Rigidity of Einstein metrics as critical points of quadratic curvature functionals on closed manifolds

AbstractWe investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to “gauge” the Euler–Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics. We show a number of compact examples are infinitesimally rigid, and consequently, are isolated critical points in the space of unit-volume Riemannian metrics. We then give examples of critical metrics which are strict local minimizers (up to diffeomorphism and scaling). A corollary is a local “reverse Bishop's inequality” for such metrics. In particular, any metric

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CHARACTERIZATIONS OF RICCI FLAT METRICS AND LAGRANGIAN SUBMANIFOLDS IN TERMS OF THE VARIATIONAL PROBLEM

Glasgow Mathematical Journal ◽

10.1017/s0017089514000536 ◽

2014 ◽

Vol 57 (3) ◽

pp. 643-651

Author(s):

TETSUYA TANIGUCHI ◽

SEIICHI UDAGAWA

Keyword(s):

Variational Problem ◽

Critical Points ◽

Einstein Metrics ◽

Lagrangian Submanifolds ◽

Volume Element ◽

Ricci Flat

AbstractGiven the pair (P, η) of (0,2) tensors, where η defines a volume element, we consider a new variational problem varying η only. We then have Einstein metrics and slant immersions as critical points of the 1st variation. We may characterize Ricci flat metrics and Lagrangian submanifolds as stable critical points of our variational problem.

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Critical associated metrics on contact manifolds. II

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700033863 ◽

1986 ◽

Vol 41 (3) ◽

pp. 404-410 ◽

Cited By ~ 9

Author(s):

D. E. Blair ◽

A. J. Ledger

Keyword(s):

Scalar Curvature ◽

Critical Points ◽

Compact Manifold ◽

Einstein Metrics ◽

Contact Structure ◽

Riemannian Metrics ◽

Contact Manifold ◽

Contact Manifolds ◽

Contact Metric Structure ◽

Sasakian Metrics

AbstractThe study of the integral of the scalar curvature, ∫MRdVg, as a function on the set of all Riemannian metrics of the same total volume on a compact manifold is now classical, and the critical points are the Einstein metrics. On a compact contact manifold we consider this and ∫M (R − R* − 4n2) dv, with R* the *-scalar curvature, as functions on the set of metrics associated to the contact structure. For these integrals the critical point conditions then become certain commutativity conditions on the Ricci operator and the fundamental collineation of the contact metric structure. In particular, Sasakian metrics, when they exist, are maxima for the second function.

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The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽

10.1017/s0094837300025707 ◽

1980 ◽

Vol 6 (02) ◽

pp. 146-160 ◽

Cited By ~ 29

Author(s):

William A. Oliver

Keyword(s):

Critical Points ◽

Scleractinian Corals ◽

Convincing Evidence ◽

Early Triassic ◽

Rugose Corals ◽

History Of ◽

The Subject ◽

Relationship Of ◽

The Relationship ◽

Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

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