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…
One Citation
On the variational properties of the prescribed Ricci curvature functional
- Mathematics
- 2021
We study the prescribed Ricci curvature problem for homogeneous metrics. Given a (0,2)-tensor field T , this problem asks for solutions to the equation Ric(g) = cT for some constant c. Our approach…
References
SHOWING 1-10 OF 24 REFERENCES
ON THE RICCI ITERATION FOR HOMOGENEOUS METRICS ON SPHERES AND PROJECTIVE SPACES
- Mathematics
- 2018
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
- Mathematics
- 2013
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
- MathematicsMathematische Nachrichten
- 2018
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*
- Mathematics
- 1978
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
- Mathematics
- 2002
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
- Mathematics
- 2001
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
- Mathematics
- 2000
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
- Mathematics
- 2002
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
- MathematicsJournal of Mathematical Analysis and Applications
- 2019