Curvature contraction flows in the sphere

We show that convex surfaces in an ambient three-sphere contract to round points in finite time under fully nonlinear, degree one homogeneous curvature flows, with no concavity condition on the speed. The result extends to convex axially symmetric hypersurfaces of the (n+1)-dimensional sphere. Using a different pinching function we also obtain the analogous results for contraction by Gauss curvature

Finite time singularities for the locally constrained Willmore flow of surfaces

In this paper we study the steepest descent L2-gradient flow of the functional Wλ1,λ2, which is the the sum of the Willmore energy, λ1-weighted surface area, and λ2-weighted enclosed volume, for surfaces immersed in R3. This coincides with the Helfrich functional with zero `spontaneous curvature\u27. Our first results are a concentration-compactness alternative and interior estimates for the flow. For initial data with small energy, we prove preservation of embeddedness, and by directly estimating the Euler-Lagrange operator from below in L2 we obtain that the maximal time of existence is finite. Combining this result with the analysis of a suitable blowup allows us to show that for such initial data the flow contracts to a round point in finite time

The potential impact on Florida-based marina and boating industries of a post-embargo Cuba: an analysis of geographic, physical, policy and industry trends

The information in this Technical Paper addresses the future of the US-Cuban marina and recreational boating industries from the geographic, physical, policy making and economic perspectives for a post-embargo Cuba. Each individual paper builds on the presentations made at the workshop, the information obtained in the subsequent trip to Cuba and presents in detailed form information which we hope is useful to all readers. (147pp.

Contraction of convex surfaces by nonsmooth functions of curvature

We consider the motion of convex surfaces with normal speed given by arbitrary strictly monotone, homogeneous degree one functions of the principal curvatures (with no further smoothness assumptions). We prove that such processes deform arbitrary uniformly convex initial surfaces to points in finite time, with spherical limiting shape. This result was known previously only for smooth speeds. The crucial new ingredient in the argument, used to prove convergence of the rescaled surfaces to a sphere without requiring smoothness of the speed, is a surprising hidden divergence form structure in the evolution of certain curvature quantities

Convexity estimates for hypersurfaces moving by convex curvature functions

We consider the evolution of compact hypersurfaces by fully non-linear, parabolic curvature ows for which the normal speed is given by a smooth, convex, degree one homoge- neous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the ow. The result extends the convexity estimate [HS99b] of Huisken and Sinestrari for the mean curvature ow to a large class of speeds, and leads to an analogous description of `type-II\u27 singularities. We remark that many of the speeds considered are positive on larger cones than the positive mean half-space, so that the result in those cases also applies to non-mean-convex initial data

Non-collapsing in fully nonlinear curvature flows

We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a non-collapsing estimate: Precisely, the function which gives the curvature of the largest interior sphere touching the hy- persurface at each point is a subsolution of the linearized flow equation if the speed is concave. If the speed is convex then there is an analogous statement for exterior spheres. In particular, if the hypersurface moves with positive speed and the speed is concave in the principal curvatures, then the curvature of the largest touching inte- rior sphere is bounded by a multiple of the speed as long as the solution exists. The proof uses a maximum principle applied to a function of two points on the evolving hypersurface. We illustrate the techniques required for dealing with such functions in a proof of the known containment principle for flows of hypersurfaces

Compliance to Bleach Disinfection Protocols among Injecting Drug Users in Miami

Bleach cleansing of injection equipment has been recommended to reduce the risk of human immunodeficiency virus (HIV) transmission associated with the reuse of injection equipment by injecting drug users (IDUs). We evaluated the recall and performance of the most commonly recommended bleach cleansing procedure of two complete fillings of the syringe with bleach, followed by two complete fillings with rinse water, and not putting used bleach and water back into source containers. IDUs were taught this procedure on enrollment in an HIV prevention demonstration project in Dade County, Florida. During follow-up session 6-12 months after initial training, the knowledge and ability of IDUs to perform bleach cleansing were assessed by trained observers using a standardized method. In 1988-90, we assessed the knowledge and ability of 450 IDUs to perform the bleach cleansing procedure taught at enrollment. More than 90% of IDUs assessed performed the basic steps. However, only 43.1% completely filled the syringe with bleach and only 35.8% completely filled the syringe with bleach at least twice. Substantial proportions of IDUs did not perform all the steps of the previously taught bleach cleansing procedure. Compliance decreased as the number of steps required was increased. This limited compliance may make bleach cleansing less effective and suggests that some IDUs may fail to adequately disinfect injection equipment and therefore sterile needles and syringes are safer than bleach-cleansed ones. Compliance testing can help assess the effectiveness of HIV prevention programs. © 1994 Raven Press, Ltd., New York

Lifespan theorem for constrained surface diffusion flows

We consider closed immersed hypersurfaces in $\R^{3}$ and $\R^4$ evolving by a class of constrained surface diffusion flows. Our result, similar to earlier results for the Willmore flow, gives both a positive lower bound on the time for which a smooth solution exists, and a small upper bound on a power of the total curvature during this time. By phrasing the theorem in terms of the concentration of curvature in the initial surface, our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows.Comment: 29 pages. arXiv admin note: substantial text overlap with arXiv:1201.657

Curvature contraction of convex hypersurfaces by nonsmooth speeds

We consider contraction of convex hypersurfaces by convex speeds, homogeneous of degree one in the principal curvatures, that are not necessarily smooth. We show how to approximate such a speed by a sequence of smooth speeds for which behaviour is well known. By obtaining speed and curvature pinching estimates for the flows by the approximating speeds, independent of the smoothing parameter, we may pass to the limit to deduce that the flow by the nonsmooth speed converges to a point in finite time that, under a suitable rescaling, is round in the C^2 sense, with the convergence being exponential