evolution by mean curvature flow of lagrangian spherical surfaces in complex euclidean plane abstract we describe the evolution under the mean curvature flow of embedded lagrangian spherical surfaces in the complex euclidean plane c 2 in particular we answer the question 4 7 addressed in 10 by a neves about finding out a condition on a starting lagrangian torus in c 2 such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the clifford torus

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

shift operators on banach spaces abstract j r holub has obtained several results for shift operators on c x in this paper we answer some questions of holub and obtain extensions of many of his results in particular we show that c x r does not admit a shift operator if x has only countably many components and each component is infinite we show that c x c does not admit a shift operator for certain special compact hausdorff spaces x we show that there exists a compact hausdorff space x which is not totally disconnected and both c x c and c x r admit shift operators if 1 p x is a finite non atomic measure space then l p r does not admit a disjointness preserving shift operator we also show that l p for 1 p is the only l p r space which admits a disjointness preserving shift operator

ergodicity of stochastic 2d navier stokes equation with levy noise abstract in this paper we deal with the 2d navier stokes equation perturbed by a levy noise force whose white noise part is non degenerate and that the intensity measure of poisson measure is finite existence and uniqueness of invariant measure for this equation is obtained two main properties of the markov semigroup associated with this equation are proved in other words strong feller property and irreducibility hold in the same space

weighted composition operators ofc0 x s abstract in this paper we prove that into isometries and disjointness preserving linear maps from c 0 x into c 0 y are essentially weighted composition operators tf h f for some continuous map and some continuous scalar valued function h

generalized backward shifts on banach spaces this paper is in part a brief survey of backward shifts however we present several new results on backward and forward shifts which have not appeared so far these results concern isomorphism invariance of backward and forward shifts and the duality between these properties

spdes driven by poisson random measure with non lipschitz coefficients existence results the article deals with spdes driven by poisson random measure with non lipschitz coefficients let a e e be a generator of an analytic semigroup on e e being a certain banach space let open image in new window be a stochastic basis carrying an e valued poisson random measure with characteristic measure and compensator let 1 p 2 our point of interest is the existence of solutions to spde s of e g the following type

optimal local holder index for density states of superprocesses with 1 β branching mechanism for 0 0 if d 1 but locally unbounded otherwise moreover in the case of continuity we determine the optimal local holder index

embedded minimal tori in s3 and the lawson conjecture we show that any embedded minimal torus in s 3 is congruent to the clifford torus this answers a question posed by h b lawson jr in 1970

stochastic partial differential equation driven by stable noise we construct a weak solution to the stochastic partial differential equation driven by a one sided stable noise without negative jumps we prove the weak existence of the solution for parameters the facts that for is not a lipschitz function and is not a gaussian noise require the development of new methods which we believe are of independent interest we also show that when the above equation gives an alternative description of super brownian motion with stable branching mechanism

mean curvature flow of monotone lagrangian submanifolds we use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone lagrangian submanifolds in mathbb c n

the existence and asymptotic behaviour of energy solutions to stochastic 2d functional navier stokes equations driven by levy processes abstract let d be a bounded or unbounded open domain of 2 dimensional euclidean space r 2 if the boundary d exists then we assume that the boundary is smooth in this paper assuming that the kinematic viscosity 0 is large enough we discuss the existence and exponential stability of energy solutions to the following 2 dimensional stochastic functional navier stokes equation perturbed by the levy process d x t x t x t x t f t x t f t x t p d t d x t g t x t d w t u k t x t y q d t d y d i v x 0 in 0 d where x t x t x is the initial function for x d and t r 0 with r 0 it is assumed that f g f and k satisfy the lipschitz condition and the linear growth condition if there exists the boundary d then x t x 0 on 0 d

isometric shifts onc k abstract several problems regarding isometric shifts proposed by gutek et al j funct anal 101 1991 97 119 are solved

a local regularity theorem for mean curvature flow this paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the gaussian densities are close to 1

backward shifts on banach spacesc x abstract j r holub conjectured that the banach spacec x of real valued continuous functions does not admit a backward shift ifxis a compact hausdorff space with an infinite connected component here the conjecture is settled by proving that for arbitrary infinite compact hausdorff spacesx c x does not admit a backward shift

isometric shifts on c0 x abstract for a linear isometry t c 0 x c 0 y of finite corank there is a cofinite subset y 1 of y such that tf y 1 h f is a weighted composition operator and x is homeomorphic to a quotient space of y 1 modulo a finite subset when x y such a t is called an isometric quasi n shift on c 0 x in this case the action of t can be implemented as a shift on a tree like structure called a t tree in m x with exactly n joints the t tree is total in m x when t is a shift with these tools we can analyze the structure of t

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

uniqueness of compact tangent flows in mean curvature flow we show for mean curvature flows in euclidean space that if one of the tangent flows at a given space time point consists of a closed multiplicity one smoothly embedded self similar shrinker then it is the unique tangent flow at that point that is the limit of the parabolic rescalings does not depend on the chosen sequence of rescalings furthermore given such a closed multiplicity one smoothly embedded self similar shrinker we show that any solution of the rescaled flow which is sufficiently close to with gaussian density ratios greater or equal to that of stays for all time close to and converges to a possibly different self similarly shrinking solution the central point in the argument is a direct application of the ojasiewicz simon inequality to huisken s monotone gaussian integral for mean curvature flow de gruyter 2014

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

blow up of solutions for semilinear stochastic delayed reaction diffusion equations with levy noise abstract this paper is concerned with a class of semilinear stochastic delayed reaction diffusion equations driven by levy noise in a separable hilbert space we establish sufficient conditions to ensure the existence of a unique positive solution moreover we study blow up of solutions in finite time in mean l p norm sense several examples are given to illustrate applications of the theory