On a fourth order equation in 3-D

Xingwang Xu; Paul C. Yang

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • 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 (2002): 1029-1042. <http://eudml.org/doc/245654>.

@article{Xu2002,
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},
language = {eng},
pages = {1029-1042},
publisher = {EDP-Sciences},
title = {On a fourth order equation in 3-D},
url = {http://eudml.org/doc/245654},
volume = {8},
year = {2002},
}

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
PY - 2002
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
UR - http://eudml.org/doc/245654
ER -

References

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  1. [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). Zbl1023.58020
  2. [2] T. Branson, Differential operators cannonically associated to a conformal structure. Math. Scand. 57 (1985) 293-345. Zbl0596.53009MR832360
  3. [3] A. Chang and P. Yang, Extremal metrics of zeta functional determinants on 4-manifolds. Ann. Math. 142 (1995) 171-212. Zbl0842.58011MR1338677
  4. [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) Zbl1031.53062
  5. [5] Y.S. Choi and X. Xu, Nonlinear biharmonic equation with negative exponent. Preprint (1999). 
  6. [6] Z. Djadli, E. Hebey and M. Ledoux, Paneitz operators and applications. Duke Math. J. 104 (2000) 129-169. Zbl0998.58009MR1769728
  7. [7] C. Fefferman and R. Graham, Conformal Invariants, in Élie Cartan et les Mathématiques d’aujourd’hui. Asterisque (1985) 95-116. Zbl0602.53007
  8. [8] E. Hebey and F. Robert, Coercivity and Struwe’s compactness for Paneitz type operators with constant coefficients. Preprint. Zbl0998.58007
  9. [9] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. Preprint (1983). Zbl1145.53053
  10. [10] X. Xu and P. Yang, Positivity of Paneitz operators. Discrete Continuous Dynam. Syst. 7 (2001) 329-342. Zbl1032.58018MR1808405

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