In this paper, we look for properties of gradient Yamabe solitons on top of warped product manifolds. Utilizing the maximum principle, we find lower bound estimates for both the potential function of the soliton and the scalar curvature of the warped product. By slightly modifying Li-Yau's technique so that we can handle drifting Laplacians, we were able to find three different gradient estimates for the warping function, one for each sign of the scalar curvature of the fiber manifold. As an application, we exhibit a nonexistence theorem for gradient Yamabe solitons possessing certain metric properties on the base of the warped product.2010 Mathematics Subject Classification. 53C21, 53C50, 53C25.
In this paper we provide a method capable of producing an infinite number of solutions for Einstein's equation on static spacetimes with perfect fluid as a matter field. All spacetimes of this type which are symmetric with respect to a given group of translations and whose spatial factor is conformally flat, are characterized. We use this method to give some exact solutions of the referred equation.
The purpose of this paper is to study gradient k-Yamabe solitons conformal to pseudo-Euclidean space. We characterize all such solitons invariant under the action of an (n − 1)-dimensional translation group. For rotational invariant solutions, we provide the classification of solitons with null curvatures. As an application, we construct infinitely many explicit examples of geodesically complete steady gradient k-Yamabe solitons conformal to the Lorentzian space.
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