The Einstein–Hilbert Functional and the Sasaki–Futaki Invariant

Boyer, Charles P.; Huang, Hongnian; Legendre, Eveline; Tønnesen-Friedman, Christina W.

doi:10.1093/imrn/rnw076

Cited by 12 publications

(32 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where a 0 (R) = V R and a 1 (R) = S R up to constants which are irrelevant to the argument. The author and his colleagues [BHLTF17] studied the stability problem through the use of the Einstein-Hilbert functional which I discuss in the next section. The Yau-Tian-Donaldson conjecture in the Sasaki case is:…”

Section: We Now Have a Results Of Collins And Székelyhidimentioning

confidence: 99%

See 1 more Smart Citation

Contact Structures of Sasaki Type and Their Associated Moduli

Boyer

2019

Complex Manifolds

Self Cite

View full text Add to dashboard Cite

This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.

show abstract

Section: We Now Have a Results Of Collins And Székelyhidimentioning

confidence: 99%

“…Theorem 5.10 ([BHLTF17]). Let (M, S) be a Sasaki manifold such that its polarized affine cone (Y, R) is K-semistable.…”

mentioning

confidence: 99%

Contact Structures of Sasaki Type and Their Associated Moduli

Boyer

2019

Complex Manifolds

Self Cite

View full text Add to dashboard Cite

show abstract

“…For general Sasakian manifolds, i.e. for Sasakian manifolds which are not necessarily transversely Fano, it is known that the convexity fails for the Einstein-Hilbert functional, and there can be several critical points, see Legendre [37], and also [9]. This fact has resemblance in the study of Einstein-Maxwell Kähler metrics as can be seen in the ambitoric examples by LeBrun [35] on the one-point-blow-up of CP 2 .…”

Section: Sasakian Geometrymentioning

confidence: 97%

Volume minimization and obstructions to solving some problems in Kähler geometry

Futaki

Ono

2018

Notices of the International Congress of Chinese Mathematicians

View full text Add to dashboard Cite

There is an obstruction to the existence of Kähler-Einstein metrics which is used to define the GIT weight for K-stability, and it has been extended to various geometric problems. This survey paper considers such extended obstructions to the existence problem of Kähler-Ricci solitons, Sasaki-Einstein metrics and (conformally) Einstein-Maxwell Kähler metrics. These three cases have a common feature that the obstructions are parametrized by a space of vector fields. We see, in these three cases, the obstructions are obtained as the derivative of suitable volume functionals. This tells us for which vector fields we should try to solve the existence problems.

show abstract

“…[16,17,35,36]. Well-known obstructions are the K-stability see [14], the transversal Futaki invariant [22] and the Einstein-Hilbert functional [28,10]. In this paper we study the latter using the Duistermaat-Heckman localization formula developping on an idea of [32].…”

Section: Introductionmentioning

confidence: 99%

“…[37]. Using the relation found in [28,10], Tian's formula should be related to the derivative of (2) in the regular case.…”

Section: Introductionmentioning

confidence: 99%

An application of the Duistermaat–Heckman theorem and its extensions in Sasaki geometry

2018

Self Cite

View full text Add to dashboard Cite

Building on an idea laid out by , we use the Duistermaat-Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein-Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Studying the leading terms we prove they are all proper. Among consequences we get that the Einstein-Hilbert functional attains its minimal value and each Sasaki cone possess at least one Reeb vector field with vanishing transverse Futaki invariant.

show abstract

The Einstein–Hilbert Functional and the Sasaki–Futaki Invariant

Cited by 12 publications

References 32 publications

Contact Structures of Sasaki Type and Their Associated Moduli

Contact Structures of Sasaki Type and Their Associated Moduli

Volume minimization and obstructions to solving some problems in Kähler geometry

An application of the Duistermaat–Heckman theorem and its extensions in Sasaki geometry

Contact Info

Product

Resources

About