properly embedded minimal planar domains in 1997 collin proved that any properly embedded minimal surface in r 3 with nite topology and more than one end has

opening nodes on horosphere packings we use bryant representation to construct constant mean curvature 1 surfaces in hyperbolic space by desingularisation of a horosphere packing

a spectral multiplier theorem for a sublaplacian on su 2 we prove a hormander type spectral multiplier theorem for a sublaplacian on su 2 with critical index determined by the euclidean dimension of the group this result is the analogue for su 2 of the result for the heisenberg group obtained by d muller and e m stein and by w hebisch

on the topology and index of minimal surfaces we show that for an immersed two sided minimal surface in r 3 there is a lower bound on the index depending on the genus and number of ends using this we show the nonexistence of an embedded minimal surface in r 3 of index 2 as conjectured by choe cho90 moreover we show that the index of an immersed two sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface

decay of weak solutions to the davey stewartson systems abstract we study the initial value problem for the davey stewartson system making use of the pseudo conformal inequality we establish lp decay estimates of weak solutions

on the davey stewartson system with competing nonlinearities this paper is concerned with the blow up solutions for the davey stewartson system with competing nonlinearities which results in the loss of scaling invariance the best constant of a new gg n type inequality is given to find the sharp threshold mass of blow up and global existence moreover under the sharp threshold mass the dynamical behavior of blow up solutions is investigated including l2 concentration l2 weak limits and limiting profile

the dynamics theorem for properly embedded minimal surfaces in this paper we prove two theorems the first one is a structure result that describes the extrinsic geometry of an embedded surface with constant mean curvature possibly zero in a homogeneously regular riemannian three manifold in any small neighborhood of a point of large almost maximal curvature we next apply this theorem and the quadratic curvature decay theorem in meeks et al j differ geom arxiv 1308 6439 to deduce compactness descriptive and dynamics type results concerning the space d m of non flat limits under dilations of any given properly embedded minimal surface m in r 3

the intrinsic hypoelliptic laplacian and its heat kernel on unimodular lie groups we present an invariant definition of the hypoelliptic laplacian on sub riemannian structures with constant growth vector using the popp s volume form introduced by montgomery this definition generalizes the one of the laplace beltrami operator in riemannian geometry in the case of left invariant problems on unimodular lie groups we prove that it coincides with the usual sum of squares we then extend a method first used by hulanicki on the heisenberg group to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular lie group of type i the main tool is the noncommutative fourier transform we then study some relevant cases su 2 so 3 sl 2 with the metrics inherited by the killing form and the group se 2 of rototranslations of the plane our study is motivated by some recent results about the cut and conjugate loci on these sub riemannian manifolds the perspective is to understand how singularities of the sub riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation

uniqueness of the riemann minimal examples we prove that a properly embedded minimal surface in r 3 of genus zero with infinite symmetry group is a plane a catenoid a helicoid or a riemann minimal example we introduce the language of hurwitz schemes to understand the underlying moduli space of surfaces in our setting

two end solutions to the allen cahn equation in r3 abstract in this paper we study axially symmetric solutions of allen cahn equation in the three dimensional euclidean space using a sophisticated continuation method we show the existence of a complete family of axially symmetric solutions with a range of logarithmic growth rates which may be regarded as the analogue of the family of catenoids and hence called two end solutions nonexistence of two end solution with a small growth rate is also shown which differs from the theory of minimal surfaces

an embedded minimal surface with no symmetries we construct embedded minimal surfaces of finite total curvature in euclidean space by gluing catenoids and planes we use weierstrass representation and we solve the period problem using the implicit function theorem as a corollary we obtain the existence of minimal surfaces with no symmetries

complete embedded minimal surfaces of finite total curvature with planar ends of smallest possible order using a method by traizet j differ geom 60 103 153 2002 which reduces the construction of minimal surfaces via the weierstras theorem and the implicit function theorem to solving algebraic equations in several complex variables we will show the existence of complete embedded minimal surfaces of finite total curvature with planar ends of least possible order

the cayley transform and uniformly bounded representations let g be a simple lie group of real rank one with iwasawa decomposition kan and bruhat big cell nman then the space g man may be almost identified with n and with k m and these identifications induce the generalised cayley transform c n k m we show that c is a conformal map of carnot caratheodory manifolds and that composition with the cayley transform combined with multiplication by appropriate powers of the jacobian induces isomorphisms of sobolev spaces h n and h k m we use this to construct uniformly bounded and slowly growing representations of g

the spinorial energy functional on surfaces this is a companion paper to ammann et al in a spinorial energy functional critical points and gradient flow arxiv 1207 3529 2012 where we introduced the spinorial energy functional and studied its main properties in dimensions equal or greater than three in this article we focus on the surface case a salient feature here is the scale invariance of the functional which leads to a plenitude of critical points moreover via the spinorial weierstras representation it relates to the willmore energy of periodic immersions of surfaces into mathbb r 3

solutions without any symmetry for semilinear elliptic problems abstract we prove the existence of infinitely many solitary waves for the nonlinear klein gordon or schrodinger equation u u u 3 0 in r 2 which have finite energy and whose maximal group of symmetry reduces to the identity

stationary isothermic surfaces and uniformly dense domains we establish a relationship between stationary isothermic surfaces and uniformly dense domains a stationary isothermic surface is a level surface of temperature which does not evolve with time a domain in the n dimensional euclidean space r n is said to be uniformly dense in a surface r n of codimension 1 if for every small r 0 the volume of the intersection of with a ball of radius r and center x does not depend on x for x we prove that the boundary of every uniformly dense domain which is bounded or whose complement is bounded must be a sphere we then examine a uniformly dense domain with unbounded boundary and we show that the principal curvatures of satisfy certain identities the case in which the surface coincides with is particularly interesting in fact we show that if the boundary of a uniformly dense domain is connected then i if n 2 it must be either a circle or a straight line and ii if n 3 it must be either a sphere a spherical cylinder or a minimal surface we conclude with a discussion on uniformly dense domains whose boundary is a minimal surface

the sharp threshold and limiting profile of blow up solutions for a davey stewartson system abstract the blow up solutions of the cauchy problem for the davey stewartson system which is a model equation in the theory of shallow water waves are investigated firstly the existence of the ground state for the system derives the best constant of a gagliardo nirenberg type inequality and the variational character of the ground state secondly the blow up threshold of the davey stewartson system is developed in r 3 thirdly the mass concentration is established for all the blow up solutions of the system in r 2 finally the existence of the minimal blow up solutions in r 2 is constructed by using the pseudo conformal invariance the profile of the minimal blow up solutions as t t blow up time is in detail investigated in terms of the ground state

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

the classification of doubly periodic minimal tori with parallel ends let k be the space of properly embedded minimal tori in quotients of r 3 by two independent translations with any fixed even number of parallel ends after an appropriate normalization we prove that k is a 3 dimensional real analytic manifold that reduces to the finite coverings of the examples defined by karcher meeks and rosenberg in 9 10 15 the degenerate limits of surfaces in k are the catenoid the helicoid and three 1 parameter families of surfaces the simply and doubly periodic scherk minimal surfaces and the riemann minimal examples

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

hermite polynomials and helicoidal minimal surfaces the main objective of this paper is to construct smooth 1 parameter families of embedded minimal surfaces in euclidean space that are invariant under a screw motion and are asymptotic to the helicoid some of these families are significant because they generalize the screw motion invariant helicoid with handles and thus suggest a pathway to the construction of higher genus helicoids as a byproduct we are able to relate limits of minimal surface families to the zero sets of hermite polynomials

from refined estimates for spherical harmonics to a sharp multiplier theorem on the grushin sphere abstract we prove a sharp multiplier theorem of mihlin hormander type for the grushin operator on the unit sphere in r 3 and a corresponding boundedness result for the associated bochner riesz means the proof hinges on precise pointwise bounds for spherical harmonics

minimal surfaces with the area growth of two planes the case of infinite symmetry we prove that a connected properly immersed minimal surface in with infinite symmetry group and area growth constant less than is a plane a catenoid or a scherk singly periodic minimal surface as a consequence the scherk minimal surfaces are the only connected periodic minimal desingularizations of the intersections of two planes

entire solutions of the allen cahn equation and complete embedded minimal surfaces of finite total curvature in r 3 we consider minimal surfaces m which are complete embedded and have finite total curvature in mathbb r 3 and bounded entire solutions with finite morse index of the allen cahn equation delta u f u 0 in mathbb r 3 here f w with w bi stable and balanced for instance w u frac 1 4 1 u 2 2 we assume that m has m ge 2 ends and additionally that m is non degenerate in the sense that its bounded jacobi fields are all originated from rigid motions this is known for instance for a catenoid and for the costa hoffman meeks surface of any genus we prove that for any small alpha gt 0 the allen cahn equation has a family of bounded solutions depending on m 1 parameters distinct from rigid motions whose level sets are embedded surfaces lying close to the blown up surface m alpha alpha 1 m with ends possibly diverging logarithmically from m alpha we prove that these solutions are l infty non degenerate up to rigid motions and find that their morse index coincides with the index of the minimal surface our construction suggests parallels of de giorgi conjecture for general bounded solutions of finite morse index

variations on a theme of heine and stieltjes an electrostatic interpretation of the zeros of certain polynomials we show a way to adapt the ideas of stieltjes to obtain an electrostatic interpretation of the zeroes of a large class of orthogonal polynomials

triply periodic constant mean curvature surfaces abstract given a closed flat 3 torus n for each h 0 and each non negative integer g we obtain area estimates for closed surfaces with genus g and constant mean curvature h embedded in n this result contrasts with the theorem of traizet 31 who proved that every flat 3 torus admits for every positive integer g with g 2 connected closed embedded minimal surfaces of genus g with arbitrarily large area

a construction of non kahler calabi yau manifolds and new solutions to the strominger system abstract we propose a new construction of compact non kahler calabi yau manifolds with balanced metrics and study the strominger system on them in particular we obtain explicit solutions to the strominger system with degeneracies on g t 4 where g is an immersed minimal surface of genus g 3 in flat t 3

the classification of singly periodic minimal surfaces with genus zero and scherk type ends given an integer k 2 let s k be the space of complete embedded singly periodic minimal surfaces in r 3 which in the quotient have genus zero and 2k scherk type ends surfaces in s k can be proven to be proper a condition under which the asymptotic geometry of the surfaces is well known it is also known that s 2 consists of the 1 parameter family of singly periodic scherk minimal surfaces we prove that for each k 3 there exists a natural one to one correspondence between s k and the space of convex unitary nonspecial polygons through the map which assigns to each m s k the polygon whose edges are the flux vectors at the ends of m a special polygon is a parallelogram with two sides of length 1 and two sides of length k 1 as consequence s k reduces to the saddle towers constructed by karcher

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

joint functional calculi and a sharp multiplier theorem for the kohn laplacian on spheres let box b be the kohn laplacian acting on 0 j forms on the unit sphere in mathbb c n in a recent paper of casarino cowling sikora and the author a spectral multiplier theorem of mihlin hormander type for box b is proved in the case 0 j n 1 here we prove an analogous theorem in the exceptional cases j 0 and j n 1 including a weak type 1 1 endpoint estimate we also show that both theorems are sharp the proof hinges on an abstract multivariate multiplier theorem for systems of commuting operators

a limit method for solving period problems on minimal surfaces we introduce a new technique to solve period problems on minimal surfaces called limit method if a family of surfaces has weierstras data converging to the data of a known example and this presents a transversal solution of periods then the original family contains a sub family with closed periods

a note on the global existence of small amplitude solutions to a generalized davey stewartson system in this paper we are interested in the cauchy problem for a generalized davey stewartson gds system we establish the global time existence of small mass solutions for the gds system in the elliptic hyperbolic hyperbolic case

on the genus of triply periodic minimal surfaces we prove the existence of embedded minimal surfaces of arbitrary genus g 3 in any at 3 torus in fact we construct a sequence of such surfaces converging to a planar foliation of the 3 torus in particular the area of the surface can be chosen arbitrarily large

handle addition for doubly periodic scherk surfaces we prove the existence of a family of embedded doubly periodic minimal surfaces of quotient genus g with orthogonal ends that generalizes the classical doubly periodic surface of scherk and the genus one scherk surface of karcher the proof of the family of immersed surfaces is by induction on genus while the proof of embeddedness is by the conjugate plateau method

quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres a sharp l p spectral multiplier theorem of mihlin hormander type is proved for a distinguished sub laplacian on quaternionic spheres this is the first such result on compact sub riemannian manifolds where the horizontal space has corank greater than one the proof hinges on the analysis of the quaternionic spherical harmonic decomposition of which we present an elementary derivation

sharp threshold of global existence and instability of standing wave for a davey stewartson system this paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave phi t x e i omega t u x for the system i phi t delta phi a phi p 1 phi b e 1 phi 2 phi 0 quad t geq 0 quad x in bf r n quad quad quad rm ds where a 0 b 0 1 0 by combining the former results

the embedded singly periodic scherk costa surfaces we give a positive answer to m traizet s open question about the existence of complete embedded minimal surfaces with scherk ends without planar geodesics in the singly periodic case these examples get close to an extension of traizet s result concerning asymmetric complete minimal submanifolds of open image in new window with finite total curvature

existence and blow up of solutions to degenerate davey stewartson equations in the present paper we consider a system of degenerate davey stewartson equations we prove the global existence of weak solutions and blow up of solutions for appropriate initial data