Boundary estimates for the Ricci flow

Abstract

Abstract. In this paper we consider the Ricci flow on manifolds with boundary with appropriate control on its mean curvature and conformal class. We obtain higher order estimates for the curvature and second fundamental form near the boundary, similar to Shi's local derivative estimates. As an application, we prove a version of Hamilton's compactness theorem in which the limit has boundary.Finally, we show that in dimension three the second fundamental form of the boundary and its derivatives are a priori controlled in terms of the ambient curvature and some non-collapsing assumptions. In particular, the flow exists as long as the curvature remains bounded, in contrast to the general case where control on the second fundamental form is also required. Then, for every positive integer j there exists C j = C(n, j) > 0 such that. Such estimates not only reveal the smoothing character of Ricci flow, but they are also an essential ingredient of a compactness theorem for sequences of Ricci flows, proven by Hamilton in [10]. This theorem allows the blow-up analysis of singularities, and is an important tool in the study of the global behaviour of the flow.When M is a manifold with boundary, although there have been several local existence results for the Ricci flow, very little is known regarding its global behaviour, even in dimension 3.Regarding local existence, Shen in [17] and Pulemotov in [16] consider natural Neumann-type boundary conditions. On the other hand, in [9] the author considers a mixed Dirichlet-Neumann boundary value problem for the Ricci flow, motivated by work of Anderson on boundary value problems for Einstein metrics in [4], where the conformal class of the boundary and its mean curvature provide an elliptic boundary value problem for the Einstein equations.

Contracting convex surfaces by mean curvature flow with free boundary on convex barriers

“…Proof of Claim: By (3.13), we have h ij ≥ 11 12 Dg ij at t = 0 provided that D is sufficiently large depending on S. Therefore, H ≥ 11 6 D at t = 0 and the claim follows from Corollary 2.7. Note that Cauchy-Schwarz inequality implies that for all t ∈ [0, T ), we have whenever D ≥ 1,…”

“…It has been a longstanding question to define Ricci flow with boundary which is well-posed for general initial data. Recently, there has been some remarkable progress made by Gianniotis [12,11]. Short-time existence and regularity were established under certain general geometric boundary conditions which are related to the boundary value problems for Einstein metrics posed by Anderson [1,2].…”

Cohomogeneity-one quasi-Einstein metrics

“…In this case, we search for a quasi-Einstein metric (g, u) on G/H × (0, 1) which can be smoothly extended to G/H ×[0, 1] so that for i = 0, 1, (g, u) coincides with (ĝ i , u(i)) when restricted to G/H × {i}, whereĝ i is a fixed G-invariant Riemannian metric on G/H, and u(i) is a fixed real number. The Dirichlet problem for Einstein metrics has been studied in [1,6], but various other boundary-value problems for equations involving the Ricci curvature have also been studied by a number of authors; for example, see [1,6,28,26,25,24,4,5,17,10,16,11].…”

The Ricci flow on domains in cohomogeneity one manifolds

“…At the same time, describing the behaviour of these solutions for large t remains a challenging open problem. In the recent work [14], Gianniotis made progress towards the resolution of this problem by producing several interesting estimates. However, a comprehensive long-time existence theorem is still out of reach.…”