Constant \(Q\)-curvature metrics on conic 4-manifolds. (English) Zbl 1501.53042
Summary: We consider the constant \(Q\)-curvature metric problem in a given conformal class on a conic 4-manifold and study related differential equations. We define subcritical, critical, and supercritical conic 4-manifolds. Following [M. Troyanov, Trans. Am. Math. Soc. 324, No. 2, 793–821 (1991; Zbl 0724.53023); S.-Y. A. Chang and P. C. Yang, Ann. Math. (2) 142, No. 1, 171–212 (1995; Zbl 0842.58011)], we prove the existence of constant \(Q\)-curvature metrics in the subcritical case. For conic 4-spheres with two singular points, we prove the uniqueness in critical cases and nonexistence in supercritical cases. We also give the asymptotic expansion of the corresponding PDE near isolated singularities.
MSC:
| 53C18 | Conformal structures on manifolds |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35R01 | PDEs on manifolds |
| 49Q10 | Optimization of shapes other than minimal surfaces |
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