The sphere theorems for manifolds with positive scalar curvature

Abstract

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if M n is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R 0 > σ n K max , where σ n ∈ ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. This gives a partial answer to Yau's conjecture on pinching theorem. Moreover, we prove that if M n (n ≥ 3) is a compact manifold whose (n − 2)-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition Ric (n−2) min > τ n (n−2)R 0 , where τ n ∈ ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker and the authors. * 2010 Mathematics Subject Classification. 53C20; 53C40. exists for all time and converges to a constant curvature metric as t → ∞. Here r g(t) denotes the mean value of the scalar curvature of g(t).Theorem B([10]). Let (M, g 0 ) be a compact, locally irreducible Riemannian manifold of dimension n(≥ 4). Assume that M × R 2 has nonnegative isotropic curvature, i.e.,for all orthonormal four-frames {e 1 , e 2 , e 3 , e 4 } and all λ, µ ∈ [−1, 1]. Then one of the following statements holds: (i) M is diffeomorphic to a spherical space form.(ii) n = 2m and the universal cover of M is a Kähler manifold biholomorphic to CP m . (iii) The universal cover of M is isometric to a compact symmetric space.On the other hand, some important work on sphere theorems for manifolds with positive Ricci curvature have been made by several geometers (see [3, 14, 21, 30, 36, 39], etc.). In 1990's, Cheeger, Colding and Petersen [14, 30] proved the following differentiable sphere theorem for manifolds with positive Ricci curvature.Theorem C. Let M n be a compact and simply connected Riemannian n-manifold with Ricci curvature Ric M ≥ n − 1. Suppose that one of the following conditions holds:where ω n = vol(S n ) and ε 1 (n) is some positive constant; (ii) λ n+1 < n + ε 2 (n), where λ n+1 is the (n + 1)-th eigenvalue of M and ε 2 (n) is some positive constant. Then M is diffeomorphic to S n . Let K(π) be the sectional curvature of M for 2-plane π ⊂ T x M , Ric(u) the Ricci curvature of M for unit vector u ∈ U x M . Set K max (x) := max π⊂TxM K(π), Ric min (x) := min u∈UxM Ric(u). Inspired by Shen's topological sphere theorem [36], the authors [44] obtained the following differentiable sphere theorem for manifolds of positive Ricci curvatures.Theorem D. Let M n be a compact Riemannian n-manifold. If Ric min > δ n (n − 1)K max , where δ n = 1 − 6 5(n−1) , then M is diffeomorphic to a spherical space form. In particular, if M is simply connected, then M is diffeomorphic to S n . Let M n be a submanifold in a Riemannian manifold M N . Denote by H and S the mean curvature and the s...

Four-Dimensional Compact Manifolds with Nonnegative Biorthogonal Curvature

“…A classical example obtained in [13] shows that n−1 n+2 is the best possible pinching for this conjecture (cf. Example 3.1 in [13]).…”

“…A classical example obtained in [13] shows that n−1 n+2 is the best possible pinching for this conjecture (cf. Example 3.1 in [13]). We also notice that if s is the scalar curvature of a Riemannian manifold M n , then the normalized scalar curvature of M n is given by s 0 = s n(n−1) .…”

Geometric, topological and differentiable rigidity of submanifolds in space forms

“…Making use of the convergence results of Hamilton and Brendle for Ricci flow and the Lawson-Simons formula for the nonexistence of stable currents, Gu and Xu [15] proved the following differentiable sphere theorem for submanifolds in space forms.…”

A result on Ricci curvature and the second Betti number

“…Finally we metion a differential sphere theorem for Ricci curvature obtained by Gu and Xu ( c.f. [3] theorem D).…”