extremal metrics
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2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Hao Fang ◽  
Biao Ma

AbstractWe 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, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 1991, 2, 793–821] and [S.-Y. A. Chang and P. C. Yang, Extremal metrics of zeta function determinants on 4-manifolds, Ann. of Math. (2) 142 1995, 1, 171–212], 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.



2020 ◽  
Vol 114 (1) ◽  
pp. 171-192
Author(s):  
Reza Seyyedali ◽  
Gábor Székelyhidi
Keyword(s):  


2019 ◽  
Vol 120 (4) ◽  
pp. 587-616
Author(s):  
Ruadhaí Dervan ◽  
Lars Martin Sektnan
Keyword(s):  


Axioms ◽  
2018 ◽  
Vol 8 (1) ◽  
pp. 2
Author(s):  
Zhuang-Dan Guan ◽  
Pilar Orellana ◽  
Anthony Van

This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. In Part II, we obtained a substantial amount of new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds N n , m when n > 15 and N n , m ′ when n > 4 . We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian–Einstein metrics. These also provide open Calabi–Yau manifolds.



2018 ◽  
Vol 340 ◽  
pp. 459-527 ◽  
Author(s):  
Bohui Chen ◽  
An-Min Li ◽  
Li Sheng


2018 ◽  
Vol 340 ◽  
pp. 363-405 ◽  
Author(s):  
Bohui Chen ◽  
An-Min Li ◽  
Li Sheng


2018 ◽  
Vol 21 (2) ◽  
pp. 585-626
Author(s):  
Jeffrey Case ◽  
Chin-Yu Hsiao ◽  
Paul Yang




2018 ◽  
Vol 370 (10) ◽  
pp. 6995-7010
Author(s):  
Vamsi Pritham Pingali
Keyword(s):  


2017 ◽  
Vol 316 ◽  
pp. 770-805 ◽  
Author(s):  
Reza Seyyedali


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