optimal rigidity estimates for nearly umbilical surfaces let r 3 be a smooth compact connected surface without boundary and denote by a its second fundamental form we prove the existence of a universal constant c such that

subelliptic p harmonic maps into spheres and the ghost of hardy spaces the story begins with the paper of muller 59 who for the sake of an application to nonlinear elasticity proved that if the jacobian determinant ju of a sobolev map u w 1 n loc rn rn is nonnegative then it belongs locally to l log l the result is quite intriguing since a priori holder inequality implies only that ju l1 and one does not suspect any higher integrability if one does not assume that ju is nonnegative then as coifman lions meyer and semmes 14 have proved ju belongs to the local hardy space h 1 loc since a nonnegative function belongs to the local hardy space if and only if it belongs locally to l log l the result of coifman lions meyer and semmes generalizes that of muller in fact coifman lions meyer and semmes proved more namely for 1 p the scalar product of a divergence free vector field e lp rn rn and a curl free vector field b lp p 1 rn rn belongs to the hardy space h 1 a priori holder inequality implies integrability of this expression only note that the jacobian is of this form the discovery that many algebraic expressions involving partial derivatives belong to the hardy space turned out to be important and widely applicable in nonlinear partial differential equations this was first shown by helein 40 41 42 let us describe briefly his result consider maps u bn s m from the n dimensional ball to the mdimensional sphere such that the p energy of u given by the functional

embedded surfaces of arbitrary genus minimizing the willmore energy under isoperimetric constraint the isoperimetric ratio of an embedded surface in mathbb r 3 is defined as the ratio of the area of the surface to power three to the squared enclosed volume the aim of the present work is to study the minimization of the willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus g geqq 0 the corresponding problem in the case of spherical surfaces that is g 0 was recently solved by schygulla see schygulla arch ration mech anal 203 901 941 2012 with different methods

on the amplitude decay of strongly non linear damped oscillators the problem considered is that of the amplitude behavior of the strongly non linear damped oscillator u 2 u u f u 0 where is small u and f u are o 1 may be arbitrarily large and f u is an odd function a simple approximation to the amplitude decay is obtained by first linearizing the static terms and then using the wkb approximation of particular interest is the influence of the non linearity on the amplitude decay several examples which illustrate this influence are provided

rigidity and non rigidity results for conformal immersions abstract in this paper we show a quantitative rigidity result for the minimizer of the willmore functional among all projective planes in r n with n 4 we also construct an explicit counterexample to a corresponding rigidity result in codimension one by showing that an enneper surface might split off during a blow up process for conformal immersions of spheres with large enough willmore energies we construct explicit counterexamples to a quantitative rigidity result and this complements the recently obtained rigidity results in 14

on the primary frequency response of strongly non linear oscillators a linearization approach abstract the problem considered is the primary frequency response of the class of strongly non linear forced oscillators u 2 u mu eg u p cos t whee g u is an odd non linear function of the displacement u t e may be arbitrarily large and m 1 0 or 1 a simple approximation to the fundamental harmonic of the response is obtained by first linearizing the static terms using the frequency relations developed in reference 1 for the autonomous conservative oscillator and then using the averaging method to solve the linearized forced problem a comparison with the numerical and other approximate analytic methods results is made which shows the advantages of the present technique in terms of computational effort and accuracy

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

chord arc surfaces with small constant ii good parameterizations abstract in a companion paper we introduced a notion for hypersurfaces in r d 1 that is analogous to the chord arc condition with small constant for planar curves in this paper we address the problem of finding good parameterizations for these surfaces in particular we show that these surfaces must be homeomorphic to r d and we build parameterizations with l p bounds on the gradient and the gradient of the inverse with p as the chord arc constant goes to 0 when d 2 we use the uniformization theorem to find a quasisymmetric parameterization we do not know how to build bilipschitz parameterizations when d 2 or quasisymmetric parameterizations when d 3 the author conjectures that they do not exist in general however we can build bilipschitz reflections in all dimensions and these reflections will satisfy certain extra estimates in terms of carleson measures

variational principles for immersed surfaces with l 2 bounded second fundamental form in this work we present new fundamental tools for studying the variations of the willmore functional of immersed surfaces into r m this approach gives for instance a new proof of the existence of a willmore minimizing embedding of an arbitrary closed surface in arbitrary codimension we explain how the same approach can solve constraint minimization problems for the willmore functional we show in particular that for a given closed surface and a given conformal class for this surface there is an immersion in r m away possibly from isolated branched points which minimizes the willmore energy among all possible lipschitz immersions in r m having an l 2 bounded second fundamental form and realizing this conformal class this branched immersion is either a smooth conformal willmore branched immersion or an isothermic branched immersion we show that branched points do not exist whenever the minimal energy in the conformal class is less than 8 and that these immersions extend to smooth conformal willmore embeddings or global isothermic embeddings of the surface in that case finally as a by product of our analysis we establish that inside a compact subspace of the moduli space the following holds weak limit of palais smale willmore sequences are conformal willmore that weak limits of palais smale sequences of conformal willmore are either conformal willmore or global isothermic and finally we observe also that weakly converging palais smale sequences of global isothermic immersions are global isothermic the analysis developped along the paper in particular these last results opens the door to the possibility of constructing new critical saddle points of the willmore functional without or with constraints using min max methods

a dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain a dynamically adaptive multilevel wavelet collocation method is developed for the solution of partial differential equations the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where sharp transitions occur the scheme handles general boundary conditions the method is applied to the solution of the one dimensional burgers equation with small viscosity a moving shock problem and a nonlinear thermoacoustic wave problem the results indicate that the method is very accurate and efficient

multiple solutions of h systems and rellich s conjecture soit x y r 2 x 2 y 2 1 on considere une fonction u r 3 satisfaisant le systeme h u 2hu x u y sur avec l une des conditions suivantes soit dirichlet u sur soit plateau u x 2 u y 2 u x u y 0 sur u et u non decroissante sur

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

quantitative stability for the first dirichlet eigenvalue in reifenberg flat domains in rn abstract in this paper we prove that if and are close enough for the complementary hausdorff distance and their boundaries satisfy some geometrical and topological conditions then 1 1 c n where 1 resp 1 is the first dirichlet eigenvalue of the laplacian in resp and is the lebesgue measure of the symmetric difference here the constant 1 could be taken arbitrary close to 1 but strictly less and c is a constant depending on a lot of parameters including dimension n and some geometric properties of the domains

a frame energy for immersed tori and applications to regular homotopy classes the paper is devoted to study the dirichelet energy of moving frames on 2 dimensional tori immersed in the euclidean 3 m dimensional space this functional called frame energy is naturally linked to the willmore energy of the immersion and on the conformal structure of the abstract underlying surface as first result a willmore conjecture type lower bound is established namely for every torus immersed in r m m 3 and any moving frame on it the frame energy is at least 2 2 and equalty holds if and only if m 4 the immersion is the standard clifford torus up to rotations and dilations and the frame is the flat one smootheness of the critical points of the frame energy is proved after the discovery of hidden conservation laws and as application the minimization of the frame energy in regular homotopy classes of immersed tori in r 3 is performed

integral menger curvature for surfaces abstract we develop the concept of integral menger curvature for a large class of nonsmooth surfaces we prove uniform ahlfors regularity and a c 1 a priori bound for surfaces for which this functional is finite in fact it turns out that there is an explicit length scale r 0 which depends only on an upper bound e for the integral menger curvature m p and the integrability exponent p and not on the surface itself below that scale each surface with energy smaller than e looks like a nearly flat disc with the amount of bending controlled by the local m p energy moreover integral menger curvature can be defined a priori for surfaces with self intersections or branch points we prove that a posteriori all such singularities are excluded for surfaces with finite integral menger curvature by means of slicing and iterative arguments we bootstrap the holder exponent up to the optimal one 1 8 p thus establishing a new geometric morrey sobolev imbedding theorem as two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral menger curvature with a uniform bound on area and of area minimizing surfaces subjected to a uniform bound on integral menger curvature

dahlberg s theorem in higher co dimension abstract in 1977 the celebrated theorem of b dahlberg established that the harmonic measure is absolutely continuous with respect to the hausdorff measure on a lipschitz graph of dimension n 1 in r n and later this result has been extended to more general non tangentially accessible domains and beyond in the present paper we prove the first analogue of dahlberg s theorem in higher co dimension on a lipschitz graph of dimension d in r n d n 1 with a small lipschitz constant we construct a linear degenerate elliptic operator l such that the corresponding harmonic measure l is absolutely continuous with respect to the hausdorff measure on more generally we provide sufficient conditions on the matrix of coefficients of l which guarantee the mutual absolute continuity of l and the hausdorff measure

on conformal surfaces of annulus type abstract let a b 0 and b a b b x x 1 x 2 r 2 b x a and assume that f is a conformal map from b a b b into r n with f 2 2 e 2 u then e 1 e 2 with e 1 e u f r and e 2 r 1 e u f is a moving frame on f b a b b and it satisfies the following equation d d e 1 e 2 0 where is the hodge star operator on r 2 with respect to the standard metric we will study the dirichlet energy of this frame and give some applications

quasisymmetric parametrizations of two dimensional metric planes the classical uniformization theorem states that any simply connected riemann surface is conformally equivalent to the disk the plane or the sphere each equipped with a standard conformal structure we give a similar uniformization for ahlfors 2 regular linearly locally connected metric planes instead of conformal equivalence we are concerned with quasisymmetric equivalence

uniformization of two dimensional metric surfaces we establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite hausdorff 2 measure applying the geometric definition of quasiconformality we give a necessary and sufficient condition for such spaces to be qc equivalent to the euclidean plane disk or sphere moreover we show that if such a qc parametrization exists then the dilatation can be bounded by 2 as an application we show that the euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2 qc parametrization this result gives a new approach to the bonk kleiner theorem on parametrizations of ahlfors 2 regular spheres by quasisymmetric maps

bi lipschitz parameterization of surfaces we consider surfaces z homeomorphic to the plane with complete possibly singular riemannian metrics if we have z k 0 and c 0 this result implies a conjecture by j fu

quasiconformal planes with bi lipschitz pieces and extensions of almost affine maps abstract a quasiplane f v is the image of an n dimensional euclidean subspace v of r n 1 n n 1 under a quasiconformal map f r n r n we give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi lipschitz n manifold and for a quasiplane to have big pieces of bi lipschitz images of r n one main novelty of these results is that we analyze quasiplanes in arbitrary codimension n n to establish the big pieces criterion we prove new extension theorems for almost affine maps which are of independent interest this work is related to investigations by tukia and vaisala on extensions of quasisymmetric maps with small distortion

analysis of non linear autonomous conservative oscillators by a time transformation method the problem considered is that of the free vibration of the class of strongly non linear oscillators u mu f u 0 where m 1 0 or 1 f u is a non linear function of the displacement u t and is not small through a continuous stretching of the time t a new time t is found in whose domain the response is simple harmonic this time transformation method allows the response u t and period to be calculated to any desired degree of accuracy however the primary objective is to develop reasonably simple accurate approximations for the oscillators considered

a theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity the energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by g kirchhoff in 1850 we show that it arises as a 0 limit of three dimensional nonlinear elasticity theory as the thickness of a plate goes to zero a key ingredient in the proof is a sharp rigidity estimate for maps v v u r n u r n we show that the l 2 distance of rv from a single rotation matrix is bounded by a multiple of the l 2 distance from

on the sobolev space of isometric immersions we prove that every w 2 2 isometric immersion from a convex regular domain of r 2 into r3 can be approximated in w 2 2 norm by smooth isometric immersions from the same domain into r3

stratification for the singular set of approximate harmonic maps the aim of this note is to extend the results in naber and valtorta ann math 2 185 131 227 https doi org 10 4007 annals 2017 185 1 3 2017 to the case of approximate harmonic maps more precisely we will proved that the singular strata mathcal s k u of an approximate harmonic map are k rectifiable and we will show effect bounds on the quantitative strata in the process we will simplify many of the arguments from naber and valtorta 2017 and in particular we produce a new main covering lemmas which vastly simplifies the older argument

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

sharp commutator estimates via harmonic extensions abstract we give an alternative proof of several sharp commutator estimates involving riesz transforms riesz potentials and fractional laplacians our methods only involve harmonic extensions to the upper half space integration by parts and trace space characterizations the commutators we investigate are jacobians more generally coifman rochberg weiss commutators chanillo s commutator with the riesz potential coifman mcintosh meyer kato ponce vega type commutators the fractional leibniz rule and the da lio riviere three term commutators we also give a new limiting l 1 estimate for a double commutator of coifman rochberg weiss type and several intermediate estimates the beauty of our method is that all those commutator estimates which are originally proven by various specific methods or by general para product arguments can be obtained purely from integration by parts and trace theorems another interesting feature is that in all these cases the cancellation effect responsible for the commutator estimate simply follows from the product rule for classical derivatives and can be traced in a precise way

effective reifenberg theorems in hilbert and banach spaces a famous theorem by reifenberg states that closed subsets of mathbb r n that look sufficiently close to k dimensional at all scales are actually c 0 gamma equivalent to k dimensional subspaces since then a variety of generalizations have entered the literature for a general measure mu in mathbb r n one may introduce the k dimensional jones beta numbers of the measure where beta k mu x r quantifies on a given ball b r x how closely in an integral sense the support of the measure is to living inside a k dimensional subspace recently it has been proven that if these beta numbers satisfy the uniform summability estimate int 0 2 beta k mu x r 2 frac dr r m then mu must be rectifiable with uniform measure bounds note that one only needs the square of the beta numbers to satisfy the summability estimate this power gain has played an important role in the applications for instance in the study of singular sets of geometric equations one may also weaken these pointwise summability bounds to bounds which are more integral in nature the aim of this article is to study these effective reifenberg theorems for measures in a hilbert or banach space for hilbert spaces we see all the results from mathbb r n continue to hold with no additional restrictions for a general banach spaces we will see that the classical reifenberg theorem holds and that a weak version of the effective reifenberg theorem holds in that if one assumes a summability estimate int 0 2 beta k mu x r 1 frac dr r m without power gain then mu must again be rectifiable with measure estimates improving this estimate in order to obtain a power gain turns out to be a subtle issue for k 1 we will see for a uniformly smooth banach space that if int 0 2 beta 1 mu x r alpha frac dr r m alpha 2 where alpha is the smoothness power of the banach space then mu is again rectifiable with uniform measure estimates

bubble tree of branched conformal immersions and applications to the willmore functional we develop a bubble tree construction and prove compactness results for w 2 2 branched conformal immersions of closed riemann surfaces with varying conformal structures whose limit may degenerate in bbb r n with uniformly bounded areas and willmore energies the compactness property is applied to construct willmore type surfaces in compact riemannian manifolds this includes a existence of a willmore 2 sphere in bbb s n with at least 2 nonremovable singular points b existence of minimizers of the willmore functional with prescribed area in a compact manifold n provided i the area is small when genus is 0 and ii the area is close to that of the area minimizing surface of schoen yau and sacks uhlenbeck in the homotopy class of an incompressible map from a surface of positive genus to n and pi 2 n is trivial c existence of smooth minimizers of the willmore functional if a douglas type condition is satisfied

invertibility and non invertibility in thin elastic structures the nonlinear elastic energy of a thin film of thickness h is given by a functional eh friesecke james and muller derived the limits as h 0 of the functionals h eh for 3 in this article we study the invertibility properties of almost minimizers of these functionals and more generally of sequences with equiintegrable energy density we show that they are invertible almost everywhere away from a thin boundary layer near the film surface moreover we obtain an upper bound for the width of this layer and a uniform upper bound on the diameter of preimages we construct examples showing that these bounds are sharp in particular for all 3 there exist lipschitz continuous low energy deformations which are not locally invertible

new monotonicity formulas for ricci curvature and applications i we prove three new monotonicity formulas for manifolds with a lower ricci curvature bound and show that they are connected to rate of convergence to tangent cones in fact we show that the derivative of each of these three monotone quantities is bounded from below in terms of the gromov hausdorff distance to the nearest cone the monotonicity formulas are related to the classical bishop gromov volume comparison theorem and perelman s celebrated monotonicity formula for the ricci flow we will explain the connection between all of these moreover we show that these new monotonicity formulas are linked to a new sharp gradient estimate for the green function that we prove this is parallel to the fact that perelman s monotonicity is closely related to the sharp gradient estimate for the heat kernel of li yau in cm4 one of the monotonicity formulas is used to show uniqueness of tangent cones with smooth cross sections of einstein manifolds finally there are obvious parallelisms between our monotonicity and the positive mass theorem of schoen yau and witten

quantitative rigidity results for conformal immersions in this paper we prove several quantitative rigidity results for conformal immersions of surfaces in with bounded total curvature we show that branched conformal immersions which are close in energy to either a round sphere a conformal clifford torus an inverted catenoid an inverted enneper s minimal surface or an inverted chen s minimal graph must be close to these surfaces in the w norm moreover we apply these results to prove a corresponding rigidity result for complete connected and non compact surfaces

chord arc surfaces with small constant i abstract we introduce a notion for hypersurfaces in r d 1 that is analogous to the chord arc condition with small constant for curves in the plane we give various equivalent operator theoretic function theoretic and geometrical characterizations of these surfaces we not know any nice potential theoretic characterizations in a companion paper we address the issue of finding good parameterizations for these surfaces

harmonic measure on locally flat domains we will review work with tatiana toro yielding a characterizati on of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation

analysis aspects of willmore surfaces a new formulation for the euler lagrange equation of the willmore functional for immersed surfaces in m is given as a nonlinear elliptic equation in divergence form with non linearities comprising only jacobians letting vec h be the mean curvature vector of the surface our new formulation reads mathcal l vec h 0 where mathcal l is a well defined locally invertible self adjoint elliptic operator several consequences are studied in particular the long standing open problem asking for a meaning to the willmore euler lagrange equation for immersions having only l 2 bounded second fundamental form is now solved the regularity of weak willmore immersions with l 2 bounded second fundamental form is also established its proof relies on the discovery of conservation laws which are preserved under weak convergence a weak compactness result for willmore surfaces with energy less than 8 the li yau condition ensuring the surface is embedded is proved via a point removability result established for wilmore surfaces in m thereby extending to arbitrary codimension the main result in ks3 finally from this point removability result the strong compactness of willmore tori below the energy level 8 is proved both in dimension 3 this had already been settled in ks3 and in dimension 4

removability of point singularities of willmore surfaces we investigate point singularities of willmore surfaces which for example appear as blowups of the willmore flow near singularities and prove that closed willmore surfaces with one unit density point singularity are smooth in codimension one as applications we get in codimension one that the willmore flow of spheres with energy less than 8 exists for all time and converges to a round sphere and further that the set of willmore tori with energy less than

the isoperimetric inequality and q curvature abstract a well known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions in dimension 2 the constant can be controlled by the integral of the positive part of the gaussian curvature in this paper we showed that on simply connected conformally flat manifolds of higher dimensions the role of the gaussian curvature can be replaced by the branson s q curvature we achieve this by exploring the relationship between a p weights and integrals of the q curvature

non separable splines and numerical computation of evolution equations by the galerkin methods we construct new non separable splines and we apply the spline sampling approximation to the computation of numerical solutions of evolution equations the non separable splines are basis functions which give a fine sampling approximation which enables us to compute numerical solutions by means of the method of lines combined with the galerkin method to demonstrate our approach we compute numerical solutions of the burgers equation and the kadomtsev petviashvili equation

estimation of the conformal factor under bounded willmore energy we prove for closed orientable surfaces in mathbb r 3 with willmore energy less that 8 pi delta and whose conformal structures are compactly contained in moduli space that after applying appropriate mobius transformations the conformal factors between the induced metrics and conformal metrics of constant curvature are uniformly bounded by constants depending only on delta 0 the genus of the surfaces and the compact subset of the moduli space secondly for a given sequence of closed orientable surfaces as above we prove that the conformal factor remains bounded without applying mobius transformations if and only if no topology is lost similar estimates hold in higher codimension

mean curvature volume and properness of isometric immersions the first author s work was partially supported by the research program of university jaume i project p1 1b2012 18 and dgi mineco grant feder mtm2013 48371 c2 2 p the second author s work was partially supported by the research program of university jaume i project p1 1b2012 18 dgi mineco grant feder mtm2013 48371 c2 2 p and generalitat valenciana grant prometeoii 2014 064

critical weak immersed surfaces within sub manifolds of the teichmuller space abstract we prove that the critical points of various energies such as the area the willmore energy the frame energy for tori etc among possibly branched immersions constrained to evolve within a smooth sub manifold of the teichmuller space satisfy the corresponding constrained euler lagrange equation we deduce that critical points of the willmore energy or the frame energy for tori are smooth analytic surfaces away possibly from isolated branched points under the condition that either the genus is at most 2 or if the sub manifold does not intersect the subspace of hyper elliptic points using a compactness result from 19 we can conclude that each closed sub manifold of the teichmuller space including points under the previous assumptions possess a possibly branched smooth willmore minimizer satisfying the conformal constrained willmore equation

energy quantization for willmore surfaces and applications we prove a bubble neck decomposition together with an energy quantization result for sequences of willmore surfaces into r m with uniformly bounded energy and nondegenerating conformal type we deduce the strong compactness of willmore closed surfaces of a given genus modulo

a highly anisotropic nonlinear elasticity model for vesicles ii derivation of the thin bilayer bending theory we study the thin shell limit of the nonlinear elasticity model for vesicles introduced in part i we consider vesicles of width 2 varepsilon downarrow 0 with elastic energy of order varepsilon 3 in this regime we show that the limit model is a bending theory for generalized hypersurfaces namely co dimension one oriented varifolds without boundary up to a positive factor the limit functional is the willmore energy in the language of it gamma convergence we establish a compactness result a lower bound result and the matching upper bound in the smooth case

willmore spheres in compact riemannian manifolds abstract the paper is devoted to the variational analysis of the willmore and other l 2 curvature functionals among immersions of 2 dimensional surfaces into a compact riemannian m manifold m m h with m 2 the goal of the paper is two fold on one hand we give the right setting for doing the calculus of variations including minmax methods of such functionals for immersions into manifolds and on the other hand we prove the existence results for possibly branched willmore spheres under various constraints prescribed homotopy class prescribed area or under curvature assumptions for m m to this aim using the integrability by compensation theory we first establish the regularity for the critical points of such functionals we then prove a rigidity theorem concerning the relation between cmc and willmore spheres then we prove that for every non null 2 homotopy class there exists a representative given by a lipschitz map from the 2 sphere into m m realizing a connected family of conformal smooth possibly branched area constrained willmore spheres as explained in the introduction this comes as a natural extension of the minimal immersed spheres in homotopy class constructed by sacks and uhlenbeck 1981 in 38 in situations when they do not exist moreover for every a 0 we minimize the willmore functional among connected families of weak possibly branched immersions of the 2 sphere having prescribed total area equal to a and we prove full regularity for the minimizer finally under a mild curvature condition on m m h we minimize the sum of the area with the square of the l 2 norm of the second fundamental form among weak possibly branched immersions of the two spheres and we prove the regularity of the minimizer

analysis of forced non linear undamped oscillators by a time transformation method the recently developed time transformation method 1 3 is extended to the analysis of forced vibration of the strongly non linear undamped oscillator u u ef u p t where f u is a non linear function of the displacement u t and e is not necessarily small two types of external loading p t are considered a step input and a simple harmonic input the frequency amplitude relation is obtained through a slight modification of the results developed in reference 2 for the unforced oscillator a comparison with the results of numerical and other analytic methods is made it can be deduced that the time transformation approach offers simple and accurate approximations for the oscillators considered

a finite difference method for the korteweg de vries and the kadomtsev petviashvili equations a linearized implicit finite difference method for the korteweg de vries equation is proposed and straightforwardly extended to the kadomtsev petviashvili equation we investigate the order of accuracy of the method and prove the method to be unconditionally linearly stable the numerical experiments for the korteweg de vries and the kadomtsev petviashvili equations are carried out with various conditions numerical results for the collision of two lump type solitary wave solutions to the kadomtsev petviashvili equation are also reported

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

asymptotically flat three manifolds contain minimal planes abstract let m g be an asymptotically flat 3 manifold containing no closed embedded minimal surfaces we prove that for every point p m there exists a complete properly embedded minimal plane in m containing p

rectifiable sets and the traveling salesman problem let k c c be a bounded set in this paper we shall give a simple necessary and sufficient condit ion for k to lie in a rectifiable curve we say that a set is a rectifiable curve if it is the image of a finite interval under a lipschitz mapping recall that for a connected set f c c f is a rectifiable curve not necessarily simple if and only if l f where l denotes one dimensional hausdorff measure this classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge but not necessarily each vertex at most twice if k is a finite set then we are essentially reduced to the classical traveling salesman problem tsp compute the length of the shortest hami l ton ian cycle which hits all points of k this is the same up to a constant multiple as asking for the inf imum of l f where f is a curve k c f such a f is called a spanning tree in tsp theory for infinite sets k we cannot hope in general to have k be a subset of a jordan curve what we should therefore look at is connected sets which conta in k let fmi n be the shortest minimal spanning tree then we cannot possibly solve our problem for sets k of infinite cardinality if we cannot find f i f c o fmin for any finite set k here and throughout the paper c co c1 c o etc denote various universal constants while there are several algorithms for computing l fml these algorithms work for finite graphs satisfying the triangle inequality and do not use the euclidean properties of k see 13 for an excellent discussion of some of these algorithms therefore these methods cannot solve our problem for general infinite k we present a method which is a minor modification of a well known algorithm farthest insert ion see 13 which yields a f with i f c o l fmi the farthest insert ion algorithm has been extensively studied with large numerical calculations on computers and is experimentally good in the sense that the f produced satisfy i f c o l f i for all examples which have

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we