Highly connected manifolds with positive Ricci curvature

Boyer, Charles P.; Galicki, Krzysztof

doi:10.2140/gt.2006.10.2219

Cited by 9 publications

(8 citation statements)

References 38 publications

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“…However, in contrast there are sphere bundles with CR Sasaki structures having no regular Reeb field in their Sasaki cone. In [BG06] infinitely many such Sasaki CR structures are given on the trivial sphere bundles S 2d × S 2d+1 for d > 1 which have Sasaki metrics of positive Ricci curvature . They are represented by Brieskorn manifolds belonging to infinitely many inequivalent contact structures [BMvK16] and it is shown in [BvC18] that their Sasaki cones admit no extremal Sasaki metrics whatsoever.…”

Section: Invariants and The Classification Of Sasaki Cr Structuresmentioning

confidence: 99%

Sasakian Geometry on Sphere Bundles

Boyer¹,

Tønnesen-Friedman²

2020

Preprint

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The purpose of this paper is to study the Sasakian geometry on odd dimensional sphere bundles over a smooth projective algebraic variety N with the ultimate, but probably unachievable goal of understanding the existence and non-existence of extremal and constant scalar curvature Sasaki metrics. We apply the fiber join construction of Yamazaki [Yam99] for K-contact manifolds to the Sasaki case. This construction depends on the choice of d + 1 integral Kähler classes [ω j ] on N that are not necessarily colinear in the Kähler cone. We show that the colinear case is equivalent to a subclass of a different join construction orginally described in [BG00a, BGO07], applied to the spherical case by the authors in [BTF14, BTF16] when d = 1, and known as cone decomposable [BHLTF18]. The non-colinear case gives rise to infinite families of new inequivalent cone indecomposable Sasaki contact CR on certain sphere bundles. We prove that the Sasaki cone for some of these structures contains an open set of extremal Sasaki metrics and, for certain specialized cases, the regular ray within this cone is shown to have constant scalar curvature. We also compute the cohomology groups of all such sphere bundles over a product of Riemann surfaces.

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Section: Invariants and The Classification Of Sasaki Cr Structuresmentioning

confidence: 99%

Sasakian Geometry on Sphere Bundles

Boyer¹,

Tønnesen-Friedman²

2020

Preprint

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“…The oriented diffeomorphism type is not known explicitly and not all possible diffeomorphism types occur. In [BG06] a formula is given for the number of diffemorphism types D n (k) obtained by our method, and tables are given for dimension 7 and 11. Nevertheless, since there is a periodicity modulo a subgroup of bP 4n+4 , we do have countably infinite families of Sasakian structures with an n + 1 dimensional Sasaki cones having no extremal Sasaki metrics.…”

Section: Explicit Examplesmentioning

confidence: 99%

Relative K-stability and extremal Sasaki metrics

Boyer

Coevering

2018

Mathematical Research Letters

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We define K-stability of a polarized Sasakian manifold relative to a maximal torus of automorphisms. The existence of a Sasaki-extremal metric in the polarization is shown to imply that the polarization is K-semistable. Computing this invariant for the deformation to the normal cone gives an extention of the Lichnerowicz obstruction, due to Gauntlett, Martelli, Sparks, and Yau, to an obstruction of Sasaki-extremal metrics. We use this to give a list of examples of Sasakian manifolds whose Sasaki cone contains no extremal representatives. These give the first examples of Sasaki cones of dimension greater than one that contain no extremal Sasaki metrics whatsoever. In the process we compute the unreduced Sasaki cone for an arbitrary smooth link of a weighted homogeneous polynomial.1991 Mathematics Subject Classification. 53C25 primary, 32W20 secondary.

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“…Higher dimensional Diffeomorphism Types. Next we consider links studied in [BG06] and discussed in Section 9.5.2 of [BG08]. These do not admit SE metrics, but they are probably easier to work with.…”

Section: 3mentioning

confidence: 99%

“…We remark that bP 4n+2 is the identity for n = 1, 3, 7, 15 and it is Z 2 when 4n + 2 = 2 i − 2 for i ≥ 3, otherwise it is unknown. It is shown in [BG06] that these manifolds admit positive Sasaki metrics. Concerning their moduli we now have Theorem 1.3.…”

Section: Introductionmentioning

confidence: 99%

Brieskorn manifolds, positive Sasakian geometry, and contact topology

2015

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Abstract. Using S 1 -equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn-Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of S 2 × S 3 and certain rational homology spheres. We then apply our result to show that on these manifolds the moduli space of classes of positive Sasakian structures has infinitely many components. We also apply our results to give lower bounds on the number of components of the moduli space of Sasaki-Einstein metrics on certain homotopy spheres. Finally a new family of Sasaki-Einstein metrics of real dimension 20 on S 5 is exhibited.

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Highly connected manifolds with positive Ricci curvature

Cited by 9 publications

References 38 publications

Sasakian Geometry on Sphere Bundles

Sasakian Geometry on Sphere Bundles

Relative K-stability and extremal Sasaki metrics

Brieskorn manifolds, positive Sasakian geometry, and contact topology

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