lifespan theorem for constrained surface diffusion flows we consider closed immersed hypersurfaces in mathbb r 3 and mathbb 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

hyperbolic mean curvature flow with a forcing term evolution of plane curves abstract in this paper we study the evolution of closed strictly convex plane curves moving by the hyperbolic mean curvature flow with a forcing term it is shown that the flow admits a unique short time smooth solution and the convexity of the curves is preserved during the evolution when the forcing term is a negative constant we prove the curves either converge to a point or a c 0 curve for a positive constant forcing term the flow has a unique smooth solution in any finite time and expands to infinity as t tends to infinity if the initial curvature is smaller than m the flow will blow up in a finite time if the initial curvature is larger than m

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

non parametric radially symmetric mean curvature flow with a free boundary we study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed smooth hypersurface in euclidean space in this paper we treat two distinct problems the first problem has a free neumann boundary only while the second has two disjoint boundaries a free neumann boundary and a fixed dirichlet height we separate the two problems and prove that under certain initial conditions we have either long time existence followed by convergence to a minimal surface or finite maximal time of existence at the end of which the graphs develop a curvature singularity we also give a rate of convergence for the singularity

curvature flow with a general forcing term in euclidean spaces abstract we consider closed convex hypersurfaces moving in euclidean spaces with normal velocity equal to h f where h h t is a nonnegative continuous function of t and f is evaluated at the principal curvatures and satisfies the standard conditions we study long time existence and convergence of the evolving hypersurfaces in three different cases which include andrews contractive case mccoy s mixed volume preserving case and the additional expanding case

on the determinant of elliptic differential and finite difference operators in vector bundles over s 1 for an elliptic differential operatora overs 1 a sum limits k 0 n a k x d k witha k x in end r and as a principal angle the regularized determinant det a is computed in terms of the monodromy mapp a associated toa and some invariant expressed in terms ofa n anda n 1 a similar formula holds for finite difference operators a number of applications and implications are given in particular we present a formula for the signature ofa whena is self adjoint and show that the determinant ofa is the limit of a sequence of computable expressions involving determinants of difference approximation ofa

flow of nonconvex hypersurfaces into spheres the flow of surfaces by functions of their principal curvatures has been intensively studied it started with the work of brakke 1 who used the formalism of geometric measure theory a more classical approach had been chosen by huisken who looked at the so called inward flow in 5 the outward flow of surfaces this term will be explained in the sequel is scaling invariant and therefore more natural than the inward flow producing more general results so far huisken 6 and urbas 8 proved that the outward flow of convex surfaces by suitable functions of their principal curvatures converges to spheres the convexity assumption is essential in their work in this paper we would like to present a different method for proving the convergence of star shaped surfaces into spheres via the outward flow let be a symmetric positive function homogeneous of degree one being defined on an open cone of rn with vertex in the origin which contains the positive diagonal i e all tuples of the form

determinants finite difference operators and boundary value problems we relate the determinants of differential and difference operators to the boundary values of solutions of the operators previous proofs of related results have involved considering one parameter families of such operators showing the desired quantities are equal up to a constant and then calculating the constant we take a more direct approach for a fixed operator we prove immediately that the two sides of our formulas are equal by using the following simple observation proposition 1 3 let u su n c write u in block form u left begin array 20 c u 11 u 12 u 21 u 22 end array right where u11and u22are square matrices then det u 11 overline det u 22

mean curvature flow of entire graphs in a half space with a free boundary 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

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

isogeometric analysis of geometric partial differential equations abstract we consider the numerical approximation of geometric partial differential equations pdes defined on surfaces in the 3d space in particular we focus on the geometric pdes deriving from the minimization of an energy functional by l 2 gradient flow we analyze two energy functionals the area which leads to the mean curvature flow a nonlinear second order pde and the willmore energy leading to the willmore flow a nonlinear fourth order pde we consider surfaces represented by single patch tensor product nurbs and discretize the pdes by means of nurbs based isogeometric analysis in the framework of the galerkin method to approximate the high order geometric pdes we use high order continuous nurbs basis functions for the time discretization of the nonlinear geometric pdes we use backward differentiation formulas bdf with extrapolation of the geometric quantities involved in the weak formulation of the problem in this manner we solve a linear problem at each time step we report numerical results concerning the mean curvature and willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme

pinching estimates and motion of hypersurfaces by curvature functions second derivative pinching estimates are proved for a class of elliptic and parabolic equations including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature the estimates imply convergence of convex hypersurfaces to spheres under these flows improving earlier results of b chow and the author the result is obtained via a detailed analysis of gradient terms in the equations satisfied by second derivatives

parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes are constructed as asymptotic limits of a geometric evolution equation in particular an alternative constructive proof is given for the existence of maximal and constant mean curvature slices

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

asymptotic behavior of flows by powers of the gaussian curvature we consider a 1 parameter family of strictly convex hypersurfaces in mathbb r n 1 moving with speed k alpha where denotes the outward pointing unit normal vector and alpha geqslant 1 n 2 for alpha gt 1 n 2 we show that the flow converges to a round sphere after rescaling in the affine invariant case alpha 1 n 2 our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling

evolution of geometric statistics a model for the description of the geometry in multiphase mechanics is derived and studied equations for the rate of change of mean and gauss curvature are derived these coupled equations show that the evolution of these curvatures depend on the velocity of the interface normal to itself examples are given to show the utility of these equations to apply these concepts to multiphase mechanical situations an ensemble average is applied to the curvature equations equations for the evolution of the volume fraction and the interfacial area density are also derived these equations also depend on the curvatures and the average interfacial velocity examples are given to show that these equations give a description of the geometry of multiphase mechanics that is superior to the present description which describes the force between materials as depending on two geometric parameters the volume fraction and the average radius of the dispersed material these examples also suggest constitutive equations to d

inverse mean curvature flow with forced term abstract in this paper we study the evolution of convex hypersurfaces by inverse mean curvature minus an external force field c we prove that the flow will preserve the convexity for any c if c 1 h on the initial surface we prove that the flow will expand the hypersurface for all time and after rescaling the hypersurface will converge to a sphere if c 1 h on the initial surface we show that the maximum existence time of the flow is finite and the hypersurface will contract to a point when approaching the final time

anisotropic flow of convex hypersurfaces by the square root of the scalar curvature abstract we show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor when the strictly convex initial hypersurface in euclidean space is suitably pinched we also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere for a general case in dimension two it is shown that with a volume preserving rescaling the limit profile satisfies a soliton equation

determinants and periodic solutions of delay equations finite difference equations may be thought of as discrete analogues of delay equations taking this point of view we give an elementary account of an algebraic determinant identity due to burghelea friedlander and kappeler which relates the determinant of a periodic difference operator to the monodromy of a fundamental solution the result is applied to a simple class of functional integral operators

genus for groups abstract we introduce a notion of a genus g c g for a class of groups c and g c it consists of isomorphism classes of groups from c having the same profinite completion as g we show finiteness results for g c g for several important families of groups including finitely generated virtually free groups we develop formulas for the number of elements in g c g in various cases by these we find interesting examples where g c g contains only one element

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

self shrinkers for the mean curvature flow in arbitrary codimension in this paper we generalize colding minicozzi s recent results about codimension 1 self shrinkers for the mean curvature flow to higher codimension in particular we prove that the sphere bf s n sqrt 2n is the only complete embedded connected f stable self shrinker in mathbf r n k with mathbf h ne 0 polynomial volume growth flat normal bundle and bounded geometry we also discuss some properties of symplectic self shrinkers proving that any complete symplectic self shrinker in mathbf r 4 with polynomial volume growth and bounded second fundamental form is a plane as a corollary we show that there is no finite time type i singularity for symplectic mean curvature flow which has been proved by chen li using different method we also study lagrangian self shrinkers and prove that for lagrangian mean curvature flow the blow up limit of the singularity may be not f stable

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

determinants of dirac boundary value problems over odd dimensional manifolds we present a canonical construction of the determinant of an elliptic selfadjoint boundary value problem for the dirac operatord over an odd dimensional manifold for 1 dimensional manifolds we prove that this coincides with the function determinant this is based on a result that elliptic self adjoint boundary conditions ford are parameterized by a preferred class of unitary isomorphisms between the spaces of boundary chiral spinor fields with respect to a decompositions 1 x 0 x 1 we explain how the determinant of a dirac type operator overs 1 is related to the determinants of the corresponding boundary value problems overx 0 andx 1

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface

flow by powers of the gauss curvature abstract we prove that convex hypersurfaces in r n 1 contracting under the flow by any power 1 n 2 of the gauss curvature converge after rescaling to fixed volume to a limit which is a smooth uniformly convex self similar contracting solution of the flow under additional central symmetry of the initial body we prove that the limit is the round sphere for 1

convexity and horizontal second fundamental forms for hypersurfaces in carnot groups we use a riemannian approximation scheme to give a characterization for smooth convex functions on a carnot group in the sense of danielli garofalo nhieu or lu manfredi stroffolini in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph