Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-18T09:22:43.967Z Has data issue: false hasContentIssue false

A convergence theorem for Riemannian manifolds and some applications

Published online by Cambridge University Press:  22 January 2016

Atsushi Kasue*
Affiliation:
Department of Mathematics, Osaka University, Toyonaka, Osaka 560, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of the present paper is first to reformulate a Lipschitz convergence theorem for Riemannian manifolds originally introduced by Gromov [17] and secondly to give some applications of the theorem to a class of open Riemannian manifolds.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

[1] Anderson, M. T., The compactification of a minimal submanifold in Euclidean space by the Gauss map, preprint.Google Scholar
[2] Ballmann, W., Gromov, M. and Schroeder, V., Manifolds of Nonpositive Curvature, Progress in Math., 61, Birkhäuser, Boston-Basel-Stuttgart, 1985.Google Scholar
[3] Brittain, D. L., A diameter pinching theorem for positive Ricci curvature, preprint.Google Scholar
[4] Cheeger, J. and Ebin, D. G., Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam-Oxford-New York, 1975.Google Scholar
[5] Cheeger, J. and Gromoll, D., The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom., 6 (1971), 119128.Google Scholar
[6] Cheeger, J. and Gromoll, D., On the structure of complete manifolds of nonnegative curvature, Ann. of Math., 96 (1974), 413443.Google Scholar
[7] Chern, S.-S., On the curvature integra in a Riemannian manifold, Ann. of Math., 46 (1945), 674684.Google Scholar
[8] Cohn-Vossen, S., Kurzeste Wege und Totalkrümung auf Flächen, Compositio Math., 2 (1935), 69133.Google Scholar
[9] Croke, C. B., Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Éc. Norm. Sup., Paris, 13 (1980), 419435.Google Scholar
[10] Fiala, F., Le problème des isopérimètres sur les surfaces ouvertes à courbure positive, Comment. Math. Helv., 13 (1940/41), 293346.CrossRefGoogle Scholar
[11] Finn, M., On a class of conformal metrics, with application to differential geometry in the large, Comment. Math. Helv., 40 (1965), 130.Google Scholar
[12] Fukaya, K., On a compactification of the set of Riemannian manifolds with bounded curvature and diameters, Curvature and Topology of Riemannian Manifolds, Lecture Notes in Math., 1201, Springer-Verlag, 1986.Google Scholar
[13] Greene, R. E. and Wu, H., C°° convex functions and manifolds of positive curvature, Acta Math., 137 (1976), 209245.CrossRefGoogle Scholar
[14] Greene, R. E. and Wu, H., Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Math., 699, Springer-Verlag, 1979.Google Scholar
[15] Greene, R. E. and Wu, H., Gap Theorems for noncompact Riemannian manifolds, Duke Math. J., 49 (1982), 731756.Google Scholar
[16] Greene, R. E. and Wu, H., Lipschitz convergence of Riemannian manifolds, Pacific J. Math., 131 (1988), 119141.Google Scholar
[17] Gromov, M., Structures métrique pour les variétés riemanniennes, redige par J. Lafontaine et P. Pansu, Textes Math. No. 1, Edic/Fernand Nathan, Paris, 1981.Google Scholar
[18] Huber, A., On the isoperimetric inequality on surfaces of variable Gaussian curvature, Ann. of Math., 60 (1954), 237247.Google Scholar
[19] Huber, A., On subharmonic functions and differential geometry in the large, Comment. Math. Helv., 32 (1957), 1372.Google Scholar
[20] Huber, A., Métrique conformes complètes et singularités isolées de fonctions sousharmoniques, C. R. Acad. des Sci., Paris, 260 (1965), 62676268.Google Scholar
[21] Jost, J., Harmonic Mappings between Riemannian manifolds, Proc. Centre of Math. Analysis, Australia Nat. Univ., 4, 1983.Google Scholar
[22] Kasue, A., Applications of Laplacian and Hessian comparison theorems, Geometry of Geodesies and Related Topics, Advanced Studies in Pure Math., 3 (1984), 333386.Google Scholar
[23] Kasue, A., On manifolds of asymptotically nonnegative curvature, preprint, M.S.R.I. Berkeley, July 1986.Google Scholar
[24] Kasue, A., A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. Ecole Norm. Sup., Paris, 21 (1988), 593622.Google Scholar
[25] Kasue, A. and Sugahara, K., Gap theorems for certain submanifolds of Euclidean spaces and Hyperbolic space forms, Osaka J. Math., 24 (1987), 679704.Google Scholar
[26] Kasue, A. and Sugahara, K. II, Curvature and Topology of Riemannian Manifolds, Lecture Notes in Math., 1201, Springer-Verlag, 1986.Google Scholar
[27] Katsuda, A., Gromov’s convergence theorem and its application, Nagoya Math. J., 100 (1985), 1148.Google Scholar
[28] Katsuda, A., A pinching problem for locally homogeneous spaces, Curvature and Topology of Riemannian Manifolds, Lecture Notes in Math., 1201, Springer-Verlag, 1986.Google Scholar
[29] Maeda, M., A geometric significance of total curvature on complete open surfaces, Geometry of Geodesies and Related Topics Advanced Studies in Pure Math., 3 (1984), 451458.CrossRefGoogle Scholar
[30] Nijenhuis-W., Woolf, B., Some integration problems in almost complex manifolds, Ann. of Math., 77 (1963), 424483.Google Scholar
[31] Nikolaev, L. G., Smoothness of the metric of spaces with two-sided bounded Aleksandrov curvature, Siberian Math. J., 24 (1983), 247263.Google Scholar
[32] Peters, S., Cheeger’s finiteness theorem for diffeomorphism classes of Riemannian manifolds, J. reine angew. Math., 349 (1984), 7782.Google Scholar
[33] Peters, S., Convergence of Riemannian manifolds, Compositio Math., 62 (1987), 316.Google Scholar
[34] Poor, W. A. Jr., Some results on nonnegatively curved manifolds, J. Differential Geom., 9 (1974), 583600.Google Scholar
[35] Shiohama, K., Total curvature and minimal areas of complete open surfaces, Proc. Amer. Math. Soc, 94 (1985), 310316.Google Scholar
[36] Shiohama, K., An integral formula for the measure of rays on complete open surfaces, J. Differential Geom., 23 (1986), 197205.CrossRefGoogle Scholar
[37] Uesu, K., The Titz metric on focal points free manifolds and its application, preprint.Google Scholar
[38] Walter, R., A generalized Allendoeffer-Weil formula and an inequality of the Cohn-Vossen type, J. Differential Geom., 10 (1975), 167180.Google Scholar