A new variational characterization of 𝑛-dimensional space forms

Abstract

Abstract. A Riemannian manifold (M n , g) is associated with a Schouten (0, 2)-tensor Cg which is a naturally defined Codazzi tensor in case (M n , g) is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional, where M is the space of smooth Riemannian metrics on a compact smooth manifold M and {σ k (Cg ), 1 ≤ k ≤ n} is the elementary symmetric functions of the eigenvalues of Cg with respect to g. We prove that if n ≥ 5 and a conformally flat metric g is a critical point of F 2 | M 1 with F 2 [g] ≥ 0, then g must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of F 2 | M 1 with F 2 [g] ≥ 0 characterized the three-dimensional space forms.

A Reilly type integral formula and its applications

“…Moreover, tr g (T (k) ) = (n − k)σ k . When the metric is locally conformally flat, then we have divT (k) = 0, for example, see [5][6][7]17]. Therefore, we have Theorem 1.8.…”

Two Kazdan–Warner-type identities for the renormalized volume coefficients and the Gauss–Bonnet curvatures of a Riemannian metric

“…Proof of Theorem 1.1 in the case 2k = n. As in [Han 2006b], we will prove that for any conformal metric…”

“…In the next section, we first provide a general proof for Theorem 1.1 by adapting an ingredient in a preprint version [Han 2006b] of [Han 2006a], and using of a variation formula for v (2k) (g) established in [Graham 2009] and [Chang and Fang 2008]. Because of the explicit expression for v (6) (g) and potential applications to other related problems in low dimensions, we provide in Section 3 a self-contained proof for Theorem 1.1 in the case k = 3.…”

On the 2-scalar curvature

“…Hence 0 is be a minimax value of M σ 2 (g)dvol(g) and the product metric g 0 would be a critical point of σ 2 on the space of all metrics. If this is true, then a result in [40] implies that g 0 is a metric of constant sectional curvature. This certainly is false.…”