Radius Estimates for Alexandrov Space with Boundary


In this note, we study the radius of a positively curved or nonnegatively curved Alexandrov space with strictly convex boundary, the convexity of which is measured by the Base-Angle defined by Alexander and Bishop. As an application of the radius estimates, we derive several rigidity theorems, which can be thought as extensions of a recent result of Grove–Petersen.

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  1. 1.

    Alexander, S.B., Bishop, R.L.: Extrinsic curvature of semiconvex subspaces in Alexandrov geometry. Ann. Glob. Anal. Geom. 37(3), 241–262 (2010)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Burago, Y., Gromov, M., Perelman, G.: Aleksandrov spaces with curvatures bounded below. Uspekhi Mat. Nauk 47(284), 3–51 (1992)

    MathSciNet  Google Scholar 

  3. 3.

    Ge, J.: Comparison theorems for manifolds with mean convex boundary. Commun. Contemp. Math. 17(5), 1550010 (2015). 12

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ge, J.: Fillings of positively curved Alexandrov spaces. Preprint (2018)

  5. 5.

    Grove, K., Petersen, P.: A lens rigidity theorem in Alexandrov geometry. Preprint (2018)

  6. 6.

    Li, M.M.: A sharp comparison theorem for compact manifolds with mean convex boundary. J. Geom. Anal. 24(3), 1490–1496 (2014)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Petrunin, A.: Semiconcave Functions in Alexandrov’s Geometry. Surveys in Differential Geometry, vol. 11, pp. 137–201. International Press, Somerville, MA (2007)

    Google Scholar 

  8. 8.

    Shi, Y., Tam, L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62(1), 79–125 (2002)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Shi, Y., Tam, L.-F.: Scalar curvature and singular metrics. Pac. J. Math. 293(2), 427–470 (2018)

    MathSciNet  Article  Google Scholar 

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The authors would like to thank Stephanie Alexander and Yuguang Shi for their interest in our work and helpful discussions. The first author would like to thank Karsten Grove for communications and discussions on the radius estimates. The authors would like to thank the anonymous referee for careful reading of our manuscript and several helpful suggestions.

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Correspondence to Jian Ge.

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J. Ge is partially supported by NSFC 11731001.

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Ge, J., Li, R. Radius Estimates for Alexandrov Space with Boundary. J Geom Anal 31, 619–630 (2021). https://doi.org/10.1007/s12220-019-00292-2

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  • Alexandrov space
  • Riemannian manifold
  • Radius
  • Rigidity

Mathematics Subject Classification

  • Primary 53C23
  • 53C20