The Ricci flow on manifolds with boundary

The Ricci flow on surfaces with boundary

“…Therefore, if the Ricci flow (1) cannot be extended past T < ∞, the curvature blows up. We invite the reader to consult the recent work of Gianniotis [17], where this property of the Ricci flow on manifolds with boundary is discussed in a more general context. 2.2.…”

“…One of the main difficulties in studying this problem arises from the fact that even trying to impose meaningful boundary conditions for the Ricci flow, for which existence and uniqueness results can be proved so interesting geometric applications can be hoped for, seems to be a challenging task. For the reader to get an idea of the difficulty of the problem, we recommend the interesting works of Y. Shen [29], S. Brendle [4], A. Pulemotov [28] and P. Gianniotis [17]. In the case of the boundary conditions imposed by Shen [29], satisfactory convergence results have given for manifolds of positive Ricci curvature and totally geodesic boundary, and also when the boundary is convex and the metric is rotationally symmetric.…”

The Ricci flow on domains in cohomogeneity one manifolds

“…For instance, it is necessary to find boundary conditions for equation (1.1) that would allow tractable analysis and admit a meaningful geometric interpretation. Doing so is difficult, as equation (1.1) is only weakly parabolic; see [27,Section 5.1], [13,Introduction] and also [2,Secion 3]. Note that the Ricci flow on surfaces with boundary appears to be more approachable.…”

“…Note that the Ricci flow on surfaces with boundary appears to be more approachable. For results in this area, consult the references in [13]. However, the higher-dimensional setting considered in the present paper encompasses a different set of difficulties and requires different techniques.…”

The Dirichlet problem for the prescribed Ricci curvature equation on cohomogeneity one manifolds

“…In a similar fashion, new knowledge about boundary-value problems for (1.1) may help answer questions about the existence and uniqueness of solutions to boundary-value problems for the Ricci flow and the Einstein equation. Such questions were investigated in [29,1,10,5,2,17,23,25] and other works. A large number still remain open.…”