examples of constant mean curvature immersions of the 3 sphere into euclidean 4 space abstract mean curvature is one of the simplest and most basic of local differential geometric invariants therefore closed hypersurfaces of constant mean curvature in euclidean spaces of high dimension are basic objects of fundamental importance in global differential geometry before the examples of this paper the only known example was the obvious one of the round sphere indeed the theorems of h hopf for immersion of s2 into e3 and a d alexandrov for imbedded hypersurfaces of en have gone a long way toward characterizing the round sphere as the only example of a closed hypersurface of constant mean curvature with some added assumptions examples of this paper seem surprising and are constructed in the framework of equivariant differential geometry

the bernstein problem for affine maximal hypersurfaces in this paper we prove the validity of the chern conjecture in affine geometry 18 namely that an affine maximal graph of a smooth locally uniformly convex function on two dimensional euclidean space r 2 must be a paraboloid more generally we shall consider the n dimensional case r n showing that the corresponding result holds in higher dimensions provided that a uniform strict convexity condition holds we also extend the notion of affine maximal to non smooth convex graphs and produce a counterexample showing that the bernstein result does not hold in this generality for dimension n 10

the principal series for a reductive symmetric space ii eisenstein integrals in this paper we develop a theory of eisenstein integrals related to the principal series for a reductive symmetric space g h here g is a real reductive group of harish chandra s class an involution of g and h an open subgroup of the group g of xed points for the group g itself is a symmetric space for the left right action of g g we refer to this setting as the group case up to a normalization our eisenstein integrals generalize those of harish chandra 18 associated with a minimal parabolic subgroup in the group case

on the construction of infinitely many mutually noncongruent examples of minimal imbeddings of s 2n 1 into cp n n geqslant 2

principal lines on surfaces immersed with constant mean curvature configurations of lines of principal curvature on constant mean curva ture immersed surfaces are studied analytical models for these configurations near general isolated umbilical points and particular types of ends are found from the existence of transversal invariant measures for the foliations by principal lines established here follows that the union of recurrent lines of principal curvature is an open set examples illustrating all possible cases are given 1 introduction this paper is devoted to the study of the possible configurations of lines of principal curvature and umbilical points of surfaces immersed in r3 with constant mean curvature except in the case where the image of the immersion is contained in a plane or a sphere the umbilical points are isolated 10 11 a very precise analytical model is found for the family of principal lines near the isolated umbilical points this model is related to a result of h hopf 10 11 which permits the computation of the index of the umbilical point the behaviour of the principal lines at infinity near particular types of ends called here elementary ends of the immersion is also described through precise analytical models in connection with the global behaviour of the principal lines it is shown that their foliations have a natural smooth transversal invariant measure this implies that the compact principal lines cycles appear in cylindrical open sets and that the closures of recurrent principal lines have nonempty interior examples illustrating all possible situations are given however the recurrent lines have been found only for immer sions of tori into spaces which are not more than locally isometric to r3 the existence of such immersions into r3 seems to be a very difficult question related to a conjecture of h hopf 24 p 684 problem 63 2 from the point of view of principal lines an immersion with constant mean curvature presents a global configuration which is quite opposite to that of the principally structurally stable immersions studied in 4 5 for a large class of this kind of immersions it is shown that the principal cycles are isolated and do not present recurrent lines 4 5 examples of immersions of the sphere and torus into

fourier transforms on a semisimple symmetric space let g h be a semisimple symmetric space that is g is a connected semisimple real lie group with an involution and h is an open subgroup of the group of xed points for in g the main purpose of this paper is to study an explicit fourier transform on g h in terms of general representation theory the abstract fourier transform of a compactly supported smooth function f 2 c 1 c g h is given by see 5 1 f f zg h f x x dx for h a unitary irreducible representation of g and 2 h an h invariant distribution vector for here dx is the invariant measure on g h thus f isa smooth vector for h depending linearly on our goal is to obtain an explicit version of the restriction of this fourier transform to representations h in the minimal unitary principal series h for g h under the assumption that the center of g is nite in a sequel 9 to this paper it will be proved that a function f 2 c 1 c g h is uniquely determined by the restriction of f to this series a priori it is known that f is determined by f

overdetermined elliptic problems in topological disks abstract we introduce a method based on the poincare hopf index theorem to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk applications to some well known nonlinear elliptic pdes are provided our result can be seen as the analogue of hopf s uniqueness theorem for constant mean curvature spheres but for the general analytic context of overdetermined elliptic problems

fourier inversion on a reductive symmetric space let x be a semisimple symmetric space in previous papers 8 and 9 we have dened an explicit fourier transform for x and shown that this transform is injective on the space c 1 c x ofcompactly supported smooth functions on x in the present paper which is a continuation of these papers we establish an inversion formula for this transform

radon transformation on reductive symmetric spaces support theorems abstract we introduce a class of radon transforms for reductive symmetric spaces including the horospherical transforms and derive support theorems for these transforms a reductive symmetric space is a homogeneous space g h for a reductive lie group g of the harish chandra class where h is an open subgroup of the fixed point subgroup for an involution on g let p be a parabolic subgroup such that p is opposite to p and let n p be the unipotent radical of p for a compactly supported smooth function on g h we define r p g to be the integral of n p n g n h over n p the radon transform r p thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions for these transforms we derive support theorems in which the support of is partially characterized in terms of the support of r p the proof is based on the relation between the radon transform and the fourier transform on g h and a paley wiener shift type argument our results generalize the support theorem of helgason for the radon transform on a riemannian symmetric space

constant mean curvature invariant surfaces and extremals of curvature energies abstract we determine the extremal curves of curvature energy functionals which generalize a variational problem studied by blaschke in r 3 the generalization is made by extending both the lagrangian energy itself and the ambient space to riemannian and lorentzian 3 space forms then we show that constant mean curvature cmc invariant surfaces in 3 space forms can be constructed locally as the evolution of the above extremals under their binormal flow with appropriate velocity moreover we see that these surfaces are intrinsically described by a warping function satisfying an ermakov milne pinney equation finally we use the previous findings to extend known results on isometric deformations of cmc surfaces and the lawson s correspondence of cmc cousins to our background spaces

non compactness of the space of minimal hypersurfaces we show that the space of min max minimal hypersurfaces is non compact when the manifold has an analytic metric of positive ricci curvature and dimension 3 le n 1 le 7 furthermore we show that bumpy metrics with positive ricci curvature admit minimal hypersurfaces with unbounded mathrm index mathrm area when combined with the recent work fo f c marques and a neves we then deduce some new properties regarding the infinitely many minimal hypersurfaces they found

topological simplicity of the cremona groups the cremona group is topologically simple when endowed with the zariski or euclidean topology in any dimension 2 and over any infinite field two elements are moreover always connected by an affine line so the group is path connected

on the construction of codimension two minimal immersions of exotic spheres into euclidean spheres it is nowadays a well known fundamental fact in differential topology that there exist many smooth manifolds which are homeomorphic to but non diffeomorphic to spheres such manifolds were christened exotic spheres by their founder milnor 223 the structures of those exotic spheres which bound some parallelizable manifolds were systematically analyzed by the techniques of surgery in 20 for example it is not difficult to show that such exotic spheres can be embedded as codimension two submanifolds of the unit spheres in differential geometry and geometric variational theory the study of closed minimal submanifolds of the euclidean n sphere s 1 is directly related to that of the local structure of singularities of minimal subvarieties in the general riemannian setting therefore closed minimal submanifolds of s 1 are not only interesting nice geometric objects by themselves but are also of general theoretical importance in the study of geometric measure theory 16 18 24 among various problems on the closed minimal submanifolds of euclidean spheres the following two problems naturally distinguish themselves as especially interesting namely

the first eigenvalue of the laplacian of an isoparametric minimal hypersurface in a unit sphere soit m n 1 une variete de riemann connexe compacte a n 1 dimensions sans bord et son laplacien agissant sur des fonctions lisses sur m a un spectre discret 0 0 m 1 m 2 m k m x soit f une immersion minimale de m n 1 dans la sphere unite a n dimensions s n 1 de l espace euclidien on etudie la premiere valeur propre du laplacien d une hypersurface isoparametrique minimale fermee dans s n 1

the plancherel decomposition for a reductive symmetric space i spherical functions we prove the plancherel formula for spherical schwartz functions on a reductive symmetric space our starting point is an inversion formula for spherical smooth compactly supported functions the latter formula was earlier obtained from the most continuous part of the plancherel formula by means of a residue calculus in the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized eisenstein integrals and of the maass selberg relations satisfied by the associated c functions

normal subgroups in the cremona group let k be an algebraically closed field we show that the cremona group of all birational transformations of the projective plane mathbb p mathbf k 2 is not a simple group the strategy makes use of hyperbolic geometry geometric group theory and algebraic geometry to produce elements in the cremona group that generate non trivial normal subgroups

stability of hypersurfaces with constant mean curvature soit m n compacte orientable et soit x m n r n 1 une immersion a courbure moyenne constante non nulle alors x est stable si et seulement si x m r n 1 est une sphere s n r n 1

on the construction of infinitely many congruence classes of imbedded closed minimal hypersurfaces in s n 1 for all n geq 3

fourier and poisson transformation associated to a semisimple symmetric space soit g un groupe de lie semisimple connexe contenu dans un groupe de lie semisimple simplement connexe dont l algebre de lie est la complexification de celle de g soit une involution non triviale de g et une involution de cartan commutant avec soit h un groupe point fixe de on etudie les representations h spheriques de dimension finie de g contenant un vecteur k fixe ou k est un sous groupe compact maximal stable de g sous representations de dimension finie de le noyau de poisson la distribution u i la transformation de fourier et poisson formules integrales

topologies and structures of the cremona groups we study the algebraic structure of the n dimensional cremona group and show that it is not an algebraic group of innite dimension ind group if n 2 we describe the obstruction to this which is of a topological nature by contrast we show the existence of a euclidean topology on the cremona group which extends that of its classical subgroups and makes it a topological group

stability of hypersurfaces of constant mean curvature in riemannian manifolds hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 display many similarities with minimal hypersurfaces of overline m n 1 they are both solutions to the variational problem of minimizing the area function for certain variations in the first case however the admissible variations are only those that leave a certain volume function fixed for precise definitions see sect 2 this isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces

a general theory of canonical forms si g est un groupe de lie compact et m une g variete riemannienne d orbites principales de codimension k alors une section ou une forme canonique pour m est une sous variete fermee lisse a k dimensions de m qui rencontre toutes les orbites de m orthogonalement

existence and uniqueness for a class of cauchy problems with characteristic initial manifolds on donne un theoreme d existence unicite pour une grande classe d equations d ordre 2 quasilineaires a donnees de cauchy sur une variete qui est caracteristique au sens ou l equation est singuliere d ordre 1 le long de la variete initiale