On the classification of warped product Einstein metrics

Abstract

In this paper we take the perspective introduced by Case-Shu-Wei of studying warped product Einstein metrics through the equation for the Ricci curvature of the base space. They call this equation on the base the m-quasi Einstein equation, but we will also call it the (λ, n + m)-Einstein equation. In this paper we extend the work of Case-Shu-Wei and some earlier work of Kim-Kim to allow the base to have non-empty boundary. This is a natural case to consider since a manifold without boundary often occurs as a warped product over a manifold with boundary, and in this case we get some interesting new canonical examples. We also derive some new formulas involving curvatures that are analogous to those for the gradient Ricci solitons. As an application, we characterize warped product Einstein metrics when the base is locally conformally flat.

Uniqueness of Warped Product Einstein Metrics and Applications

“…This shows that if w is a positive solution to (1.1) and m > 1, then there is a warped product Einstein metric with base (M, g M ). When w vanishes on ∂ M, one also always obtains a smooth Einstein metric; see Proposition 1.1 in [8].…”

ON THE SOLUTIONS OF THE (λ, n + m)-EINSTEIN EQUATION

“…Note that on the whole manifold M , the metric tensor g and the functions f,f are real analytic by Proposition 2.4 of [5]. In particular, ϕ is real analytic, implying that the set ∇ϕ = 0 is not open unless ϕ is trivial.…”

“…We remark that if λ > 0 and M is complete, possibly with non-empty boundary, M is known to be compact by [5].…”

“…Recently, there has been an increasing interest in the study of the extension of the Ricci tensor called the m-BakryEmery Ricci tensor (c.f. [3], [5]). It is given by It is well known that when m is a positive integer, the m-quasi Einstein metrics correspond to certain warped product Einstein metrics (c.f.…”

“…It is given by It is well known that when m is a positive integer, the m-quasi Einstein metrics correspond to certain warped product Einstein metrics (c.f. [3], [5]). It is clear that if we take m to infinity, we obtain the gradient Ricci soliton equation…”