We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative and satisfies the equation classically. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed
Mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. If the hypersurface is in general or generic position, then we explain what singularities can occur under the flow, what the flow looks like near these singularities, and what this implies for the structure of the singular set. At the end, we will briefly discuss how one may be able to use the flow in lowdimensional topology.
more » « less Award ID(s):
 1408398
 NSFPAR ID:
 10532055
 Publisher / Repository:
 American Mathematical Society
 Date Published:
 Journal Name:
 Bulletin of the American Mathematical Society
 Volume:
 52
 Issue:
 2
 ISSN:
 02730979
 Page Range / eLocation ID:
 297 to 333
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract C ^{1}manifold with cylindrical singularities. © 2017 Wiley Periodicals, Inc. 
Abstract We study the mean curvature flow in 3dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimensiontwo mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer untrapped initial surface, a condition which resembles the meanconvexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a global foliation of the past of an outermost MOTS, provided the null hypersurface admits an untrapped foliation asymptotically.

We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3‑manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman’s conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of 3‑manifolds—in particular to the generalized Smale conjecture—which will appear in a subsequent paper.more » « less

When a fluid stream in a conduit splits in order to pass around an obstruction, it is possible that one branch will be critically controlled while the other remains not so. This is apparently the situation in Pacific Ocean abyssal circulation, where most of the northward flow of Antarctic bottom water passes through the Samoan Passage, where it is hydraulically controlled, while the remainder is diverted around the Manihiki Plateau and is not controlled. These observations raise a number of questions concerning the dynamics necessary to support such a regime in the steady state, the nature of upstream influence and the usefulness of rotating hydraulic theory to predict the partitioning of volume transport between the two paths, which assumes the controlled branch is inviscid. Through the use of a theory for constant potential vorticity flow and accompanying numerical model, we show that a steadystate regime similar to what is observed is dynamically possible provided that sufficient bottom friction is present in the uncontrolled branch. In this case, the upstream influence that typically exists for rotating channel flow is transformed into influence into how the flow is partitioned. As a result, the partitioning of volume flux can still be reasonably well predicted with an inviscid theory that exploits the lack of upstream influence.more » « less

Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated singularities, especially in the locally complete intersection (lci) case. First, we reprove the fact that a
$k$ rational isolated singularity is$k$ Du Bois without any lci assumption. For isolated lci singularities, we give a complete characterization of the$k$ Du Bois and$k$ rational singularities in terms of standard invariants of singularities. In particular, we show that$k$ Du Bois singularities are$(k1)$ rational for isolated lci singularities. In the course of the proof, we establish some new relations between invariants of isolated lci singularities and show that many of these vanish. The methods also lead to a quick proof of an inversion of adjunction theorem in the isolated lci case. Finally, we discuss some results specific to the hypersurface case.