Uniqueness of constant scalar curvature Sasakian metrics

Jin, Xuecheng; Zhang, Xi

doi:10.1007/s10455-015-9493-x

Cited by 4 publications

(7 citation statements)

References 22 publications

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“…Assume that c 2 dη − (n − 1)χ is a transverse Kähler form. Then, the Sasaki J-flow converges C ∞ to a smooth critical metric.The proof is based on the long time existence of the flow and the uniform lower bound on the second derivatives of a solution to the flow established in [23], and on the estimates developed in sections 3 and 4 of the present paper, which are obtained applying the maximum principle and a Moser iteration argument (see the proof of Prop 4.6).It is worth pointing out that as immediate corollary we get the C ∞ convergence of the flow to a critical metric on compact 5-dimensional Sasaki manifolds under the assumption c 2 dη − χ > 0.As application, we highlight the relation between the Sasaki J-flow and the Mabuchi Kenergy, introduced in the Sasakian context by [9] (see also [14]), proving a lower bound for the K-energy map under the existence of a critical metric (see Theorem 5.2 at the end of the paper).…”

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confidence: 62%

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On the convergence of the Sasaki J-flow

Zedda

2016

Ann Glob Anal Geom

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mentioning

confidence: 62%

“…As application, we highlight the relation between the Sasaki J-flow and the Mabuchi Kenergy, introduced in the Sasakian context by [9] (see also [14]), proving a lower bound for the K-energy map under the existence of a critical metric (see Theorem 5.2 at the end of the paper).…”

mentioning

confidence: 89%

On the convergence of the Sasaki J-flow

Zedda

2016

Ann Glob Anal Geom

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“…We shall prove this theorem following the lines closely as in [24]. Given this result, one can then extend the K-energy to E 1 (M, ξ, ω T ), and keep it still convex along geodesics, see [2,41,51,5]. We actually generalize almost all related results in [24] to Sasaki setting, building up on profound results in pluripotential theory by many and geodesic equation [32].…”

Section: Introductionmentioning

confidence: 80%

Scalar curvature and properness on Sasaki manifolds

2018

Preprint

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We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of 16,17] on existence of Kähler metrics with constant scalar curvature (csck) on compact Kähler manifolds. Following their strategy, we prove that given a Sasaki structure (with Reeb vector field and complex structure on its cone fixed ), there exists a Sasaki structure with transverse constant scalar curvature (cscs) if and only if the K-energy is reduced proper modulo the identity component of the automorphism group which preserves both the Reeb vector field and transverse complex structure. Technically, the proof mainly consists of two parts. The first part is a priori estimates for scalar curvature type equations which are parallel to Chen-Cheng's results in [16,17] in Sasaki setting. The second part is geometric pluripotential theory on a compact Sasaki manifold, building up on profound results in geometric pluripotential theory on Kähler manifolds. There are notable, and indeed subtle differences in Sasaki setting (compared with Kähler setting) for both parts (PDE and pluripotential theory). The PDE part is an adaption of deep work of Chen-Cheng [15,16,17] to Sasaki setting with necessary modifications, where Chen's continuity path plays a very important role. While the geometric pluripotential theory on a compact Sasaki manifold has new difficulties, compared with geometric pluripotential theory in Kähler setting which is very intricate. We shall present the details of geometric pluripotential on Sasaki manifolds in a separate paper [40] (joint work with Jun Li).

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“…Theorem 2 is the counterpart of main results in [30] in Sasaki setting. An important notion in the study of csck is the convexity of K-energy along C 1, 1 geodesics [3] (see also [26]), which was generalized to Sasaki setting by [51,57]. Given the results above, one can then extend K-energy to E 1 -class and keep its convexity along finite energy geodesics as in [7].…”

Section: Introductionmentioning

confidence: 99%

Geometric Pluripotential Theory on Sasaki Manifolds

2019

J Geom Anal

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We extend profound results in pluripotential theory on Kähler manifolds [31] to Sasaki setting via its transverse Kähler structure. As in Kähler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature (cscs) in terms of properness of K-energy, considered by the first named author in [49]. One main result is to generalize T. Darvas' theory on the geometric structure of the space of Kähler potentials in Sasaki setting. Along the way we extend most of corresponding results in pluripotential theory to Sasaki setting via its transverse Kähler structure.

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Uniqueness of constant scalar curvature Sasakian metrics

Cited by 4 publications

References 22 publications

On the convergence of the Sasaki J-flow

On the convergence of the Sasaki J-flow

Scalar curvature and properness on Sasaki manifolds

Geometric Pluripotential Theory on Sasaki Manifolds

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