Geom. 103(1), 59–82 (2016)

    Article  MathSciNet  Google Scholar 

  7. Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14, 333–340 (1962)

    Article  MathSciNet  Google Scholar 

  8. Reilly, R.: Applications of the Hessian operator in a Riemannian manifold. Indiana Univ. Math. J. 26(3), 459–472 (1977)

    Article  MathSciNet  Google Scholar 

  9. Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65, 45–76 (1979)

    Article  MathSciNet  Google Scholar 

  10. Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971)

    Article  MathSciNet  Google Scholar 

  11. Witten, E.: A new proof of the positive energy theorem. Comm. Math. Phys. 80, 381–402 (1981)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China

    Mijia Lai

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  1. Mijia Lai
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Correspondence to Mijia Lai.

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Lai’s research is supported in part by National Natural Science Foundation of China No. 11871331.

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Lai, M. A note on Hang-Wang’s hemisphere rigidity theorem. Math. Z. 296, 901–909 (2020). https://doi.org/10.1007/s00209-020-02469-w

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