Previous |  Up |  Next


MSC: 52A38, 53C20, 53C21
Full entry | Fulltext not available (moving wall 24 months)      Feedback
spherically symmetric manifolds; radial Ricci curvature; radial sectional curvature; volume comparison
In this paper, for complete Riemannian manifolds with radial Ricci or sectional curvature bounded from below or above, respectively, with respect to some point, we prove several volume comparison theorems, which can be seen as extensions of already existing results. In fact, under this radial curvature assumption, the model space is the spherically symmetric manifold, which is also called the generalized space form, determined by the bound of the radial curvature, and moreover, volume comparisons are made between annulus or geodesic balls on the original manifold and those on the model space.
[1] Abresch, U.: Lower curvature bounds, Toponogov's theorem, and bounded topology. Ann. Sci. Éc. Norm. Supér. (4) 18 (1985), 651-670. MR 0839689
[2] Barroso, C. S., Bessa, G. Pacelli: Lower bounds for the first Laplacian eigenvalue of geodesic balls of spherically symmetric manifolds. Int. J. Appl. Math. Stat. 6 (2006), Article No. D06, 82-86. MR 2338140
[3] Chavel, I.: Eigenvalues in Riemannian Geometry. Pure and Applied Mathematics 115 Academic Press, Orlando (1984). MR 0768584
[4] Cheeger, J., Yau, S. T.: A lower bound for the heat kernel. Commun. Pure Appl. Math. 34 (1981), 465-480. DOI 10.1002/cpa.3160340404 | MR 0615626
[5] Cheng, S.-Y.: Eigenfunctions and eigenvalues of Laplacian. Differential Geometry Proc. Sympos. Pure Math. 27, Stanford Univ., Stanford, Calif., 1973, Part 2 Amer. Math. Soc., Providence (1975), 185-193 S. S. Chern et al. MR 0378003
[6] Cheng, S.-Y.: Eigenvalue comparison theorems and its geometric applications. Math. Z. 143 (1975), 289-297. DOI 10.1007/BF01214381 | MR 0378001
[7] Elerath, D.: An improved Toponogov comparison theorem for nonnegatively curved manifolds. J. Differ. Geom. 15 (1980), 187-216. MR 0614366
[8] Freitas, P., Mao, J., Salavessa, I.: Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds. Calc. Var. Partial Differ. Equ. 51 (2014), 701-724. DOI 10.1007/s00526-013-0692-7 | MR 3268868
[9] Gage, M. E.: Upper bounds for the first eigenvalue of the Laplace-Beltrami operator. Indiana Univ. Math. J. 29 (1980), 897-912. DOI 10.1512/iumj.1980.29.29061 | MR 0589652
[10] Hu, Z., Jin, Y., Xu, S.: A volume comparison estimate with radially symmetric Ricci curvature lower bound and its applications. Int. J. Math. Math. Sci. 2010 (2010), Article ID 758531, 14 pages. MR 2629591
[11] Itokawa, Y., Machigashira, Y., Shiohama, K.: Maximal diameter theorems for manifolds with restricted radial curvature. Proc. of the 5th Pacific Rim Geometry Conf., Tôhoku University, Sendai, Japan, 2000 Tohoku Math. Publ. 20 Tôhoku University, Sendai (2001), 61-68 S. Nishikawa. MR 1864887
[12] Kasue, A.: A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold. Jap. J. Math., New Ser. 8 (1982), 309-341. MR 0722530
[13] Katz, N. N., Kondo, K.: Generalized space forms. Trans. Am. Math. Soc. 354 (2002), 2279-2284. DOI 10.1090/S0002-9947-02-02966-5 | MR 1885652
[14] Klingenberg, W.: Contributions to Riemannian geometry in the large. Ann. Math. (2) 69 (1959), 654-666. DOI 10.2307/1970029 | MR 0105709
[15] Mao, J.: Eigenvalue inequalities for the {$p$}-Laplacian on a Riemannian manifold and estimates for the heat kernel. J. Math. Pures Appl. (9) 101 (2014), 372-393. DOI 10.1016/j.matpur.2013.06.006 | MR 3168915
[16] Mao, J.: Eigenvalue Estimation and some Results on Finite Topological Type. Ph.D. thesis Mathematics department of Instituto Superior Técnico-Universidade Técnica de Lisboa (2013).
[17] Petersen, P.: Riemannian Geometry. Graduate Texts in Mathematics 171 Springer, New York (2006). MR 2243772 | Zbl 1220.53002
[18] Shiohama, K.: Comparison theorems for manifolds with radial curvature bounded below. Differential Geometry, Proc. of the First International Symposium, Josai University, Saitama, Japan, 2001, Josai Math. Monogr. 3 Josai University, Graduate School of Science, Saitama (2001), 81-91 Q.-M. Cheng. MR 1824602
[19] Zhu, S.: The comparison geometry of Ricci curvature. Comparison Geometry, Berkeley, 1993-1994 Math. Sci. Res. Inst. Publ. 30 Cambridge University, Cambridge (1997), 221-262 K. Grove et al. MR 1452876
Partner of
EuDML logo