| The study of Riemannian functionals has long history, and variational problems are important in Differential Geometry. In this paper, we study the variational problems of curvature functionals of four kinds on an n-dimension closed Riemannian manifold (M, g). We calculate the first and second variational formulas, and study the stability of the critical metrics of these functionals. The main results are the following four parts:1. When n≥5, we calculate the first variational formulas of the functionals Fa=∫M(σ2-ασ1)dv restricted on M1, and we prove:if the critical metirc of Fα|M1is locally conformally flat, then it satisfies that b:=σ2(g)-(n-2)/n-4ασ1(g) is constant, and if b+n/2(n-1)a2>0, then it is a metric of constant sectional curvature.2. Using the first variational formulas of Ft=V4/n-1fM(|Ric|2+tR2)dv by Gursky, Viaclovsky[1], we calculate the second variational formulas at the critical metrics on So, which is the subspace of S2(M)(the vector space of all symmetric (0,2) tensor fields). And we get critical metrics of Ft on torus, with the second variational formulas, we prove that for some special t, the second variation of Ft is non-negative on some subspace of S0.3. When n≥4, we calculate the first variational formula of the Weyl functional W=2/n∫M|W|n/2dv, and get critical metrics of’W on torus. We calculate the sec-ond variational formulas of W at the above critical metrics on torus, further, we get that the standard Einstein metric of S3(1)×S3(1)is strictly stable.4. We give a new proof on calculation of the first variational formulas of functionals (?)(p)=V(2p)/n-1∫Lpdv (1<p<n/2), where Lp are the p-th Gauss-Bonnet curvatures: When p=2, at the critical metric, we calculate the second variational formula of (?)(2) on S0, as an application of the formula, we prove that at the standard metric of the sphere go, the second variation of (?)(2) is non-positive on<S0(g0).We consider the variational problems of (?)(p) restricted on the conformal class [g], we compute the second variational formulas (p)|[g]at the critical metrics, as an application, weprove that at the standard metric of the sphere g0, the second variation of (p)|[g0]isnon-negative on [g0]. |
