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

 
 

 

A critical metric for the $L^2$-norm
of the curvature tensor on $S^3$


Author: François Lamontagne
Journal: Proc. Amer. Math. Soc. 126 (1998), 589-593
DOI: https://doi.org/10.1090/S0002-9939-98-04171-9
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Abstract | References | Additional Information

Abstract: The $L^2$-norm of the curvature tensor

\begin{displaymath}{\cal R}(g)=\frac{1}{(\mathrm{Vol}(M))^{\frac{n-4}{n}}}\int _{M}\mid\! R\! \mid ^{2}dvol_{g} \end{displaymath}

defines a Riemannian functional on the space of metrics. This work exhibits a metric on $S^3$ which is of Berger type but not of constant ricci curvature, and yet is critical for ${\cal R}$.


References [Enhancements On Off] (What's this?)

  • 1. S. Helgason Differential Geometry, Lie Groups, and Symmetric Spaces Academic Press. MR 26:2986
  • 2. Francois Lamontagne Ph.D. Thesis S.U.N.Y. at Stony Brook.
  • 3. J. Milnor Curvature of Left Invariant Metrics on Lie Groups Advances in Mathematics 21 p.293-329 (1976). MR 54:12970
  • 4. R. Palais The Principle of Symmetric Criticality Communications in Mathematical Physics 69 p.19-30 (1979). MR 81c:58026


Additional Information

François Lamontagne
Affiliation: Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec, Canada
Email: lamontaf@crm.umontreal.ca

DOI: https://doi.org/10.1090/S0002-9939-98-04171-9
Keywords: Critical metrics, Hopf fibration, Berger sphere
Received by editor(s): December 7, 1995
Received by editor(s) in revised form: July 31, 1996
Communicated by: Christopher Croke
Article copyright: © Copyright 1998 American Mathematical Society

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