Variational formulae of some functionals by the modified Schouten tensor

Abstract

In this paper, we mainly study variational formulae of functionals determined by the k-curvature of the modified Schouten tensor which is defined on the space ${\mathcal {G}}(M)$ of Riemannian metrics on a compact manifold M. Firstly, we consider the functional ${\mathcal {F}}_k^{\tau }$ given by

$$\begin{aligned} {\mathcal {F}}_k^{\tau }(g): =V^{-\frac{n-2k}{n}} \int \limits _M\sigma _k(P^{\tau })\, dv_g,\quad g\in {\mathcal {G}}(M), \end{aligned}$$

where $\sigma _k(P^{\tau })$ is the k-curvature of the modified Schouten tensor $P^{\tau }$ with $\tau \in {\mathbb {R}}$, and then prove that, for a critical metric g of the functional ${\mathcal {F}}_k^{\tau }$, $k\ge 3$, the k-curvature $\sigma _k(P^{\tau })$ is constant if the corresponding Cotton tensor vanishes. Secondly, we study a more general functional ${\mathcal {F}}_3^{\tau ,\theta }$ also given by the modified Schouten tensor $P^{\tau }$. As the result, we establish the corresponding Euler–Lagrange equation and then compute the second variational formulae for the functional ${\mathcal {F}}_3^{\tau ,\theta }$ restricted to a given conformal metric class.