Osaka Journal of Mathematics

Rigidity of manifolds with boundary under a lower Ricci curvature bound

Yohei Sakurai

Abstract

We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric neighborhoods of the boundaries. We conclude several rigidity theorems. As one of them, we obtain a volume growth rigidity theorem. We also show a splitting theorem of Cheeger-Gromoll type under the assumption of the existence of a single ray.

Article information

Source
Osaka J. Math., Volume 54, Number 1 (2017), 85-119.

Dates
First available in Project Euclid: 3 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1488531786

Mathematical Reviews number (MathSciNet)
MR3619750

Zentralblatt MATH identifier
1383.53031

Citation

Sakurai, Yohei. Rigidity of manifolds with boundary under a lower Ricci curvature bound. Osaka J. Math. 54 (2017), no. 1, 85--119. https://projecteuclid.org/euclid.ojm/1488531786

References

• A.L. Besse: Einstein Manifolds, Springer-Verlag, New York, 1987.
• R. Bishop and R. Crittenden: Geometry of Manifolds, Academic Press, 1964.
• D. Burago, Y. Burago and S. Ivanov: A Course in Metric Geometry, Graduate Studies in Math. 33, Amer. Math. Soc., 2001.
• E. Calabi: An extention of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1957), 45–56.
• I. Chavel: Eigenvalues in Riemannian Geometry, Academic Press, 1984.
• J. Cheeger: A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis, a symposium in honor of S. Bochner, Princeton University Press, Princeton, 1970, 195–199.
• J. Cheeger and T.H. Colding: Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. 144 (1996), 189–237.
• J. Cheeger and D. Gromoll: The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971), 119–128.
• C. Croke and B. Kleiner: A warped product splitting theorem, Duke Math. J. 67 (1992), 571–574.
• M.P. do Carmo and C. Xia: Rigidity theorems for manifolds with boundary and nonnegative curvature, Result. Math. 40 (2001), 122–129.
• J. Eschenburg and E. Heintze: An elementary proof of the Cheeger-Gromoll splitting theorem, Ann. Global Anal. Geom. 2 (1984), 141–151.
• H. Federer: Geometric Measure Theory, Springer-Verlag, New York, 1969.
• H. Federer and W.H. Fleming: Normal and integral currents, Ann. of Math. 72 (1960), 458–520.
• J. Ge: Comparison theorems for manifold with mean convex boundary, Comm. Contemp. Math (2014), online.
• D. Gilbarg and N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.
• M. Gromov: Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), 179–195.
• M. Gromov: Structures metriques pour les varieties Riemanniennes, Cedic-Fernand Nathan, Paris, 1981.
• E. Heintze and H. Karcher: A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. Ecole Norm. Sup. 11 (1978), 451–470.
• S. Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces, Academic press, 1978.
• R. Ichida: Riemannian manifolds with compact boundary, Yokohama Math. J. 29 (1981), 169–177.
• A. Kasue: On Laplacian and Hessian comparison theorems, Proc. Japan Acad. 58 (1982), 25–28.
• A. Kasue: A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold, Japanese J. Math. New Series 8 (1982), 309–341.
• A. Kasue: Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary, J. Math. Soc. Japan 35 (1983), 117–131.
• A. Kasue: On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold, Ann. Sci. Ecole Norm. Sup. 17 (1984), 31–44.
• A. Kasue: Applications of Laplacian and Hessian Comparison Theorems, Advanced Studies in Pure Math. 3 (1984), 333–386.
• S. Kawai and N. Nakauchi: The first eigenvalue of the $p$-Laplacian on a compact Riemannian manifold, Nonlinear Anal. 55 (2003), 33–46.
• H. Li and Y. Wei: Rigidity theorems for diameter estimates of compact manifold with boundary, International Mathematics Research Notices (2014), rnu052, 18pages.
• M. Li: A sharp comparison theorem for compact manifolds with mean convex boundary, J. Geom. Anal. 24 (2014), 1490–1496.
• P. Li: Geometric Analysis, Cambridge University Press, 2012.
• P. Li and S.T. Yau: Estimates of eigenvalues of a compact Riemannian manifold, Proc. Symp. Pure Math. 36 (1980), 205–239.
• S. Ohta: On the measure contraction property of metric measure spaces, Comm. Math. Helv. 82 (2007), 805–828.
• S. Ohta: Products, cones, and suspensions of spaces with the measure contraction property, J. Lond. Math. Soc. 76 (2007), 225–236.
• R. Perales: Volumes and limits of manifolds with Ricci curvature and mean curvature bound, arXiv preprint arXiv:1404.0560v3 (2014).
• T. Sakai: Riemannian Geometry, Translations of Mathematical Monographs 149, Amer. Math. Soc, 1996.
• K.-T. Sturm: Diffusion processes and heat kernels on metric spaces, Ann. Prob. 26 (1998), 1–55.
• C. Xia: Rigidity of compact manifolds with boundary and nonnegative Ricci curvature, Proc. Amer. Math. Soc. 125 (1997), 1801–1806.
• H. Zhang: Lower bounds for the first eigenvalue of the $p$-Laplace operator on compact manifolds with positive Ricci curvature, Nonlinear. Anal. 67 (2007), 795–802.
• H. Zhang: Lower bounds for the first eigenvalue of the $p$-Laplace operator on compact manifolds with nonnegative Ricci curvature, Adv. Geom. 7 (2007), 145–155.