A compactness theorem for a fully nonlinear Yamabe problem under a lower Ricci curvature bound

Abstract

We prove compactness of solutions of a fully nonlinear Yamabe problem satisfying a lower Ricci curvature bound, when the manifold is not conformally diffeomorphic to the standard sphere. This allows us to prove the existence of solutions when the associated cone Γ satisfies µ + Γ ≤ 1, which includes the σ k −Yamabe problem for k not smaller than half of the dimension of the manifold.

Rigidity Theorems for Diameter Estimates of Compact Manifold with Boundary

“…Firstly, for convenience of readers, we include the proof of (1.2) here (see [12]). For any point x ∈ N , since N is complete, there exists a geodesic γ : [0, d] → N parametrized by arc length with γ(0) = x, γ(d) ∈ ∂N and d = d(x, ∂N ).…”

“…The proof of Theorem 1.1 will be given in section 2. We first include the proof of the distance bound (1.2) for convenience of readers (see [12]). Then we consider the rigidity result when the equality occurs in (1.2).…”

“…Assume that the Ricci curvature of N has a negative lower bound Ric ≥ −(n − 1)c 2 for some c > 0, and the mean curvature of the boundary ∂N satisfies H ≥ (n − 1)c0 > (n − 1)c for some c0 > c > 0. Then a known result (see [12]) says that sup x∈N d(x, ∂N ) ≤ 1 c coth −1 c 0 c . In this paper, we prove that if the boundary ∂N is compact, then the equality holds if and only if N is isometric to the geodesic ball of radius 1 c coth −1 c 0 c in an n-dimensional hyperbolic space H n (−c 2 ) of constant sectional curvature −c 2 .…”

Volumes and limits of manifolds with Ricci curvature and mean curvature bounds

“…Then define the function π : M \ cut(∂M) → ∂M which assigns to each point p in the domain the unique point in the boundary that equals r(p). Then we prove the theorem of Li and Li-Nguyen, [12] and [13], respectively, that gives an upper bound on r when H ∂M ≤ H < 0. With that bound we get an upper estimate of the diameter of M in Remark 2.6.…”

“…Thus, the diameter of the manifold is bounded below by this distance. Li and Li-Nguyen in [12] and [13], respectively, proved that if (M n , g) is a complete connected Riemannian manifold with smooth boundary such that Ric(M \ ∂M) ≥ 0 and H ∂M ≤ H < 0 then r ≤ −(n − 1)/H. Hence, Diam(M) can be bounded in terms of −(n − 1)/H and Diam(∂M).…”