A rigidity phenomenon on riemannian manifolds with reverse excess pinching

Abstract

The authors introduce the Hausdorff convergence to discuss the differentiable sphere theorem with excess pinching. Finally, a type of rigidity phenomenon on Riemannian manifolds is derived.

Keywords

Volume comparison theorem / Hausdorff convergence / Differentiable sphere theorem / Harmonic coordinate / Harmonic radius

References

[1] Cheeger J., Ebin D. G.. Comparison Theorems in Riemannian Geometry, 1975, Amesterdan: North-Holland

[2] Coghlan L., Itokawa Y.. A sphere theorem for reverse volume pinching on even-dimension manifolds. Proc. Amer. Math. Soc., 1991, 111(3): 815-819

[3] Wu J. Y.. Hausdorff convergence and sphere theorem. Proc. Sym. Pure Math., 1993, 54: 685-692

[4] Wen Y.. A note on pinching sphere theorem. C. R. Acad. Sci. Paris, Ser. I, 2004, 338: 229-234

[5] Grove K., Shiohama K.. A generalized sphere theorem. Ann. of Math., 1977, 106: 201-211

[6] Pawel K.. On the spectral gap for compact manifolds. J. Diff. Geom., 1992, 36(2): 315-330

[7] Pawel L., Andreis T.. Applications of eigenvalue techniques to geometry, Contemporary Geometry. Univ. Ser. Math., 1991, New York: Plenum 21-52

[8] Sakai T.. Riemannian Geometry, 1992, Tokyo: Shokabo

[9] Shiohama K.. A sphere theorem for manlfolds of positive Ricci curvature. Tans. Amer. Math. Soc., 1983, 27: 811-819

[10] Wu J. Y.. A diameter pinching sphere theorem. Proc. Amer. Math. Soc., 1990, 197(3): 796-802

[11] Yang D. G.. Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature. Pacific J. Math., 1999, 190(2): 383-398

[12] Yang H. C.. Estimates of the first eigenvalue for a compact Riemann manifold. Sci. in Chin., Ser. A, 1990, 33(1): 39-51

[13] Zhong J. Q., Yang H. C.. On the estimates of the first eigenvalue of a compact Riemannian manifold. Sci. Sinica, Ser. A, 1984, 27(12): 1265-1273

[14] Xia C. Y.. Rigidity and sphere theorem for manifolds with positive Ricci curvature. Manuscript Math., 1994, 85: 79-87

[15] Grove K., Petersen V. P.. Manifolds near the boundary of existence. J. Diff. Geom., 1991, 33: 379-394

[16] Peters S.. Convergence of Riemannian manifolds. Compositio Math., 1987, 62: 3-16

[17] Gromov M.. Groups of polynomial growth and expanding maps. Inst. Hautes Etudes Sci. Pub. Math., 1981, 53: 183-215

[18] Gromov M.. Metric Structures for Riemannian and Non-Riemannian Spaces, 1999, Boston: Birkhäuser

[19] Hartman P.. Oscillation criteria for self-adjoint second-oder differential systems and principle sectional curvature. J. Diff. Eqs., 1979, 34: 326-338

[20] Grove K., Wilhelm F.. Hard and soft packing radius theorems. Ann. of Math., 1995, 142(2): 213-237

[21] Hebey E.. Nonlinear Analysisi on Manifolds: Sobolev Space and Inequalities, Courant Lecture Notes in Math., 1998, New York: New York University

[22] Anderson M.. Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math., 1990, 102: 429-445

[23] Anderson M., Cheeger J.. C ^α-compactness for manifolds with Ricci curvature and injective radius bounded below. J. Diff. Geom., 1992, 35(2): 265-281

[24] Cai M.. Rigidity of manifolds with large volume. Math. Z., 1992, 213: 17-31

[25] Rong, X. C., Introductin to the convergence and collapse theory in Riemannian Geometry, Lectures given in Fudan University by Xiaochun Rong, 2007.

[26] Petersen P.. Riemannian Geometry, 1998, New York: Springer

[27] Petersen P.. Convergence theorems in Riemannian geometry. Comparison Geom. MSRI Publ., 1997, 30: 167-202

[28] Tuschmann W.. Smooth diameter and eigenvalue rigidity in positive Ricci curvature. Proc. Amer. Math. Soc., 2001, 30: 303-306

[29] Sakai T.. On continuity of injectivity radius function. Math. J. Okayama Univ., 1983, 25: 91-97

[30] Gilbarg D., Trudinger N. S.. Elliptic partial differential equations of second order. Grundlehoen der Mathematischen Wissenschaften, 1983 Second Edition Berlin: Springer-Verlag

[31] Kazdan J. L.. Yau S. T.. An isoperimetric inequality and Wiedersehen manifolds, Seminar on Differential Geometry. Ann. Math. Stud., 1982, Princeton: Princeton University Press 143-157

[32] Xin Y. L.. Mean curvature flow with convex Gauss image. Chin. Ann. Math., 2008, 29B(2): 121-134

[33] Xin Y. L.. Ricci curvature and fundamental group. Chin. Ann. Math., 2006, 27B(2): 121-142

[34] Wang P.. A gap phenomenon on Riemannian manifolds with reverse volume pinching. Acta Math. Hungarica, 2007, 115(1–2): 133-144

[35] Wang P.. A differential sphere theorem on manifolds with reverse volume pinching. Acta Math. Hungarica, 2008, 119(1–2): 63-69

[36] Wang P., Wen Y.. A rigidity phenomenon on Riemannian manifolds with reverse volume pinching. Ann. Glob. Anal. Geo., 2008, 34: 69-76

[37] Wang, P. and Wen, Y., A differentiable sphere theorem with positive Ricci curvature and reverse volume pinching, J. Math. Anal. Appl., to appear.