Prescribing Ricci curvature on a product of spheres

@article{Buttsworth2020PrescribingRC,
  title={Prescribing Ricci curvature on a product of spheres},
  author={Timothy Buttsworth and Anusha M. Krishnan},
  journal={Annali di Matematica Pura ed Applicata (1923 -)},
  year={2020},
  volume={201},
  pages={1-36}
}
We prove an existence result for the prescribed Ricci curvature equation for certain doubly warped product metrics on $$\mathbb {S}^{d_1+1}\times \mathbb {S}^{d_2}$$ S d 1 + 1 × S d 2 , where $$d_i \ge 2$$ d i ≥ 2 . If T is a metric satisfying certain curvature assumptions, we show that T can be scaled independently on the two factors so as to itself be the Ricci tensor of some metric. This is the first global existence result for the prescribed Ricci curvature problem on closed cohomogeneity… 
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References

SHOWING 1-10 OF 25 REFERENCES

ON THE RICCI ITERATION FOR HOMOGENEOUS METRICS ON SPHERES AND PROJECTIVE SPACES

We study the Ricci iteration for homogeneous metrics on spheres and complex projective spaces. Such metrics can be described in terms of modifying the canonical metric on the fibers of a Hopf

The Dirichlet problem for the prescribed Ricci curvature equation on cohomogeneity one manifolds

Let M be a domain enclosed between two principal orbits on a cohomogeneity one manifold $$M_1$$M1. Suppose that T and R are symmetric invariant (0, 2)-tensor fields on M and $$\partial M$$∂M,

The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair

On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the

Cohomogeneity one manifolds with positive Ricci curvature

One of the central problems in Riemannian geometry is to determine how large the classes of manifolds with positive/nonnegative sectional -, Ricci or scalar curvature are (see [Gr]). For scalar

Ricci curvature in the neighborhood of rank-one symmetric spaces

We study the Ricci curvature of a Riemannian metric as a differential operator acting on the space of metrics close (in a weighted functional spaces topology) to the standard metric of a rank-one

The initial value problem for cohomogeneity one Einstein metrics

The PDE Ric(g) = λ · g for a Riemannian Einstein metric g on a smooth manifold M becomes an ODE if we require g to be invariant under a Lie group G acting properly on M with principal orbits of

Studies of Some Curvature Operators in a Neighborhood of an Asymptotically Hyperbolic Einstein Manifold

Abstract On an asymptotically hyperbolic Einstein manifold ( M , g 0 ) for which the Yamabe invariant of the conformal structure on the boundary at infinity is nonnegative, we show that the operators

Cohomogeneity-one quasi-Einstein metrics