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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Integral estimates of conformal metrics

Author: Xingwang Xu
Journal: Proc. Amer. Math. Soc. 124 (1996), 315-324
MSC (1991): Primary 58G25, 53C21
MathSciNet review: 1301536
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Abstract: In this article we show that there exists a rational number $\mu $, depending only on the dimension $n\ (\ge 5)$ of the manifold such that the $\mu $th-power of the conformal factor is bounded in $H^2_2$ norm in terms of volume bound and the square norm bound of the scalar curvature of the conformal metrics. Some applications are also given.

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  • An M. Anderson, Remarks on the compactness of isospectral sets in lower dimensions, Duke Math. J. 63 (1991), 699--711, MR 92m:58140.
  • Au T. Aubin, Nonlinear analysis on manifolds, Monge-Ampère equations, Springer-Verlag, New York, 1982, MR 85j:58002.
  • BGM M. Berger, P. Gauduchon, and E. Mazet, Le spectre d'une varieté Riemannienne, Lecture Notes in Math., vol. 194, Springer-Verlag, New York, 1971, MR 43:8025.
  • B A. Besse, Einstein manifolds, Springer-Verlag, New York, 1987, MR 88f:53087.
  • BCY T. Branson, S.Y.A. Chang, and P.C. Yang, Estimates and extremals for zeta function determinants on four manifolds, Comm. Math. Phys. 149 (1992), 241--262, MR 93m:58116.
  • BG R. Brooks and C. Gordon, Isospectral conformally equivalent Riemannian metrics, Bull. Amer. Math. Soc. (N.S.) 23 (1990), 433--436, MR 91a:58188.
  • BPY R. Brooks, P. Perry, and P. Yang, Isospectral sets of conformally equivalent metrics, Duke Math. J. 58 (1989), 131--150, MR 90i:58193.
  • CY1 A. S. Y. Chang and P. Yang, Isospectral conformal metrics on 3-manifolds, J. Amer. Math. Soc. 3 (1990), 117--145, MR 91c:58140.
  • CY2 ------, Compactness of isospectral conformal metrics on $S^3$, Comment. Math. Helv. 64 (1989), 363--374, MR 90c:58181.
  • C J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61--74, MR 41:7697.
  • Ga1 Z. Y. Gao, Einstein metrics, J. Differential Geom. 32 (1990), 155--183.
  • Ga2 ------, Convergence of Riemannian manifolds, Ricci and $L^\frac{n}{2}$ curvature pinching, J. Differential Geom. 32 (1990), 349--381.
  • GW R. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 139 (1988), 119--141, MR 89g:53063.
  • GLP M. Gromov, J. Lafontaine, and P. Pansu, Structures metriques pour les varietes Riemanniennes, Cedic-Fernand/Nathan, Paris, 1981.
  • G M. Gursky, Compactness of conformal metrics with integral bounds on curvature, thesis, Cal Tech., 1991.
  • Mo M. Min-Oo, Almost Einstein manifolds of negative Ricci curvature, J. Differential Geom. 32 (1990), 457--472, MR 91g:53047.
  • P S. Peter, Convergence of Riemannian manifolds, Compositio Math. 62 (1987), 3--16.
  • PR T. Parker and S. Rosenberg, Invariants of conformal Laplacians, J. Differential Geom. 25 (1988), 199--222, MR 89e:58118.
  • X X. Xu, On compactness of isospectral conformal metrics of 4-manifolds, Nagoya Math. J. (to appear).

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Additional Information

Xingwang Xu
Affiliation: address Department of Mathematics, National University of Singapore, Singapore 0511

Received by editor(s): June 22, 1994
Communicated by: \commby Peter Li
Article copyright: © Copyright 1996 American Mathematical Society

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