## Tohoku Mathematical Journal 2015 June SECOND SERIES VOL. 67, NO. 2

 Tohoku Math. J. 67 (2015), 281-295

Title ON THE STRUCTURE OF LINEARIZATION OF THE SCALAR CURVATURE

Author Gabjin Yun, Jeongwook Chang and Seungsu Hwang

(Received September 17, 2013, revised April 14, 2014)
Abstract. For a compact \$n\$-dimensional manifold a critical point metric of the total scalar curvature functional satisfies the critical point equation (1) below, if the functional is restricted to the space of constant scalar curvature metrics of unit volume. The right-hand side in this equation is nothing but the adjoint operator of the linearization of the total scalar curvature acting on functions. The structure of the kernel space of the adjoint operator plays an important role in the geometry of the underlying manifold.
In this paper, we study some geometric structure of a given manifold when the kernel space of the adjoint operator is nontrivial. As an application, we show that if there are two distinct solutions satisfying the critical point equation mentioned above, then the metric should be Einstein. This generalizes a main result in \cite{HCY1} to arbitrary dimension.

Mathematics Subject Classification. Primary 53C25; Secondary 58E11.

Key words and phrases. Total scalar curvature functional, critical point metric, Einstein metric.