Abstract
In this paper, on 4-spheres equipped with Riemannian metrics we study some integral conformal invariants, the sign and size of which under Ricci flow characterize the standard 4-sphere. We obtain a conformal gap theorem, and for Yamabe metrics of positive scalar curvature with L^2 norm of the Weyl tensor of the metric suitably small, we establish the monotonic decay of the L^p norm for certain p>2 of the reduced curvature tensor along the normalized Ricci flow, with the metric converging exponentially to the standard 4-sphere.
Highlights
We start by recalling some earlier works of Chang–Gursky–Yang [5]
Under what conditions on the curvature can we conclude that a smooth, closed Riemannian manifold is diffeomorphic to the sphere? A result which addresses this question is usually referred to as a sphere theorem, and the literature abounds with examples
In an earlier work of Chang–Gursky–Yang, they have reformulated the result of Margerin by showing that the smooth four-sphere is characterized by an integral curvature condition
Summary
We start by recalling some earlier works of Chang–Gursky–Yang [5]. Under what conditions on the curvature can we conclude that a smooth, closed Riemannian manifold is diffeomorphic (or homeomorphic) to the sphere? A result which addresses this question is usually referred to as a sphere theorem, and the literature abounds with examples. If the Gauss curvature of the surface (M 2, g) satisfies KdA > 0, M 2 is diffeomorphic to S2 or RP 2 In addition to this topological classification, the uniformization theorem implies that (M 2, g) is conformal to a surface of constant curvature, which is covered isometrically by. In an earlier work of Chang–Gursky–Yang, they have reformulated the result of Margerin by showing that the smooth four-sphere is characterized by an integral curvature condition.
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