Sharp H\"older continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

We prove a new kind of estimate that holds on any manifold with lower Ricci bounds. It relates the geometry of two small balls with the same radius, potentially far apart, but centered in the interior of a common minimizing geodesic. It reveals new, previously unknown, properties that all generalized spaces with a lower Ricci curvature bound must have and it has a number of applications. This new kind of estimate asserts that the geometry of small balls along any minimizing geodesic changes in a H\"older continuous way with a constant depending on the lower bound for the Ricci curvature, the dimension of the manifold, and the distance to the end points of the geodesic. We give examples that show that the H\"older exponent, along with essentially all the other consequences that we show follow from this estimate, are sharp. The unified theme for all of these applications is convexity. Among the applications is that the regular set is convex for any non-collapsed limit of Einstein metrics. In the general case of potentially collapsed limits of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and $a.e.$ convex, that is almost every pair of points can be connected by a minimizing geodesic whose interior is contained in the regular set. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group; the key point for this is to rule out small subgroups. The other asserts that the dimension of any limit space is the same everywhere. Finally, we show that a Reifenberg type property holds for collapsed limits and discuss why this indicate further regularity of manifolds and spaces with Ricci curvature bounds.Comment: 48 page

Regularity of the Level Set Flow

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 C[superscript 1] manifold with cylindrical singularities. © 2017 Wiley Periodicals, Inc.National Science Foundation (U.S.) (Grant DMS 1404540)National Science Foundation (U.S.) (Grant DMS 1206827

Effect of fluid forces on rotor stability of centrifugal compressors and pumps

A simple two dimensional model for calculating the rotordynamic effects of the impeller force in centrifugal compressors and pumps is presented. It is based on potential flow theory with singularities. Equivalent stiffness and damping coefficients are calculated for a machine with a vaneless volute formed as a logarithmic spiral. It is shown that for certain operating conditions, the impeller force has a destablizing effect on the rotor

The min--max construction of minimal surfaces

In this paper we survey with complete proofs some well--known, but hard to find, results about constructing closed embedded minimal surfaces in a closed 3-dimensional manifold via min--max arguments. This includes results of J. Pitts, F. Smith, and L. Simon and F. Smith.Comment: 42 pages, 13 figure

The singular set of mean curvature flow with generic singularities

A mean curvature flow starting from a closed embedded hypersurface in $R^{n+1}$ must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact embedded $(n-1)$-dimensional Lipschitz submanifolds plus a set of dimension at most $n-2$. If the initial hypersurface is mean convex, then all singularities are generic and the results apply. In $R^3$ and $R^4$, we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time-slice. For $2$ or $3$-convex hypersurfaces in all dimensions, the same arguments lead to the same conclusion: the flow is completely smooth at almost all times and connected components of the singular set are contained in time-slices. A key technical point is a strong {\emph{parabolic}} Reifenberg property that we show in all dimensions and for all flows with only generic singularities. We also show that the entire flow clears out very rapidly after a generic singularity. These results are essentially optimal