scholarly journals Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics

2011 ◽  
Vol 147 (5) ◽  
pp. 1613-1634 ◽  
Author(s):  
Eveline Legendre
Keyword(s):  
Finite Number ◽  
Scalar Curvature ◽  
Vector Fields ◽  
Existence Result ◽  
Constant Scalar Curvature ◽  
Contact Structures ◽  
Toric Geometry ◽  
Contact Manifolds ◽  
Reeb Vector ◽  
Toric Contact Manifolds

AbstractWe study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

scholarly journals Conformal vector fields and Yamabe solitons

2019 ◽  
Vol 16 (04) ◽  
pp. 1950053
Author(s):  
Nasser Bin Turki ◽  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Scalar Curvature ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Constant Scalar Curvature ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Yamabe Solitons

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are Yamabe metrics.

scholarly journals Contact homology of good toric contact manifolds

2011 ◽  
Vol 148 (1) ◽  
pp. 304-334 ◽  
Author(s):  
Miguel Abreu ◽  
Leonardo Macarini
Keyword(s):  
Homotopy Class ◽  
Chern Class ◽  
Einstein Metrics ◽  
Contact Manifold ◽  
Infinite Family ◽  
Contact Structures ◽  
Contact Homology ◽  
Contact Manifolds ◽  
First Chern Class ◽  
Toric Contact Manifolds

AbstractIn this paper we show that any good toric contact manifold has a well-defined cylindrical contact homology, and describe how it can be combinatorially computed from the associated moment cone. As an application, we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of Gauntlett et al. on Sasaki–Einstein metrics. We show in particular that these give rise to a new infinite family of non-equivalent contact structures on S2×S3 in the unique homotopy class of almost contact structures with vanishing first Chern class.

scholarly journals Killing fields generated by multiple solutions to the Fischer–Marsden equation

2015 ◽  
Vol 26 (04) ◽  
pp. 1540006 ◽  
Author(s):  
Paul Cernea ◽  
Daniel Guan
Keyword(s):  
Scalar Curvature ◽  
Multiple Solutions ◽  
Vector Fields ◽  
Killing Vector ◽  
Solution Space ◽  
Constant Scalar Curvature ◽  
Killing Fields ◽  
Sectional Curvatures ◽  
Almost All ◽  
Scalar Curvature Functional

In the process of finding Einstein metrics in dimension n ≥ 3, we can search critical metrics for the scalar curvature functional in the space of the fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by O. Kobayashi and J. Lafontaine. However, almost all the counter-examples are homogeneous. Multiple solutions to this system yield Killing vector fields. We show that the dimension of the solution space W can be at most n+1, with equality implying that (M, g) is a sphere with constant sectional curvatures. Moreover, we show that the identity component of the isometry group has a factor SO(W). We also show that geometries admitting Fischer–Marsden solutions are closed under products with Einstein manifolds after a rescaling. Therefore, we obtain a lot of non-homogeneous counter-examples to the Fischer–Marsden conjecture. We then prove that all the homogeneous manifold M with a solution are in this case. Furthermore, we also proved that a related Besse conjecture is true for the compact homogeneous manifolds.

scholarly journals Contact Structures of Sasaki Type and Their Associated Moduli

2019 ◽  
Vol 6 (1) ◽  
pp. 1-30
Author(s):  
Charles P. Boyer
Keyword(s):  
Scalar Curvature ◽  
Present Article ◽  
Einstein Metrics ◽  
Contact Structure ◽  
Constant Scalar Curvature ◽  
Contact Geometry ◽  
Contact Structures ◽  
Sasakian Geometry ◽  
Sasakian Structures

Abstract 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.

Yamabe solitons and gradient Yamabe solitons on three-dimensional N(k)-contact manifolds

2020 ◽  
Vol 17 (12) ◽  
pp. 2050177
Author(s):  
Young Jin Suh ◽  
Uday Chand De
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Three Dimensional ◽  
Constant Scalar Curvature ◽  
Infinitesimal Transformation ◽  
Contact Metric Manifold ◽  
Contact Manifolds ◽  
Flow Vector ◽  
Yamabe Solitons ◽  
Yamabe Soliton

If a three-dimensional [Formula: see text]-contact metric manifold [Formula: see text] admits a Yamabe soliton of type [Formula: see text], then the manifold has a constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, either [Formula: see text] has a constant curvature [Formula: see text] or the flow vector field [Formula: see text] is a strict contact infinitesimal transformation. Also, we prove that if the metric of a three-dimensional [Formula: see text]-contact metric manifold [Formula: see text] admits a gradient Yamabe soliton, then either the manifold is flat or the scalar curvature is constant. Moreover, either the potential function is constant or the manifold is of constant sectional curvature [Formula: see text]. Finally, we have given an example to verify our result.

scholarly journals Killing fields generated by multiple solutions to the Fischer–Marsden equation II

2016 ◽  
Vol 27 (10) ◽  
pp. 1650080
Author(s):  
Daniel Guan ◽  
Paul Cernéa
Keyword(s):  
Scalar Curvature ◽  
Multiple Solutions ◽  
Vector Fields ◽  
Killing Vector ◽  
Solution Space ◽  
Constant Scalar Curvature ◽  
Standard Sphere ◽  
Dimension Formula ◽  
Killing Fields ◽  
Sectional Curvatures

In the process of finding Einstein metrics in dimension [Formula: see text], we can search metrics critical for the scalar curvature among fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that, if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by Kobayashi and Lafontaine, and by our first paper. Multiple solutions to this system yield Killing vector fields. We showed in our first paper that the dimension of the solution space [Formula: see text] can be at most [Formula: see text], with equality implying that [Formula: see text] is a sphere with constant sectional curvatures. Moreover, we also showed there that the identity component of the isometry group has a factor [Formula: see text]. In this second paper, we apply our results in the first paper to show that either [Formula: see text] is a standard sphere or the dimension of the space of Fischer–Marsden solutions can be at most [Formula: see text].

scholarly journals Characterization of Almost Yamabe Solitons and Gradient Almost Yamabe Solitons with Conformal Vector Fields

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2362
Author(s):  
Ali H. Alkhaldi ◽  
Pişcoran Laurian-Ioan ◽  
Abimbola Abolarinwa ◽  
Akram Ali
Keyword(s):  
Scalar Curvature ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Constant Scalar Curvature ◽  
Topological Constraints ◽  
Conformal Vector ◽  
Conformal Vector Fields ◽  
Yamabe Solitons

In this paper, some sufficient conditions of almost Yamabe solitons are established, such that the solitons are Yamabe metrics, by which we mean metrics of constant scalar curvature. This is achieved by imposing fewer topological constraints. The properties of the conformal vector fields are exploited for the purpose of establishing various necessary criteria on the soliton vector fields of gradient almost Yamabe solitons so as to obtain Yamabe metrics.

On K-Polystability of cscK Manifolds with Transcendental Cohomology Class

2018 ◽  
Vol 2020 (9) ◽  
pp. 2769-2817 ◽  
Author(s):  
Zakarias Sjöström Dyrefelt
Keyword(s):  
Scalar Curvature ◽  
Cohomology Class ◽  
Vector Fields ◽  
Constant Scalar Curvature ◽  
Energy Functional ◽  
Kahler Manifolds ◽  
Kähler Metric ◽  
Holomorphic Vector Fields ◽  
Singularity Type ◽  
Kahler Metric

Abstract In this paper we study K-polystability of arbitrary (possibly non-projective) compact Kähler manifolds admitting holomorphic vector fields. As a main result we show that existence of a constant scalar curvature Kähler (cscK) metric implies geodesic K-polystability, in a sense that is expected to be equivalent to K-polystability in general. In particular, in the spirit of an expectation of Chen–Tang [28] we show that geodesic K-polystability implies algebraic K-polystability for polarized manifolds, so our main result recovers a possibly stronger version of results of Berman–Darvas–Lu [10] in this case. As a key part of the proof we also study subgeodesic rays with singularity type prescribed by singular test configurations and prove a result on asymptotics of the K-energy functional along such rays. In an appendix by R. Dervan it is moreover deduced that geodesic K-polystability implies equivariant K-polystability. This improves upon the results of [39] and proves that existence of a cscK (or extremal) Kähler metric implies equivariant K-polystability (resp. relative K-stability).

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

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