Geometric Pluripotential Theory on Sasaki Manifolds

He, Weiyong; Li, Jun

doi:10.1007/s12220-019-00257-5

Cited by 11 publications

(33 citation statements)

References 52 publications

(163 reference statements)

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“…Since then a number of important works have appeared building on the topics presented in this work: Chen-Cheng cracked the PDE theory of the csck equation [36,37], allowing to fully prove a converse of a theorem by Berman-Darvas-Lu [14]. Very recently He-Li pointed out that the contents of this survey generalize to Sasakian manifolds [77], paving the way to existence theorems for canonical metrics in that context as well. Lemma 1.1.…”

Section: Ricω = λωmentioning

confidence: 99%

Geometric pluripotential theory on Kähler manifolds

Darvas¹

2019

Advances in Complex Geometry

106

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Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Ampère type arising in Kähler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of infinite dimensional convex optimization, yielding existence results for solutions of the underlying complex Monge-Ampère equations. The purpose of this survey is to describe these developments from basic principles. Smith, V. Tosatti and the anonymous referees for their suggested corrections, careful remarks, and precisions. Also, I thank the students of MATH868D at the University of Maryland for their intriguing questions and relentless interest throughout the Fall of 2016. Chapter 3 is partly based on work done as a graduate student at Purdue University, and I am indebted to L. Lempert for encouragement and guidance. Chapter 4 surveys to some extent joint work with Y. Rubinstein, and I am grateful for his mentorship over the years.

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Section: Ricω = λωmentioning

confidence: 99%

Geometric pluripotential theory on Kähler manifolds

Darvas¹

2019

Advances in Complex Geometry

106

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“…M K is the relative Mabuchi energy acting on the space of (normalized) Sasaki structures in (N, K, J K ), studied in [34,35,56]. Secondly, in Lemma 2.17 we compute the derivative of Θ, which enables us to show (see Lemma 4.6) that it is bilipschitz with respect to d 1 .…”

Section: Introductionmentioning

confidence: 99%

“…These tools allow us to define a notion of properness for M Ǩ,κ with respect to the natural action of T C which directly translates via Θ to a corresponding notion of properness for M K (see Corollary 4.11). This allows us to use the extension of M K to the d 1 -completion and the regularity of its minimizers (obtained in [34,35]) to show the properness of M K (see Theorem 4.12) and hence of M Ǩ,κ . A key result for this to work is the transitivity of the action of T C on the space of T-invariant K-extremal Kähler metrics in 2πc 1 (L), which has been obtained in [41] using the approach in [16].…”

Section: Introductionmentioning

confidence: 99%

Weighted K-stability of polarized varieties and extremality of Sasaki manifolds

Apostolov¹,

Calderbank²,

Legendre³

2020

Preprint

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We use the correspondence between extremal Sasaki structures and weighted extremal Kähler metrics defined on a regular quotient of a Sasaki manifold, established by the first two authors in [1], and Lahdili's theory of weighted K-stability [40] in order to define a suitable notion of (relative) weighted K-stability for compact Sasaki manifolds of regular type. We show that the (relative) weighted K-stability with respect to a maximal torus is a necessary condition for the existence of a (possibly irregular) extremal Sasaki metric. We also compare weighted K-stability to the K-stability of the corresponding polarized affine cone (introduced by Collins-Székelyhidi in [18]), and prove that they agree on the class of test configurations we consider. As a byproduct, we strengthen the obstruction to the existence of a scalar-flat Kähler cone metric found in [18] from the K-semistability to the K-stability on these test configurations. We use our approach to give a characterization of the existence of a compatible extremal Sasaki structure on a principal S 1 -bundle over an admissible ruled manifold in the sense of [4], expressed in terms of the positivity of a single polynomial of one variable over a given interval.

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“…Again, it is not our intention to historically describe the history and significance of the use of metric geometry within Kähler geometry but to mention a few more articles [7,26,27,29,45,46,54], and then we refer to [25,32,53] for a better historical account and an overall picture. It is of interest here to mention the work of He and Li [34] on geometric pluripotential theory on Sasakian manifolds.…”

Section: Introductionmentioning

confidence: 99%

Geodesics in the space of $m$-subharmonic functions with bounded energy

Åhag¹,

Czyż²

2021

Preprint

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We raise our cups to Urban Cegrell, gone but not forgotten, gone but ever here.Until we meet again in Valhalla!Abstract. With inspiration from the Kähler geometry, we introduce a metric structure on the energy class, E 1,m , of m-subharmonic functions with bounded energy and show that it is complete. After studying how the metric convergence relates to the accepted convergences in this Caffarelli-Nirenberg-Spruck model, we end by constructing geodesics in a subspace of our complete metric space.

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Geometric Pluripotential Theory on Sasaki Manifolds

Cited by 11 publications

References 52 publications

Geometric pluripotential theory on Kähler manifolds

Geometric pluripotential theory on Kähler manifolds

Weighted K-stability of polarized varieties and extremality of Sasaki manifolds

Geodesics in the space of $m$-subharmonic functions with bounded energy

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