On a Fourth Order Equation in 3-D

Xingwang Xu; Paul C. Yang

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 1029-1042
  • ISSN: 1292-8119

Abstract

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In this article we study the positivity of the 4-th order Paneitz operator for closed 3-manifolds. We prove that the connected sum of two such 3-manifold retains the same positivity property. We also solve the analogue of the Yamabe equation for such a manifold.

How to cite

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Xu, Xingwang, and Yang, Paul C.. "On a Fourth Order Equation in 3-D." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 1029-1042. <http://eudml.org/doc/90640>.

@article{Xu2010,
abstract = { In this article we study the positivity of the 4-th order Paneitz operator for closed 3-manifolds. We prove that the connected sum of two such 3-manifold retains the same positivity property. We also solve the analogue of the Yamabe equation for such a manifold. },
author = {Xu, Xingwang, Yang, Paul C.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Paneitz operator; conformal invariance; Sobolev inequality; connected sum.; connected sum},
language = {eng},
month = {3},
pages = {1029-1042},
publisher = {EDP Sciences},
title = {On a Fourth Order Equation in 3-D},
url = {http://eudml.org/doc/90640},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Xu, Xingwang
AU - Yang, Paul C.
TI - On a Fourth Order Equation in 3-D
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 1029
EP - 1042
AB - In this article we study the positivity of the 4-th order Paneitz operator for closed 3-manifolds. We prove that the connected sum of two such 3-manifold retains the same positivity property. We also solve the analogue of the Yamabe equation for such a manifold.
LA - eng
KW - Paneitz operator; conformal invariance; Sobolev inequality; connected sum.; connected sum
UR - http://eudml.org/doc/90640
ER -

References

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  1. M. Ahmedou, Z. Djadli and A. Malchiodi, Prescribing a fourth order conformal invariant on the standard sphere. Part I: Perturbation Result. Comm. Comtemporary Math. (to appear).  
  2. T. Branson, Differential operators cannonically associated to a conformal structure. Math. Scand.57 (1985) 293-345.  
  3. A. Chang and P. Yang, Extremal metrics of zeta functional determinants on 4-manifolds. Ann. Math.142 (1995) 171-212.  
  4. A. Chang, M. Gursky and P. Yang, An equation of Monge-Ampere type in conformal geometry and four-manifolds of positive Ricci curvature. Ann. Math. (to appear)  
  5. Y.S. Choi and X. Xu, Nonlinear biharmonic equation with negative exponent. Preprint (1999).  
  6. Z. Djadli, E. Hebey and M. Ledoux, Paneitz operators and applications. Duke Math. J.104 (2000) 129-169.  
  7. C. Fefferman and R. Graham, Conformal Invariants, in Élie Cartan et les Mathématiques d'aujourd'hui. Asterisque (1985) 95-116.  
  8. E. Hebey and F. Robert, Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients. Preprint.  
  9. S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. Preprint (1983).  
  10. X. Xu and P. Yang, Positivity of Paneitz operators. Discrete Continuous Dynam. Syst.7 (2001) 329-342.  

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