finite groups whose element orders are consecutive integers there are various characterizations of groups by conditions on the orders of its elements for example in 8 1 the finite groups all of whose elements have prime power order have been classified in 2 16 181 some simple groups have been characterized by conditions on the orders of its elements and b h neumann 12 has determined all groups whose elements have orders 1 2 and 3 the latter groups are oc3 groups in the sense of the following

li yau gradient bounds on compact manifolds under nearly optimal curvature conditions abstract we prove li yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the ricci flow in the former case the curvature condition is r i c l p for some p n 2 or sup m m r i c 2 y d 2 n x y d y where n is the dimension of the manifold in the later case one only needs scalar curvature being bounded we will explain why the conditions are nearly optimal and give an application the li yau bound for the heat equation on manifolds with fixed metric seems to be the first one allowing ricci curvature not bounded from below

generic mean curvature flow i generic singularities it has long been conjectured that starting at a generic smooth closed embedded surface in r 3 the mean curvature ow remains smooth until it arrives at a singularity in a neighborhood of which the ow looks like concentric spheres or cylinders that is the only singularities of a generic ow are spherical or cylindrical we will address this conjecture here and in a sequel the higher dimensional case will be addressed elsewhere the key to showing this conjecture is to show that shrinking spheres cylinders and planes are the only stable self shrinkers under the mean curvature ow we prove this here in all dimensions an easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away

curvature decay estimates of graphical mean curvature flow in higher codimensions we derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions to the best of our knowledge this is the first such estimates without assuming smallness of first derivatives of the defining map an immediate application is a convergence theorem of the mean curvature flow of the graph of an area decreasing map between flat riemann surfaces

on the type iib solutions to mean curvature flow abstract in this paper we study the type iib mean curvature flow we first prove that if the convex entire graph y u y over r n n 2 satisfying there exist positive constants c and n such that u r c r for r n the longtime solution to mean curvature flow with initial data y u y must be type iib we also study the asymptotic behavior of type iib mean curvature flow and show that the limit of suitable rescaling sequence for mean convex type iib mean curvature flow satisfying andrews noncollapsing condition is translating soliton

matrix li yau hamilton inequality for the cr heat equation in pseudohermitian 2n 1 manifolds in this paper we first derive the cr analogue of matrix li yau hamilton inequality for the positive solution to the cr heat equation in a closed pseudohermitian 2n 1 manifold with nonnegative bisectional curvature and bitorsional tensor we then obtain the cr li yau gradient estimate in the heisenberg group we apply this cr gradient estimate and extend the cr matrix li yau hamilton inequality to the case of the heisenberg group as a consequence we derive the hessian comparison property for the heisenberg group

mean curvature flow in a ricci flow background following work of ecker comm anal geom 15 1025 1061 2007 we consider a weighted gibbons hawking york functional on a riemannian manifold with boundary we compute its variational properties and its time derivative under perelman s modified ricci flow the answer has a boundary term which involves an extension of hamilton s differential harnack expression for the mean curvature flow in euclidean space we also derive the evolution equations for the second fundamental form and the mean curvature under a mean curvature flow in a ricci flow background in the case of a gradient ricci soliton background we discuss mean curvature solitons and huisken monotonicity

differential harnack estimates for time dependent heat equations with potentials in a closed spherical cr 3 manifold abstract in this paper we derive differential harnack estimates for time dependent heat equations with potentials under the cr yamabe flow in a closed spherical cr 3 manifold with positive tanaka webster curvature and vanishing torsion it can be served as a generalization of li yau hamilton inequality for the cr yamabe flow the differential harnack estimate will play an important role in the study of the cr yamabe flow

gradient estimates and harnack inequalities of a parabolic equation under geometric flow abstract in this paper we consider a manifold evolving by a general geometric flow and study parabolic equation q x t t u x t a u x t x t m 0 t we establish space time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation by integrating the gradient estimates we derive the corresponding harnack inequalities finally as applications we give gradient estimates of some specific parabolic equations

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

harnack inequality and pinching estimates for anisotropic curvature flow of hypersurfaces abstract we obtain a differential harnack inequality for anisotropic curvature flow of convex hypersurfaces in euclidean space with its speed given by a curvature function of homogeneity degree one in a certain class and restrictions depending only on the initial data and the anisotropic factor which reflects the influence of the ambient space moreover the pinching estimate for such flows is derived from the maximum principle for tensors

a deterministic control based approach to fully nonlinear parabolic and elliptic equations we show that a broad class of fully nonlinear second order parabolic or elliptic pde s can be realized as the hamilton jacobi bellman equations of deterministic twoperson games more precisely given the pde we identify a deterministic discretetime two person game whose value function converges in the continuous time limit to the viscosity solution of the desired equation our game is roughly speaking a deterministic analogue of the stochastic representation recently introduced by cheridito soner touzi and victoir comm pure appl math 60 2006 1081 1110 in the parabolic setting with no u dependence it amounts to a semidiscrete numerical scheme whose timestep is a min max our result is interesting because the usual control based interpretations of second order pde s involve stochastic rather than deterministic control

uniqueness of convex ancient solutions to mean curvature flow in mathbb r 3 r 3 a well known question of perelman concerns the classification of noncompact ancient solutions to the ricci flow in dimension 3 which have positive sectional curvature and are kappa noncollapsed in this paper we solve the analogous problem for mean curvature flow in mathbb r 3 and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in mathbb r 3 which is strictly convex and noncollapsed

heat kernel and curvature bounds in ricci flows with bounded scalar curvature abstract in this paper we analyze ricci flows on which the scalar curvature is globally or locally bounded from above by a uniform or time dependent constant on such ricci flows we establish a new time derivative bound for solutions to the heat equation based on this bound we solve several open problems 1 distance distortion estimates 2 the existence of a cutoff function 3 gaussian bounds for heat kernels and 4 a backward pseudolocality theorem which states that a curvature bound at a later time implies a curvature bound at a slightly earlier time using the backward pseudolocality theorem we next establish a uniform l 2 curvature bound in dimension 4 and we show that the flow in dimension 4 converges to an orbifold at a singularity we also obtain a stronger e regularity theorem for ricci flows this result is particularly useful in the study of kahler ricci flows on fano manifolds where it can be used to derive certain convergence results

essential spectrum and heat kernel abstract upper bounds are derived for the large time behavior of the heat kernel on riemannian manifolds whose essential spectrum has a positive lower bound applications include a relative signature theorem and the construction of bounded harmonic functions

convex solutions to the mean curvature flow in this paper we study the classication of ancient convex solutions to the mean curvature ow in r n 1 an open problem related to the classication of type ii singularities is whether a convex translating solution is k rotationally symmetric for some integer 2 k n namely whether its level set is a sphere or cylinder s k 1 r n k in this paper we give an afrmative answer for entire solutions in dimension 2 in high dimensions we prove that there exist nonrotationally symmetric entire convex translating solutions but the blow down in space of any entire convex translating solution is k rotationally symmetric we also prove that the blow down in space time of an ancient convex solution which sweeps the whole space r n 1 is a shrinking sphere or cylinder

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

superlinear elliptic inequalities on manifolds abstract let m be a complete non compact riemannian manifold and let be a radon measure on m we study the problem of existence or non existence of positive solutions to a semilinear elliptic inequality u u q in m where q 1 we obtain necessary and sufficient criteria for existence of positive solutions in terms of green function of in particular explicit necessary and sufficient conditions are given when m has nonnegative ricci curvature everywhere in m or more generally when green s function satisfies the 3g inequality

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

mean curvature flows and isotopy of maps between spheres let f be a smooth map between unit spheres of possibly different dimensions we prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions a corollary is that any area decreasing map between unit spheres of possibly different dimensions is isotopic to a constant map c 2004 wiley periodicals inc

on quadratic approximations for hamilton jacobi bellman equations we study simple quadratic approximations for general hamilton jacobi bellman equations the theoretical error bounds are shown to be composed of the time discretization errors and quadratic approximation ones at each time step several numerical experiments show that the quadratic approximations work for the equations with the boundary and hamiltonian well approximated by the quadratic functions

harnack inequalities for curvature flows in riemannian and lorentzian manifolds we obtain harnack estimates for a class of curvature flows in riemannian manifolds of constant nonnegative sectional curvature as well as in the lorentzian minkowski and de sitter spaces furthermore we prove a harnack estimate with a bonus term for mean curvature flow in locally symmetric riemannian einstein manifolds of nonnegative sectional curvature using a concept of duality for strictly convex hypersurfaces we also obtain a new type of inequality so called pseudo harnack inequality for expanding flows in the sphere and in the hyperbolic space

the li yau hamilton estimate and the yang mills heat equation on manifolds with boundary the paper pursues two connected goals firstly we establish the li yau hamilton estimate for the heat equation on a manifold m with nonempty boundary results of this kind are typically used to prove monotonicity formulas related to geometric flows secondly we establish bounds for a solution t of the yang mills heat equation in a vector bundle over m the li yau hamilton estimate is utilized in the proofs our results imply that the curvature of t does not blow up if the dimension of m is less than 4 or if the initial energy of t is sufficiently small

expanding solitons with non negative curvature operator coming out of cones we consider ricci flow of complete riemannian manifolds which have bounded non negative curvature operator non zero asymptotic volume ratio and no boundary we prove scale invariant estimates for these solutions using these estimates we show that there is a limit solution obtained by scaling down this solution at a fixed point in space this limit solution is an expanding soliton coming out of the asymptotic cone at infinity

singularities of the area preserving curve shortening flow with a free boundary condition we consider the area preserving curve shortening flow with neumann free boundary conditions outside of a convex domain or at a straight line we give a criterion on initial curves that guarantees the appearance of a singularity in finite time we prove that the singularity is of type ii furthermore if these initial curves are convex then an appropriate rescaling at the finite maximal time of existence yields a grim reaper or half a grim reaper as limit flow we construct examples of initial curves satisfying the mentioned criterion

uniqueness results for second order bellman isaacs equations under quadratic growth assumptions and applications in this paper we prove a comparison result between semicontinuous viscosity sub and supersolutions growing at most quadratically of second order degenerate parabolic hamilton jacobi bellman and isaacs equations as an application we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation

monotone volume formulas for geometric flows we consider a closed manifold m with a riemannian metric gij t evolving by t gij 2sij where sij t is a symmetric two tensor on m g t we prove that if sij satises the tensor inequality d sij x 0 for all vector elds x on m where d sij x is dened in 1 6 then one can construct a forwards and a backwards reduced volume quantity the former being non increasing the latter being non decreasing along the ow t gij 2sij in the case where sij rij the ricci curvature of m the result corresponds to perelman s well known reduced volume monotonicity for the ricci

differential harnack inequalities on riemannian manifolds i linear heat equation abstract in the first part of this paper we get new li yau type gradient estimates for positive solutions of heat equation on riemannian manifolds with ricci m k k r as applications several parabolic harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with ricci curvature bounded from below in the second part we establish a perelman type li yau hamilton differential harnack inequality for heat kernels on manifolds with ricci m k which generalizes a result of l ni 2004 2006 20 21 as applications we obtain new harnack inequalities and heat kernel estimates on general manifolds we also obtain various entropy monotonicity formulas for all compact riemannian manifolds

consistency of generalized finite difference schemes for the stochastic hjb equation we analyze a class of numerical schemes for solving the hjb equation for stochastic control problems which enters the framework of markov chain approximations and generalizes the usual finite difference method the latter is known to be monotonic and hence valid only if the scaled covariance matrix is dominant diagonal we generalize this result by given the set of neighboring points allowed to enter the scheme showing how to compute effectively the class of covariance matrices that is consistent with this set of points we perform this computation for several cases in dimensions 2 3 and 4

a matrix li yau hamilton estimate for kahler ricci flow in this paper we prove a new matrix li yau hamilton lyh estimate for kahler ricci flow on manifolds with nonnegative bisectional curvature the form of this new lyh estimate is obtained by the interpolation consideration originated in ch by chow this new inequality is shown to be connected with perelman s entropy formula through a family of differential equalities in the rest of the paper we show several applications of this new estimate and its corresponding estimate for linear heat equation these include a sharp heat kernel comparison theorem generalizing the earlier result of li and tian lt a manifold version of stoll s theorem st on the characterization of algebraic divisors and a localized monotonicity formula for analytic subvarieties which sharpens the bishop volume comparison theorem motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of perelman p we prove a sharp lower bound heat kernel estimate for the time dependent heat equation which is in a certain sense dual to perelman s monotonicity of the reduced volume as an application of this new monotonicity formula we show that the blow down limit of a certain type long time solution is a gradient expanding soliton we also illustrate the connection between the new lyh estimate and the hessian comparison theorem of fin on the forward reduced distance local monotonicity formulae on entropy and forward reduced volume are also derived

plurisubharmonic functions and the kahler ricci flow in this paper a sharp linear trace li yau hamilton inequality for kahler ricci flow is proved the new inequality extends the previous trace harnack inequality obtained by h d cao we also establish sharp gradient estimates for the positive solution of the time dependent heat equation for some cases finally we apply this new linear trace li yau hamilton inequality to study the liouville properties of the plurisubharmonic functions on complete kahler manifolds with bounded nonnegative holomorphic bisectional curvature

gradient estimates for the heat equation under the ricci flow the paper considers a manifold m evolving under the ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on m

singularity of connection ricci flow for three manifolds we study the long time behavior of the solution to the connection ricci flow on closed three manifolds the connection ricci flow is a generalization of the ricci flow to connection with torsion we compare the evolving metrics under the connection ricci flow and ricci flow for some special cases we obtain the blow up rate of the ricci tensor under the connection ricci flow based on the result we give the classification of the maximal solutions to the flow

interpolating between constrained li yau and chow hamilton harnack inequalities for a nonlinear parabolic equation abstract we establish a one parameter family of harnack inequalities connecting the constrained trace li yau differential harnack inequality for a nonlinear parabolic equation to the constrained trace chow hamilton harnack inequality for this nonlinear equation with respect to evolving metrics related to the ricci flow on a 2 dimensional closed manifold this result can be regarded as a nonlinear version of the previous work of y zheng and the author j y wu y zheng interpolating between constrained li yau and chow hamilton harnack inequalities on a surface arch math 94 2010 591 600

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

semi lagrangian schemes for linear and fully non linear diffusion equations for linear and fully non linear diffusion equations of bellmanisaacs type we introduce a class of monotone approximation schemes relying on monotone interpolation as opposed to classical numerical methods these schemes converge for degenerate diffusion equations having general nondiagonal dominant coefficient matrices such schemes have to have a wide stencil in general besides providing a unifying framework for several known first order accurate schemes our class of schemes also includes more efficient versions and a new second order scheme that converges only for essentially monotone solutions the methods are easy to implement and analyze and they are more efficient than some other known schemes we prove stability and convergence of the schemes in the general case and provide error estimates in the convex case which are robust in the sense that they apply to degenerate equations and non smooth solutions the methods are extensively tested

finite topology self translating surfaces for the mean curvature flow in r3 abstract finite topology self translating surfaces for the mean curvature flow constitute a key element in the analysis of type ii singularities from a compact surface because they arise as limits after suitable blow up scalings around the singularity we prove the existence of such a surface m r 3 that is orientable embedded complete and with three ends asymptotically paraboloidal the fact that m is self translating means that the moving surface s t m t e z evolves by mean curvature flow or equivalently that m satisfies the equation h m e z where h m denotes mean curvature is a choice of unit normal to m and e z is a unit vector along the z axis this surface m is in correspondence with the classical three end costa hoffman meeks minimal surface with large genus which has two asymptotically catenoidal ends and one planar end and a long array of small tunnels in the intersection region resembling a periodic scherk surface this example is the first non trivial one of its kind and it suggests a strong connection between this problem and the theory of embedded complete minimal surfaces with finite total curvature

differential harnack estimates for backward heat equations with potentials under the ricci flow abstract in this paper we derive a general evolution formula for possible harnack quantities as a consequence we prove several differential harnack inequalities for positive solutions of backward heat type equations with potentials including the conjugate heat equation under the ricci flow we shall also derive perelman s harnack inequality for the fundamental solution of the conjugate heat equation under the ricci flow

a monotonicity formula on complete kahler manifolds with nonnegative bisectional curvature in y yau proposed to study the uniformization of complete kahler manifolds with nonnegative curvature in particular one wishes to determine whether or not a complete kahler manifold m with positive bisectional curvature is biholomorphic to cm see also gw si for this sake it was further asked in y whether or not the ring of the holomorphic functions with polynomial growth which we denote by op m is finitely generated and whether or not the dimension of the spaces of holomorphic functions of polynomial growth is bounded from above by the dimension of the corresponding spaces of polynomials on cm this paper addresses the latter questions we denote by 0 m the space of holomorphic functions of polynomial growth with degree d see section 3 for the precise definition then op m jd 0 od m in this paper we show that

harnack estimates for nonlinear backward heat equations in geometric flows let m be a closed riemannian manifold with a family of riemannian metrics gij t evolving by a geometric flow tgij 2sij where sij t is a family of smooth symmetric two tensors we derive several differential harnack estimates for positive solutions to the nonlinear backward heat type equation f t f flog f asf where a and are constants and s gijsij is the trace of sij our abstract formulation provides a unified framework for some known results proved by various authors and moreover leads to new harnack inequalities for a variety of geometric flows

hyperbolic mean curvature flow abstract in this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations is strictly hyperbolic and based on this we show that this flow admits a unique short time smooth solution and possesses the nonlinear stability defined on the euclidean space with dimension larger than 4 we derive nonlinear wave equations satisfied by some geometric quantities related to the hyperbolic mean curvature flow moreover we also discuss the relation between the equations for hyperbolic mean curvature flow and the equations for extremal surfaces in the minkowski space time

mean curvature flows of lagrangian submanifolds with convex potentials this article studies the mean curvature flow of lagrangian submanifolds in particular we prove the following global existence and convergence theorem if the potential function of a lagrangian graph in t 2n is convex then the flow exists for all time and converges smoothly to a flat lagrangian submanifold we also discuss various conditions on the potential function that guarantee global existence and convergence

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

idempotent expansions for continuous time stochastic control max plus methods have previously been used to solve deterministic control problems the methods are based on max plus or min plus expansions and can yield curse of dimensionality free numerical methods in this paper we explore min plus methods for continuous time stochastic control on a finite time horizon we first approximate the original value function via time discretization by generalizing the min plus distributive property to continuum spaces we obtain an algorithm for recursive computation of the time discretized values which we refer to as the idempotent distributed dynamic programming principle iddpp under the iddpp the value function at each step can be represented as an infimum of functions in a certain class this is a min plus expansion for the value function for the specific class of problems considered here we see that the class can be taken as that consisting of the quadratic functions a means for reducing the numbers of constituent quadratic functions is discussed

constrained matrix li yau hamilton estimates on kahler manifolds we derive a family of constrained matrix li yau hamilton estimates on kahler manifolds as a result we first get a constrained matrix li yau hamilton estimate for heat equation on a kahler manifold with fixed kahler metric secondly we get such estimate for forward conjugate heat equation on kahler manifolds with time dependent kahler metrics evolving under the kahler ricci flow

convexity estimates for surfaces moving by curvature functions we consider the evolution of compact surfaces by fully nonlinear parabolic curvature ows for which the normal speed is given by a smooth degree one homogeneous function of the principal curvatures of the evolving surface under no further restrictions on the speed function we prove that initial surfaces on which the speed is positive become weakly convex at a singularity of the flow this generalises the corresponding result 26 of huisken and sinestrari for the mean curvature ow to the largest possible class of degree one homogeneous surface flows

sharp differential estimates of li yau hamilton type for positive p p forms on kahler manifolds in this paper we study the heat equation of hodge laplacian deformation of p p forms on a kahler manifold after identifying the condition and establishing that the positivity of a p p form solution is preserved under such an invariant condition we prove the sharp differential harnack in the sense of liyau hamilton estimates for the positive solutions of the hodge laplacian heat equation we also prove a nonlinear version coupled with the kahler ricci flow and some interpolating matrix differential harnack type estimates for both the kahler ricci flow and the ricci flow 2011 wiley periodicals inc

finite groups whose all irreducible character degrees are hall numbers abstract we call m is a hall number for g if m is the order of a hall subgroup of g that is gcd g m m 1 the aim of this paper is to investigate the structure of the finite group g whose all irreducible character degrees are hall numbers for g

complete riemannian manifolds with pointwise pinched curvature an analogous bonnet myers theorem is obtained for a complete and positively curved n dimensional n 3 riemannian manifold m n we prove that if n 4 and the curvature operator of m n is pointwise pinched or if n 3 and the ricci curvature of m 3 is pointwise pinched then m n is compact

remarks on li yau inequality on graphs abstract in this paper we study li yau gradient estimates for the solutions u to the heat equation t u u on graphs under the curvature condition c d n k which is introduced by bauer et al in 5 as applications we derive harnack inequalities and heat kernel estimates on graphs also we present a type of hamilton gradient estimates

mean curvature flow with surgeries of two convex hypersurfaces we consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow it is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded the purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure compared with the notions of weak solutions existing in the literature the flow with surgeries has the advantage that it keeps track of the changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold our construction is inspired by the procedure originally introduced by hamilton for the ricci flow and then employed by perelman in the proof of thurston s geometrization conjecture in this paper we consider initial hypersurfaces which have dimension at least three and are two convex that is such that the sum of the two smallest principal curvatures is nonnegative everywhere under these assumptions we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology as a corollary we obtain a classification up to diffeomorphism of the hypersurfaces under consideration

self similar solutions of a 2 d multiple phase curvature flow this article studies self similar shrinking stationary and expanding solutions of a two dimensional motion by curvature equation modelling evolution of grain boundaries in polycrystals here the interfacial energy densities are assumed to depend only on the grains and the herring condition is used for triple junctions the intersections of three grain boundaries in particular in the isotropic case a total of six configurations are classified as the only self similar shrinking solutions

finite groups with consecutive nonlinear character degrees abstract the aim of this paper is to investigate structure of the finite groups whose nonlinear irreducible character degrees are consecutive integers