tangent lines inflections and vertices of closed curves we show that every smooth closed curve immersed in euclidean space r3 satisfies the sharp inequality 2 p i v 6 which relates the numbers p of pairs of parallel tangent lines i of inflections or points of vanishing curvature and v of vertices or points of vanishing torsion of we also show that 2 p i v 4 where p is the number of pairs of concordant parallel tangent lines the proofs which employ curve shortening flow with surgery are based on corresponding inequalities for the numbers of double points singularities and inflections of closed curves in the real projective plane rp2 and the sphere s2 which intersect every closed geodesic these findings extend some classical results in curve theory from works of mobius fenchel and segre including arnold s tennis ball theorem

invariant geometric evolutions of surfaces and volumetric smoothing the study of geometric flows for smoothing multiscale representation and analysis of two and three dimensional objects has received much attention in the past few years in this paper we first survey the geometric smoothing of curves and surfaces via geometric heat type flows which are invariant under the groups of euclidean and affine motions second using the general theory of differential invariants we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group as an application we present the simplest affine invariant flow for convex surfaces in three dimensional space which like the affine invariant curve shortening flow will be of fundamental importance in the processing of three dimensional images

singularities of lagrangian mean curvature flow zero maslov class case we study singularities of lagrangian mean curvature flow in n when the initial condition is a zero maslov class lagrangian we start by showing that in this setting singularities are unavoidable more precisely we construct lagrangians with arbitrarily small lagrangian angle and lagrangians which are hamiltonian isotopic to a plane that nevertheless develop finite time singularities under mean curvature flow

motion of a set by the curvature of its boundary abstract the study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary often these equations are anisotropic indicating the preferred directions of the crystal structure in the isotropic case this equation is called the curve shortening or the mean curvature flow equation and has been studied by differential geometric tools in general there are no classical solutions to these equations in this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions our approach is closely related to that of osher and sethian chen giga and goto and evans and spruck who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation although we use their results in an essential way we give an intrinsic definition our main results are the existence of a solution large time asymptotics of this solution and its connection to the level set solution of osher and sethian chen giga and goto and evans and spruck in general there is no uniqueness even for classical solutions but we prove a uniqueness result under restrictive assumptions we also construct a class of explicit solutions which are dilations of wulff crystals

curvature evolution of nonconvex lens shaped domains we study the curvature flow of planar nonconvex lens shaped domains con sidered as special symmetric networks with two triple junctions we show that the evolv ing domain becomes convex in finite time then it shrinks homothetically to a point as proved in 22 our theorem is the analog of the result of grayson 13 for curvature flow of closed planar embedded curves

the dynamics on three dimensional strictly convex energy surfaces we show that a hamiltonian flow on a three dimensional strictly convex energy surface s c r4 possesses a global surface of section of disc type it follows in particular that the number of its periodic orbits is either 2 or oc by a recent result of j franks on area preserving homeomorphisms of an open annulus in the plane the construction of this surface of section is based on partial differential equations of cauchy riemann type for maps from punctured riemann surfaces into r x s3 equipped with special almost complex structures

cubulating graphs of free groups with cyclic edge groups we prove that if g is a group that splits as a finite graph of finitely generated free groups with cyclic edge groups and g has no non euclidean baumslag solitar subgroups then g is the fundamental group of a compact nonpositively curved cube complex in addition if g is also word hyperbolic i e if g contains no baumslag solitar subgroups of any type we show that g is linear in fact is a subgroup of sl n z

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

sturm attractors for quasilinear parabolic equations abstract the goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations as it was done for the semilinear case by brunovsk and fiedler 1986 8 and generalized by fiedler and rocha 1996 13 in particular we construct heteroclinic connections between hyperbolic equilibria stating necessary and sufficient conditions for heteroclinics to occur such conditions can be computed through a permutation of the equilibria lastly an example is computed yielding the well known chafee infante attractor

commensurability of groups quasi isometric to raags let g be a right angled artin group with defining graph gamma and let h be a finitely generated group quasi isometric to g we show if g satisfies that 1 its outer automorphism group is finite 2 gamma does not contain any induced 4 cycles 3 gamma is star rigid then h is commensurable to g we show condition 2 is sharp in the sense that if gamma contains an induced 4 cycle then there exists an h quasi isometric to g but not commensurable to g moreover one can drop condition 1 if h is a uniform lattice acting on the universal cover of the salvetti complex of g as a consequence we obtain a conjugation theorem for such uniform lattices the ingredients of the proof include a blow up building construction in huang and kleiner duke math j 167 3 537 602 2018 https doi org 10 1215 00127094 2017 0042 and a haglund wise style combination theorem for certain class of special cube complexes however in most of our cases relative hyperbolicity is absent so we need new ingredients for the combination theorem

optical flow a curve evolution approach a novel approach for the computation of optical flow based on an l sup 1 type minimization is presented it is shown that the approach has inherent advantages since it does not smooth the flow velocity across the edges and hence preserves edge information a numerical approach based on computation of evolving curves is proposed for computing the optical flow field computations are carried out on a number of real image sequences in order to illustrate the theory as well as the numerical approach

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

behavioral analysis of anisotropic diffusion in image processing in this paper we analyze the behavior of the anisotropic diffusion model of perona and malik 1990 the main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem so its behavior can be analyzed based on the shape of the corresponding energy surface we show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem it is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space we give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima the mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator and the other that flows normally to the edges and thus acts as an enhancement operator

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

a stable manifold theorem for the curve shortening equation on presente une famille de solutions homothetiques de l equation pour une courbe planaire x kn et on demontre l existence de varietes non lineaires stables et instables autour de telles solutions

uniqueness of self similar shrinkers with asymptotically conical ends in this paper we show the uniqueness of smooth embedded selfshrinkers asymptotic to generalized cylinders of infinite order also we construct non rotationally symmetric self shrinking ends asymptotic to generalized cylinders with rate as fast as any given polynomial

parabolic equations for curves on surfaces 2 intersections blow up and generalized solutions we describe a theory for parabolic equations for immersed curves on surfaces which generalizes the curve shortening or flow by mean curvature problem as well as several models in the theory of phase transitions in two dimensions a class of equations is described for which the initial value problem is well posed for rough initial data for which one can give a description of the way a smooth solution becomes singular and for which one can define generalized solutions i e solutions which are smooth except at a discrete set of times the methods which are used in this paper are more geometrical than those of part i by comparing arbitrary solutions with certain special solutions and by considering the way they intersect we derive estimates for the curvature and the tangent which allow one to study the initial value problem and the way solutions become singular

repeatedly smooting discrete scale space evolution and dominant point detection in this paper the fourier analysis is used to derive the properties of an evolving curve an arbitrary kernel repeatedly smoothing akrs evolution of a curve is then introduced it is shown that when the repeated number is large the akrs evolution of a curve is an approximately discrete implementation of the scale based evolution of this curve in the euclidean space as a special case an exponential repeatedly smoothing is proposed to implement the scale based evolution it is shown that in addition to its simple implementation and its desired approximation to a gaussian kernel an exponential function is a function that when it is selected as a repeatedly smoothing kernel the motion both magnitude and direction of a point on a curve from the i 1 th to ith instant is equal to the curvature of the ith smoothed curve at this point finally a perimeter controlled evolution method is proposed to extract dominant points it is shown experimentally that the proposed method is robust to noise object rotation and object changes in sizes

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

the strong atiyah conjecture for virtually cocompact special groups we provide new conditions for the strong atiyah conjecture to lift to finite group extensions in particular we show that fundamental groups of compact special cube complexes satisfy these conditions so the conjecture holds for finite extensions of these groups

shortening three dimensional curves via two dimensional flows in this paper a curve evolution approach for the computation of geodesic curves on 3d surfaces is presented the algorithm is based on deforming via the curve shortening flow an arbitrary initial curve ending at two given surface points the 3d curve shortening flow is first transformed into an equivalent 2d one this 2d flow is implemented using an efficient numerical algorithm for curve evolution with fixed end points

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

discrete active models and applications abstract optimization processes based on active models play central roles in many areas of computational vision as well as computational geometry unfortunately current models usually require highly complex and sophisticated mathematical machinery and at the same time they suffer from a number of limitations which impose restrictions on their applicability in this paper a simple class of discrete active models called migration processes mps is presented the processes are based on iterated averaging over neighborhoods defined by constant geodesic distance it is demonstrated that the mp model a system of self organizing active particles has a number of advantages over previous models both parametric active models snakes and implicit contour evolution models due to the generality of the mp model the process can be applied to derive natural solutions to a variety of optimization problems including defining minimal surface patches given their boundary curves finding shortest paths joining sets of points and decomposing objects into primitive parts

anisotropic flows for convex plane curves where v and k are respectively the normal velocity and curvature of the interface equation 1 is sometimes called the curve shortening problem because it is the negative l2 gradient flow of the length of the interface by drawing pictures one can be easily convinced that a simple closed curve stays simple and smooth along 1 and shrinks to a point in finite time with the limiting shape of a circle however a mathematical treatment of 1 turned out to be rather delicate in fact a rigorous study did not exist until the early 1980s when differential geometers considered 1 as a tool in the search for simple closed geodesics on surfaces it was also regarded as a model case for more general curvature flows which are believed to be important in the topological classification of low dimensional manifolds as a first attempt gage 10 proved that the isoperimetric ratio decreases along convex curves then in gage and hamilton 12 it was shown that a convex curve stays convex and shrinks to a point in finite time moreover if one normalizes the flow by dilating it so that the enclosing area is constant the normalized flow converges smoothly to a circle finally grayson 14 completed this line of investigation by showing that a simple curve evolves into a convex one before shrinks to a point for an alternative approach to this result one may consult hamilton 18 recently mullins s theory was generalized by gurtin 15 16 and by angenent and gurtin 4 5 see also the monograph of gurtin 17 to include anisotropy and the possibility of a difference in bulk energies between phases anisotropy is indispensable in dealing with crystalline materials for perfect conductors the temperatures in both phases are constant the equation becomes

symmetry reductions and exact solutions of the affine heat equation lie symmetry group method is applied to study the affine heat equation for surface its symmetry groups and corresponding optimal systems are determined and group invariant solutions associated to the symmetries are obtained and classified

geodesics on s2 and periodic points of annulus homeomorphisms we show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points a consequence is the theorem that every smooth riemannian metric ons2 with positive gaussian curvature has infinitely many distinct closed geodesics

a poincare birkhoff theorem for tight reeb flows on s 3 we consider reeb flows on the tight 3 sphere admitting a pair of closed orbits forming a hopf link if the rotation numbers associated to the transverse linearized dynamics at these orbits fail to satisfy a certain resonance condition then there exist infinitely many periodic trajectories distinguished by their linking numbers with the components of the link this result admits a natural comparison to the poincare birkhoff theorem on area preserving annulus homeomorphisms an analogous theorem holds on so 3 and applies to geodesic flows of finsler metrics on s 2

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

dynamic active contours for visual tracking visual tracking using active contours is usually set in a static framework the active contour tracks the object of interest in a given frame of an image sequence a subsequent prediction step ensures good initial placement for the next frame this approach is unnatural the curve evolution gets decoupled from the actual dynamics of the objects to be tracked true dynamical approaches exist all being marker particle based and thus prone to the shortcomings of such particle based implementations in particular topological changes are not handled naturally in this framework the now classical level set approach is tailored for evolutions of manifolds of codimension one however dynamic curve evolution is at least a codimension two problem we propose an efficient level set based approach for dynamic curve evolution which addresses the artificial separation of segmentation and prediction while retaining all the desirable properties of the level set formulation it is based on a new energy minimization functional which for the first time puts dynamics into the geodesic active contour framework

a dynamical characterization of universally tight lens spaces we give necessary and sufficient conditions for a closed connected co orientable contact 3 manifold m to be a standard lens space based on assumptions on the reeb flow associated to a defining contact form our methods also provide rational global surfaces of section for non degenerate reeb flows on l p q std with prescribed binding orbits

a sharp lower bound for the entropy of closed hypersurfaces up to dimension six the entropy is a natural geometric quantity which measures the complexity of a hypersurface in mathbb r n 1 it is non increasing along the mean curvature flow and so plays a significant role in analyzing the dynamics of this flow in colding et al j differ geom 95 1 53 69 2013 colding ilmanen minicozzi white showed that within the class of closed smooth self shrinking solutions of the mean curvature flow in mathbb r n 1 the entropy is uniquely minimized at the round sphere they conjectured that for 2 le n le 6 the round sphere minimizes the entropy among all closed hypersurfaces using an appropriate weak mean curvature flow we prove their conjecture for these dimensions our approach also gives a new proof of the main result of colding et al j differ geom 95 1 53 69 2013 and extends its conclusions to compact singular self shrinking solutions

residual finiteness growths of virtually special groups let g be a virtually special group then the residual finiteness growth of g is at most linear this result cannot be found by embedding g into a special linear group indeed the special linear group mathrm sl k mathbb z for k 2 has residual finiteness growth n k 1

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

mixed 3 manifolds are virtually special letm be a compact oriented irreducible 3 manifold which is neither a graph manifold nor a hyperbolic manifold we prove that 1m is virtually special

limit groups are subgroup conjugacy separable abstract a group g is called subgroup conjugacy separable if for every pair of non conjugate finitely generated subgroups of g there exists a finite quotient of g where the images of these subgroups are not conjugate we prove that limit groups are subgroup conjugacy separable we also prove this property for one relator groups of the form r a 1 a n w n with n w the property is also proved for infinite virtual retracts equivalently for infinite quasiconvex subgroups of hyperbolic virtually special groups

the bianchi groups are separable on geometrically finite subgroups let d be a square free positive integer and od the ring of integers in q p id the main result of this paper is that the groups psl 2 od are

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

evolution of plane curves driven by a nonlinear function of curvature and anisotropy in this paper we study evolution of plane curves satisfying a geometric equation v beta k nu where v is the normal velocity and k and nu are the curvature and tangential angle of a plane curve gamma we follow the direct approach and we analyze the so called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional alpha we show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities from an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta k nu gamma nu k m 0 m le 2 and the governing system of equations includes a nontrivial tangential velocity functional

minimal surfaces a geometric three dimensional segmentation approach a novel geometric approach for three dimensional object segmentation is presented the scheme is based on geometric deformable surfaces moving towards the objects to be detected we show that this model is related to the computation of surfaces of minimal area local minimal surfaces the space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected the general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework the scheme is stable robust and automatically handles changes in the surface topology during the deformation results related to existence uniqueness stability and correctness of the solution to this geometric deformable model are presented as well based on an efficient numerical algorithm for surface evolution we present a number of examples of object detection in real and synthetic images

coxeter groups are virtually special abstract in this paper we prove that every finitely generated coxeter group has a finite index subgroup that is the fundamental group of a special cube complex some consequences include every f g coxeter group is virtually a subgroup of a right angled coxeter group every word hyperbolic coxeter group has separable quasiconvex subgroups

what is the best causal scale space for three dimensional images we study the unique affine invariant morphological scale space in three dimensions we discuss its properties and show that it improves the dynamic shape model we explain the algorithms and display the first numerical experiments

symmetries and invariant solutions for the geometric heat flows we study lie symmetries and invariant solutions of the geometric heat flows the basic similarity reductions for the ghe are performed reduced equations and exact solutions associated with the symmetries are obtained group invariant solutions and reductions for the affine case are also discussed in a special case

finite time singularities for lagrangian mean curvature flow given any embedded lagrangian on a four dimensional compact calabi yau we find another lagrangian in the same hamiltonian isotopy class that develops a finite time singularity under mean curvature flow this contradicts a weaker version of the thomas yau conjecture regarding long time existence and convergence of lagrangian mean curvature flow

a combination theorem for special cube complexes we prove that certain compact cube complexes have special nite covers this means they have nite covers whose fundamental groups are quasiconvex subgroups of right angled artin groups as a result we obtain linearity and the separability of quasiconvex subgroups for the groups we consider our result applies in particular to a compact negatively curved cube complex whose hyperplanes do not self intersect for a cube complex with word hyperbolic fundamental group we show that it is virtually special if and only if its hyperplane stabilizers are separable in a nal application we show that the fundamental groups of every simple type uniform arithmetic hyperbolic manifolds are cubical and virtually special

finsler active contours in this paper we propose an image segmentation technique based on augmenting the conformal or geodesic active contour framework with directional information in the isotropic case the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest typically edges is small the conformal factor that is chosen depends only upon position and is in this sense isotropic although directional information has been studied previously for other segmentation frameworks here we show that if one desires to add directionality in the conformal active contour framework then one gets a well defined minimization problem in the case that the factor defines a finsler metric optimal curves may be obtained using the calculus of variations or dynamic programming based schemes finally we demonstrate the technique by extracting roads from aerial imagery blood vessels from medical angiograms and neural tracts from diffusion weighted magnetic resonance imagery

area and length preserving geometric invariant scale spaces in this paper area preserving multi scale representations of planar curves are described this allows smoothing without shrinkage at the same time preserving all the scale space properties the representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant when the euclidean geometric heat flow is used the resulting representation is euclidean invariant and similarly it is affine invariant when the affine one is used the flows are geometrically intrinsic to the curve and exactly satisfy all the basic requirements of scale space representations in the case of the euclidean heat flow it is completely local as well the same approach is used to define length preserving geometric flows a similarity scale invariant geometric heat flow is studied as well in this work

tits alternatives for graph products we discuss various types of tits alternative for subgroups of graph products of groups and prove that under some natural conditions a graph product of groups satisfies a given form of tits alternative if and only if each vertex group satisfies this alternative as a corollary we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large as another corollary we prove that every non abelian subgroup of a right angled artin group has an epimorphism onto the free group of rank 2 in the course of the paper we develop the theory of parabolic subgroups which allows to describe the structure of subgroups of graph products that contain no non abelian free subgroups we also obtain a number of results regarding the stability of some group properties under taking graph products

on the stability of m fold circles and the dynamics of generalized curve shortening flows abstract in this paper we study the asymptotic and exponential stability of the m fold circle as a solution of the p curve shortening flow p 1 an integer

multiplicity of the second schrodinger eigenvalue on closed surfaces we improve roughly by a factor 2 the known bound on the multiplicity of the second eigenvalue of schrodinger operators i e laplace plus potential on closed surfaces this gives four new topological types of surfaces for which colin de verdiere s conjecture relating the maximal multiplicity to the chromatic number of the surface is verified the proof goes by defining a space cal p of nodal splittings of the surface equipped with a double covering to which a borsuk ulam type theorem is applied

a characterization of the tight 3 sphere ii recall that every closed oriented connected 3 manifold m admits a contact form we shall give conditions on a periodic orbit p0 of the reeb vector field defined by that allows the construction of an open book decomposition of m p0 into embedded planes asymptotic to p0 such that p0 is the binding orbit of the decomposition it follows that m is the tight 3 sphere as a by product a new dynamical criterion for the tightness of a contact structure is derived in the proof the results extend earlier results 1999 john wiley sons inc

order structures and the heat equation abstract in this paper we shall introduce the notion of order structure and use it to study properties of a solution u x t of a scalar linear parabolic equation it is well known that in one space dimension the number of zeros of u t is nonincreasing with t this zero number is an example of an order structure and the zero number property is very useful in unfolding the structure of the global attractor of the semiflow generated by a scalar parabolic equation we shall prove that in one space dimension the zero number is the only order structure preserved by linear parabolic equations in two dimensions the only order structure preserved by linear parabolic equations is the order structure induced by the comparison principle for second order equations consequently in two dimensions there does not exist a fine decomposition into equivalence classes as in the one dimensional case