We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat–Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki–Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a–maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler–Einstein metric.

/pdf/sasaki-einstein-manifolds-and-volume-minimisation-klv3fs597v.pdf

Sasaki-Einstein manifolds and volume minimisation

for metricsg in the conformal class of g0, where we use the metric g to view the tensor as an endomorphism of the tangent bundle and where σk d notes the trace of the induced map on the kth exterior power; that is, σk is the kth elementary symmetric function of the eigenvalues. The case k = 1,R = constant is known as the Yamabe problem, and it has been studied in great depth (see [11] and [17]). We let M1 denote the set of unit volume metrics in the conformal class [g0]. We show that these equations have the following variational properties. Theorem 1. If k 6= n/2 and (N, [g0]) is locally conformally flat, then a metric g ∈M1 is a critical point of the functional Fk : g 7→ ∫

Conformal geometry, contact geometry, and the calculus of variations

In this article we obtain a priori estimates for solutions to the prescribed scalar curvature equation on 2- and 3-spheres under a nondegeneracy assumption on the curvature function. Using this estimate, we use the continuity method to demonstrate the existence of solutions to this equation when a map associated to the given curvature function has non-zero degree.

The scalar curvature equation on 2- and 3-spheres

﻿ A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kahler A similar result is proved for 8-manifolds HyperKahler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kahler manifold (indefinite if the scalar curvature is negative) This construction is compatible with the quaternionic Kahler and hyperKahier quotient constructions and allows quaternionic Kahler geometry to be subsumed into the theory of hyperKahler manifolds It is shown that the hyperKahler metrics that arise admit a certain type of SU (2)- action, possess functions which are Kahler potentials for each of the complex structures simultaneously and determine quaternionic Kahler structures via a variant of the moment map construction Quaternionic Kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKahler metrics with circle actions on complex line bundles over Kahler-Einstein (complex) contact manifolds Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKahler metrics defined by Kronheimer, are shown to give rise to quaternionic Kahler metrics and various examples of these metrics are identified It is shown that any quaternionic Kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds The twistor space structure of the projectivised nilpotent orbits is studied

Hyperkähler and quaternionic Kähler geometry

In conformal geometry, the Sobolev inequality at a critical exponent has received much attention. In particular, the determination of the best constants has played a crucial role in the Yamabe problem. In dimension two the analogous problem deals with the Moser-Trudinger inequality: on a compact Riemann surface M2, there exists a constant c = c(M) so that f e47w2 < c(M) if f IVw12 < 1 and f w = 0. The connection of this inequality with geometry comes through the zeta functional determinant of the Laplacian as defined by Ray-Singer: for a Riemannian metric g, let 0 < A1 < A2 < ... be the spectrum

Extremal metrics of zeta function determinants on 4-manifolds

/pdf/certain-conditions-for-a-riemannian-manifold-to-be-isometric-1yybwpny41.pdf

Certain conditions for a Riemannian manifold to be isometric with a sphere

https://www.jstage.jst.go.jp/article/jjm1924/26/0/26_0_43/_pdf

Affine connections on manifolds with almost complex, quaternion or Hermitian structure

A Lorentzian (n+1)-manifold $\left( {{{\tilde M}^{n + 1}},\tilde g} \right)$ is called (globally) static [1], [2] if $\tilde M$ is a product space ℝ x M of ℝ with an n-manifold Mn and the metric $\tilde g$ has the form 

$$\tilde g = - f{(x)^2}d{t^2} + x*g,$$

(0.1)

where $t:\tilde M \to $ and $x:\tilde M \to$ are the natural projections, g a Riemannian metric on M, and f a positive function on M. We consider Einstein’s equation on ($(\tilde M,\tilde g)$) with perfect fluid as a matter field, i.e., 

$$\mathop {Ric}\limits^ - \frac{1}{2}\tilde R\tilde g = (\mu + p)\eta \otimes \eta + p\tilde g,$$

(0.2)

where n is a l-form with $\tilde g\left( {\eta ,\eta } \right) = - 1$, whose associated vector field represents the flux of the fluid, and μ and p are functions on $\tilde M$ which are called the energy density and the pressure, respectively [l]. In other words, (0.2) says that, at each point of $\tilde M$, the Ricci tensor $\widetilde {Ric}$ has at most two distinct eigenvalues with multiplicities l and n. It is known [2] that under the condition (0.1), (0.2) is equivalent to the following equation on (M,g); 

$$Ric - \frac{{Hess\;f}}{f} = \frac{1}{n}\left( {R - \frac{{\Delta f}}{f}} \right)g$$

(0.3)

and then, there are relations; 

$$\mu = \frac{R}{2}$$



$$p = \frac{{n - 1}}{n}\left( {\frac{{\Delta f}}{f} - \frac{{n - 2}}{{2\left( {n - 1} \right)}}R} \right)$$

(0.4)

.

Conformally-Flatness and Static Space-Time

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1261413488

On $n$ -dimensional homogeneous spaces of Lie groups of dimension greater than $n(n-1)/2$ .

/pdf/hermitian-manifolds-with-quaternion-structure-af0fy2a63e.pdf

Morio Obata

Papers

Certain conditions for a Riemannian manifold to be isometric with a sphere

Affine connections on manifolds with almost complex, quaternion or Hermitian structure

Conformally-Flatness and Static Space-Time

On $n$ -dimensional homogeneous spaces of Lie groups of dimension greater than $n(n-1)/2$ .

Hermitian manifolds with quaternion structure