We study a flow of G2-structures that all induce the same Riemannian metric. This isometric flow is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor T along the flow. We show that at a finite-time singularity the torsion must blow up, so the flow will exist as long as the torsion remains bounded. We prove a Cheeger-Gromov type compactness theorem for the flow. We describe an Uhlenbeck-type trick which together with a modification of the underlying connection yields a nice reaction-diffusion equation for the torsion along the flow. We define a scale-invariant quantity Θ for any solution of the flow and prove that it is almost monotonic along the flow. Inspired by the work of Colding-Minicozzi [CM12] on the mean curvature flow, we define an entropy functional and after proving an ε-regularity theorem, we show that low entropy initial data lead to solutions of the flow that exist for all time and converge smoothly to a G2-structure with divergence-free torsion. We also study finite-time singularities and show that at the singular time the flow converges to a smooth G2-structure outside a closed set of finite 5-dimensional Hausdorff measure. Finally, we prove that if the singularity is of Type-I then a sequence of blow-ups of a solution admits a subsequence that converges to a shrinking soliton for the flow.

We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial data. Finally, we establish that under suitable control of the boundary data the flow exists as long as the ambient curvature and the second fundamental form of the boundary remain bounded.

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.

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In a singular Type I Ricci flow, we consider a stratification of the set where there is curvature blow-up, according to the number of the Euclidean factors split by the tangent flows. We then show that the strata are characterized roughly in terms of the decay rate of their volume, which in our context plays the role of a dimension estimate.