Non-parametric radially symmetric mean curvature flow with a free boundary Non-parametric radially symmetric mean curvature flow with a free boundary

The Helfrich functional, denoted by Hc0, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise Hc0 among all possible configurations. The functional integrates a spontaneous curvature parameter c0 together with the mean curvature of the bilayer and constraints on area and volume, either through an inclusion of osmotic pressure difference and tensile stress or otherwise. Using the mathematical concept of embedded orientable surface to represent the configuration of the bilayer, one might expect to be able to adapt methods from differential geometry and the calculus of variations to perform a fine analysis of bilayer configurations in terms of the parameters that it depends upon. In this article we focus upon the case of spherical red blood cells with a view to better understanding spherocytes and spherocytosis. We provide a complete classification of spherical solutions in terms of the parameters in the Helfrich model. We additionally present some further analysis on the rigidity and stability of spherocytes. Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise H c 0 among all possible configurations. The functional integrates a spontaneous curvature parameter c 0 together with the mean curvature of the bilayer and constraints on area and volume, either through an inclusion of osmotic pressure difference and tensile stress or otherwise. Using the mathematical concept of embedded orientable surface to represent the configuration of the bilayer, one might expect to be able to adapt methods from differential geometry and the calculus of variations to perform a fine analysis of bilayer configurations in terms of the parameters that it depends upon. In this article we focus upon the case of spherical red blood cells with a view to better understanding spherocytes and spherocytosis. We provide a complete classification of spherical solutions in terms of the parameters in the Helfrich model. We additionally present some further analysis on the rigidity and stability of spherocytes. Disciplines Engineering | Science and Technology Studies

Recently, fully nonlinear curvature flow of a certain class of axially symmetric hypersurfaces with boundary conditions time of existence was obtained, in the case of convex speeds ( J. A. McCoy et al., Annali di Matematica Pura ed Applicata 1-13, 2013). In this paper we remove the convexity condition on the speed in the case it is homogeneous of degree one in the principal curvatures and the boundary conditions are pure Neumann. Moreover, we classify the singularities of the flow of a larger class of axially symmetric hypersurfaces as Type I. Our approach to remove the convexity requirement on the speed is based upon earlier work of Andrews for evolving convex surfaces (B. H. Andrews, Invent Math 138(1):151-161, 1999; Calc Var Partial Differ Equ 39(3-4):649-657, 2010); these arguments for obtaining a 'curvature pinching estimate' may be adapted to this setting due to axial symmetry. As further applications of curvature pinching in this setting, we show that closed, convex, axially symmetric hypersurfaces contract under the flow to round points, and hypersurfaces contracting self-similarly are necessarily spheres. These results are new for n ≥ 3. Abstract. Recently, fully nonlinear curvature flow of a certain class of axially symmetric hypersurfaces with boundary conditions was considered and a partial characterisation of the finite maximal time of existence was obtained, in the case of convex speeds [MMW]. In this paper we remove the convexity condition on the speed in the case it is homogeneous of degree one in the principal curvatures and the boundary conditions are pure Neumann. Moreover, we classify the singularities of the flow of a larger class of axially symmetric hypersurfaces as Type I. Our approach to remove the convexity requirement on the speed is based upon earlier work of Andrews for evolving convex surfaces [An2, An4]; these arguments for obtaining a 'curvature pinching estimate' may be adapted to this setting due to axial symmetry. As further applications of curvature pinching in this setting, we show that closed, convex, axially symmetric hypersurfaces contract under the flow to round points, and hypersurfaces contracting self-similarly are necessarily spheres. These results are new for n ≥ 3.Mathematics Subject Classification (2010). Primary 35K55, 35R35, 53C44; Secondary 35K60.

We study the mean curvature flow of graphs with prescribed contact angle on a fixed, smooth hyperplane in Euclidean space. We obtain long time existence and convergence to a self similar solution of the mean curvature flow orthogonal to the fixed hyperplane.Abstract. We study the mean curvature flow of graphs with prescribed contact angle on a fixed, smooth hyperplane in Euclidean space. We obtain long time existence and convergence to a self similar solution of mean curvature flow orthogonal to the fixed hyperplane.2000 Mathematics Subject Classification. 53C44 and 58J35.

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

Abstract. The purpose of this paper is twofold: firstly, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface, and secondly, to illustrate the application of Killing vector fields in the preservation of graphicality for the mean curvature flow with free boundary. To this end we focus on the mean curvature flow of a topological annulus with inner boundary meeting a standard n-sphere in R n+1 perpendicularly and outer boundary fixed to an n − 1-sphere with radius R > 1 translated by a vector he n+1 for h ∈ R where {e i } i=1,...,n+1 is the standard basis of R n+1 . We call this the sphere problem. Our work is set in the context of graphical mean curvature flow with either symmetry or mean concavity/convexity restrictions. For rotationally symmetric initial data we obtain, depending on the exact configuration of the initial graph, either long time existence and convergence to a minimal hypersurface with boundary or the development of a finite-time curvature singularity. With reflectively symmetric initial data we are able to use Killing vector fields to preserve graphicality of the flow and uniformly bound the mean curvature pointwise along the flow. Finally we prove that the mean curvature flow of an initially mean concave/convex graphical surface exists globally in time and converges to a piece of a minimal surface.

In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such suppport hypersurfaces; the results obtained this way do not require a-priori any symmetry or topological restrictions.2000 Mathematics Subject Classification. 53C44 and 58J35.

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