A local curvature estimate for the Ricci flow

Abstract

We show that the norm of the Riemann curvature tensor of any smooth solution to the Ricci flow can be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensor, and the elapsed time. This provides a new, direct proof of a result ofŠesǔm, which asserts that the curvature of a solution on a compact manifold cannot blow up while the Ricci curvature remains bounded, and extends its conclusions to the noncompact setting. We also prove that the Ricci curvature must blow up at least linearly along a subsequence at a finite time singularity.

Local curvature estimates for the Laplacian flow

“…As remarked in [25], to prove Theorem 1.3, it suffices to establish the following integral estimate. Theorem 1.4 Let M be a smooth 7-manifold and ϕ(t), t ∈ [0, T ), where T < ∞, be a solution to the flow (1.1) for closed G 2 -structures with associated metric g(t) = g ϕ(t) for each t. Assume that there exist constants A, K > 0 and a point x 0 ∈ M such that the geodesic ball B g(0) (x 0 , A/ √ K ) is compactly contained in M and that…”

“…In this paper, we give a new elementary proof of Theorem 1.2, based on the idea of [25] and the structure of the Eq. (1.1).…”

Mixed Integral Norms for Ricci Flow

“…Sesum (for closed manifolds) and later Ma and Cheng (for complete manifolds with bounded curvature) showed that indeed a bound on the Ricci curvature rather than the full Riemannian curvature tensor suffices to extend the flow, see [13,15]. See also [8] for a local version of the result. In their paper, Ma and Cheng also proved the extensibility of a complete Ricci flow under the assumption of bounded scalar curvature and Weyl tensor.…”

Consensus of Multiagent System in the Sense of Curvature and Torsion

“…In the last decades, curvature and torsion are taken as the efficient tools to solve some problems in different communities, such as fluid dynamics [1,2] and mathematics physics [3]. In recent years, the further research on curvature and torsion has been investigated in [4][5][6][7][8], and the results provide more ways to solve the consensus problems. However, as the efficient tools, curvature and torsion have not attracted much attention in the control field of multiagent system (MAS).…”