second eigenvalue of schrodinger operators and mean curvature let m be a compact immersed submanifold of the euclidean space the hyperbolic space or the standard sphere for any continuous potential q on m we give a sharp upper bound for the second eigenvalue of the operator q in terms of the total mean curvature of m and the mean value of q moreover we analyze the case where this bound is achieved as a consequence of this result we obtain an alternative proof for the alikakos fusco conjecture concerning the stability of the interface in the allen cahn reaction diffusion model

automorphisms of the nottingham group abstract let k be a finite field of characteristic p 5 the nottingham group n over k is the group of normalised automorphisms of the local field k t in this paper we determine the automorphism group of n in particular we show that every automorphism of the nottingham group is standard we also give a complete description of the conjugacy classes of elements of order p in n

hausdorff dimension pro p groups and kac moody algebras every finitely generated profinite group can be given the structure of a metric space and as such it has a well defined hausdorff dimension function in this paper we study hausdorff dimension of closed subgroups of finitely generated pro p groups g we prove that if g is p adic analytic and h c g is a closed subgroup then the hausdorff dimension of h is dimh dimg where the dimensions are of h and g as lie groups letting the spectrum spec g of g denote the set of hausdorff dimensions of closed subgroups of g it follows that the spectrum of p adic analytic groups is finite and consists of rational numbers we then consider some non p adic analytic groups g and study their spectrum in particular we investigate the maximal hausdorff dimension of nonopen subgroups of g and show that it is equal to 1 1 d 1 in the case of g sld fp t where p 2 and to 1 2 if g is the so called nottingham group where p 5 we also determine the spectrum of sl2 fp t p 2 completely showing that it is equal to 0 2 3 1 some of the proofs rely on the description of maximal graded subalgebras of kac moody algebras recently obtained by the authors in joint work with e i zelmanov

how paper folds bending with local constraints a variational framework is introduced to describe how a surface bends when it is subject to local constraints on its geometry this framework is applied to describe the patterns of a folded sheet of paper the unstretchability of paper implies a constraint on the surface metric bending is penalized by an energy quadratic in mean curvature the local lagrange multipliers enforcing the constraint are identified with a conserved tangential stress that couples to the extrinsic curvature of the sheet the framework is illustrated by examining the deformation of a flat sheet into a generalized cone

environmental bias and elastic curves on surfaces the behavior of an elastic curve bound to a surface will reflect the geometry of its environment this may occur in an obvious way the curve may deform freely along directions tangent to the surface but not along the surface normal however even if the energy itself is symmetric in the curve s geodesic and normal curvatures which control these modes very distinct roles are played by the two if the elastic curve binds preferentially on one side or is itself assembled on the surface not only would one expect the bending moduli associated with the two modes to differ binding along specific directions reflected in spontaneous values of these curvatures may be favored the shape equations describing the equilibrium states of a surface curve described by an elastic energy accommodating environmental factors will be identified by adapting the method of lagrange multipliers to the darboux frame associated with the curve the forces transmitted to the surface along the surface normal will be determined features associated with a number of different energies both of physical relevance and of mathematical interest are described the conservation laws associated with trajectories on surface geometries exhibiting continuous symmetries are also examined

embedding properties of hereditarily just infinite profinite wreath products abstract we study infinitely iterated wreath products of finite permutation groups w r t product actions in particular we prove that for every non empty class of finite simple groups x there exists a finitely generated hereditarily just infinite profinite group w with composition factors in x such that any countably based profinite group with composition factors in x can be embedded into w additionally we investigate when infinitely iterated wreath products of finite simple groups w r t product actions are co hopfian or non co hopfian

pro p groups with few normal subgroups abstract motivated by the study of pro p groups with finite coclass we consider the class of pro p groups with few normal subgroups this is not a well defined class and we offer several different definitions and study the connections between them furthermore we propose a definition of periodicity for pro p groups thus providing a general framework for some periodic patterns that have already been observed in the existing literature we then focus on examples and show that strikingly all the interesting examples not only have few normal subgroups but in addition have periodicity in the lattice of normal subgroups

abstract commensurators of profinite groups in this paper we initiate a systematic study of the abstract com mensurators of profinite groups the abstract commensurator of a profinite group g is a group comm g which depends only on the commensurability class of g we study various properties of comm g in particular we find two natural ways to turn it into a topological group we also use comm g to study topological groups which contain g as an open subgroup all such groups are totally disconnected and locally compact for instance we con struct a topologically simple group which contains the pro 2 completion of the grigorchuk group as an open subgroup on the other hand we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups several celebrated rigidity theorems like pink s analogue of mostow s strong rigidity theorem for simple algebraic groups defined over local fields and the neukirch uchida theorem can be re formulated as structure theorems for the commensurators of certain profinite groups

the nottingham group for p 2 abstract in this paper i consider the nottingham group over a finite field of characteristic 2 i obtain a number of results previously known only for nottingham groups in odd characteristic for example i obtain information on the derived series and i prove that the nottingham group is hereditarily just infinite as a pro 2 group as a consequence the nottingham group over any finite field of characteristic 2 is not analytic over any pro 2 ring

on otsuki tori and their willmore energy abstract otsuki tori in the three dimensional unit sphere are studied from calculations of the willmore energy the lower and upper bound estimates for the otsuki tori are established invariance of the otsuki tori under antipodal maps is also considered and in such cases better lower bound estimates of their willmore energies are obtained

extendability of kirchhoff elastic rods in complete riemannian manifolds the kirchhoff elastic rod is a classical mathematical model of equilibrium configurations of thin elastic rods and is defined to be a solution of the euler lagrange equations associated to the energy with the effect of bending and twisting we consider the initial value problem for the euler lagrange equations in a riemannian manifold in a previous paper the author proved the existence and uniqueness of global solutions of the initial value problem in the case where the ambient space is a space form in the present paper we extend this result to the case where the ambient space is a general complete riemannian manifold this implies that an arbitrary kirchhoff elastic rod of finite length in a complete riemannian manifold extends to that of infinite length

new just infinite pro p groups of finite width and subgroups of the nottingham group we use the action of the nottingham group on the completion of its lie algebra to construct new examples of just infinite pro p groups of finite width whose graded lie algebras with respect to the lower central series are isomorphic to sl2 fp tfp t the groups constructed are non linear and thus answer the question posed in comm algebra 30 2002 1293 the proof of nonlinearity uses a new description of the centralizers of elements of order p in the nottingham group and the concept of hausdorff dimension

new properties and examples of incomplete lorentzian tori a new procedure to construct incomplete metrics on a two torus is introduced relations between curvature and completeness are explored and examples showing a wide independence among them are given incompleteness appears as a usual property of the lorentzian tori and the density of the incomplete metrics in the set of all the lorentzian ones is discussed this article can be seen as a partial study of compact space times and the conformal lorentzian moduli space mod t2 as claimed by yurtsever j math phys 31 3064 1990 and as a continuation of the authors previous article j math phys 34 3768 1993

on the commensurator of the nottingham group let p 5 be a prime number we prove that the abstract commensurator of the nottingham group n f p coincides with its automorphism group which is known to be a finite extension of n f p as a corollary we deduce that the nottingham group cannot be embedded as an open subgroup of a topologically simple group

a criterion for reduction of variables in the willmore chen variational problem and its applications we exhibit a criterion for a reduction of variables for willmorechen submanifolds in conformal classes associated with generalized kaluzaklein metrics on flat principal fibre bundles our method relates the variational problem of willmore chen with an elasticity functional defined for closed curves in the base space the main ideas involve the extrinsic conformal invariance of the willmore chen functional the large symmetry group of generalized kaluza klein metrics and the principle of symmetric criticality we also obtain interesting families of elasticae in both lens spaces and surfaces of revolution riemannian and lorentzian we give applications to the construction of explicit examples of isolated willmore chen submanifolds discrete families of willmore chen submanifolds and foliations whose leaves are willmore chen submanifolds

index subgroups of the nottingham group abstract it is known that every countably based pro p group embeds as a closed subgroup into the nottingham group j j f p at the same time the subgroup structure of j remains rather mysterious in this paper we introduce the class of so called index subgroups of j and study their fundamental properties two main applications are given i we prove that a certain infinite subclass of index subgroups provides new examples of hereditarily just infinite pro p groups of finite width ii we determine which hausdorff dimensions occur for index subgroups and subsequently show that the hausdorff spectrum of j begins with a proper interval

some hereditarily just infinite subgroups of j abstract this work examines the commutator structure of some closed subgroups of the wild group of automorphisms of a local field with perfect residue field a group we call j in particular we establish a new approach to evaluating commutators in j and using this method investigate the normal subgroup structure of some classes of index subgroups of j as introduced by klopsch we provide new proofs of fesenko s results that lead to a proof that the torsion free group t t k 1 a k t qk 1 a k f p is hereditarily just infinite and by extending his work we also demonstrate the existence of a new class of hereditarily just infinite subgroups of j which have non trivial torsion

relativistic particles with rigidity generating non standard examples of willmore chen hypersurfaces we study a natural extension to higher dimensions of the nambu goto polyakov action in particular those dynamical objects evolving with so 3 symmetry in four dimensions we show that this problem is strongly related to that of relativistic particles with rigidity of order three in a hyperbolic plane the moduli space of solitonic solutions is completely determined in terms of the so called rotation number a quantization principle for closed solutions is also obtained and this gives a rational one parameter family of willmore chen hypersurfaces in the standard conformal structure of dimension four moreover these are the first non standard examples of this kind of hypersurfaces

deforming convex curves with fixed elastic energy abstract this paper presents a curve flow which preserves the elastic energy of the evolving curve if the initial curve is a planar simple and smooth curve with positive curvature then the local and global existence of the flow is proved under this flow the evolving curve will converge to a finite circle in the c metric as time goes to infinity

equivariant willmore surfaces in conformal homogeneous three spaces abstract the complete classification of homogeneous three spaces is well known for some time of special interest are those with rigidity four which appear as riemannian submersions with geodesic fibres over surfaces with constant curvature consequently their geometries are completely encoded in two values the constant curvature c of the base space and the so called bundle curvature r in this paper we obtain the complete classification of equivariant willmore surfaces in homogeneous three spaces with rigidity four all these surfaces appear by lifting elastic curves of the base space once more the qualitative behaviour of these surfaces is encoded in the above mentioned parameters c r the case where the fibres are compact is obtained as a special case of a more general result that works via the principle of symmetric criticality for bundle like conformal structures in circle bundles however if the fibres are not compact a different approach is necessary we compute the differential equation satisfied by the equivariant willmore surfaces in conformal homogeneous spaces with rigidity of order four and then we reduce directly the symmetry to obtain the euler lagrange equation of 4 r 2 elasticae in surfaces with constant curvature c we also work out the solving natural equations and the closed curve problem for elasticae in surfaces with constant curvature it allows us to give explicit parametrizations of willmore surfaces and willmore tori in those conformal homogeneous 3 spaces