On families of Lagrangian submanifolds

We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ω _t are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ _t of diffeomorphisms of X such that ω _t =Φ _t ^*(ω₀). If L⊂X is a Lagrangian submanifold for (X,ω₀), L _t =Φ _t ^-1(L) is thus a Lagrangian submanifold for (X,ω _t ). Here we show that if we simply assume that L is compact and ω _t | _L is exact for every t, a family L _t as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi–Yau structure. Received: 29 May 2001/ Revised version: 17 October 2001     

A unimodularly invariant theory for immersions into the affine space

We develop a unimodularly invariant theory for immersions with higher codimension into the affine space. Received: 6 September 2001; in final form: 22 November 2001 / Published online: 29 April 2002 RID="*" ID="*" Supported by the Deutsche Forschungsgemeinschaft     

Almost invariant submanifolds for compact group actions

We define a C ¹ distance between submanifolds of a riemannian manifold M and show that, if a compact submanifold N is not moved too much under the isometric action of a compact group G, there is a G-invariant submanifold C ¹-close to N. The proof involves a procedure of averaging nearby submanifolds of riemannian manifolds in a symmetric way. The procedure combines averaging techniques of Cartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney’s idea of realizing submanifolds as zeros of sections of extended normal bundles. Received September 14, 1999 / final version received November 29, 1999     

A variational problem for submanifolds in a sphere

Let be an n-dimensional submanifold in an (n + p)-dimensional unit sphere S ^{n + p} , M is called a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the Willmore functional , where is the square of the length of the second fundamental form, H is the mean curvature of M. In [11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S ^{n + p} . In this paper, we discover that a similar integral inequality of Simons’ type still holds for the critical submanifolds of the functional . Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation.     

An affine analogue of the Hartman-Nirenberg cylinder theorem

Let X be a smooth, complete, connected submanifold of dimension in a complex affine space , and r is the rank of its Gauss map . The authors prove that if and in the pencil of the second fundamental forms of X, there are two forms defining a regular pencil all eigenvalues of which are distinct, then the submanifold X is a cylinder with -dimensional plane generators erected over a smooth, complete, connected submanifold Y of rank r and dimension r. This result is an affine analogue of the Hartman-Nirenberg cylinder theorem proved for and r = 1. For and , there exist complete connected submanifolds that are not cylinders. Received: 20 October 2000 / Revised version: 18 April 2001 / Published online: 18 January 2002     

Moebius geometry of submanifolds in ? n

In this paper we define a Moebius invariant metric and a Moebius invariant second fundamental form for submanifolds in ? ⁿ and show that in case of a hypersurface with n≥ 4 they determine the hypersurface up to Moebius transformations. Using these Moebius invariants we calculate the first variation of the moebius volume functional. We show that any minimal surface in ? ⁿ is also Moebius minimal and that the image in ? ⁿ of any minimal surface in ℝ ⁿ unter the inverse of a stereographic projection is also Moebius minimal. Finally we use the relations between Moebius invariants to classify all surfaces in ?³ with vanishing Moebius form. Received: 18 November 1997     

The scalar curvature of CR submanifolds of maximal CR dimension of complex projective space

We treat n-dimensional compact minimal submanifolds of complex projective space when the maximal holomorphic tangent subspace is (n − 1)-dimensional and we give a sufficient condition for such submanifolds to be tubes over totally geodesic complex subspaces. Authors’ addresses: Mirjana Djorić, Faculty of Mathematics, University of Belgrade, Studentski trg 16, pb. 550, 11000 Belgrade, Serbia; Masafumi Okumura, 5-25-25 Minami Ikuta, Tama-ku, Kawasaki, Japan     

On the area of constant mean curvature discs and annuli with circular boundaries

It is still an open question whether a constant mean curvature (CMC) disc which is bounded by a circle is necessarily a spherical cap or a flat disc. The authors together with López [1] recently showed that the only stable CMC discs which are bounded by a circle are spherical caps. In this paper we derive lower bounds for the area of constant mean curvature discs and annuli with circular boundaries in 3-dimensional space forms. Received November 8, 1999; in final form January 18, 2000 / Published online March 12, 2001     

On the zeros of the partial sums of \cos(z) and \sin(z)

Summary. Let denote the -th partial sum of the exponential function. Carpenter et al. (1991) [1] studied the exact rate of convergence of the zeros of the normalized partial sums to the so-called Szeg?-curve Here we apply parts of the results found by Carpenter et al. to the zeros of the normalized partial sums of and . Received August 11, 1995     

Spherical submanifolds in a Euclidean space

In this paper, we are interested in extending the study of spherical curves in R ³ to the submanifolds in the Euclidean space R ^n+p . More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R ^n+p lies on the hypersphere S ^n+p−1(c) (standardly imbedded sphere in R ^n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S ⁿ (c) (cf. Theorem 4.1).     

On the Range of the Chébli–Trimèche Transform

We first characterise the L²-Schwartz functions whose image under the Chébli–Trimèche transform are compactly supported smooth functions. We then generalise a theorem by H. H. Bang, characterising the smooth L^p-functions whose (distributional) transform have compact support.The author is supported by a research grant from the Australian Research Council.     

On tangent cones of Alexandrov spaces with curvature bounded below

The tangent cones of an inner metric Alexandrov space with finite Hausdorff dimension and a lower curvature bound are always inner metric spaces with nonnegative curvature. In this paper we construct an infinite-dimensional inner metric Alexandrov space of nonnegative curvature which has in one point a tangent cone whose metric is not an inner metric. Received: 20 October 1999 / Revised version: 8 May 2000     

A Möbius characterization of submanifolds in real space forms with parallel mean curvature and constant scalar curvature

In this paper, we give a Möbius characterization of submanifolds in real space forms with parallel mean curvature vector fields and constant scalar curvatures, generalizing a theorem of H. Li and C.P. Wang in [LW1].Supported by NSF of Henan, P. R. China     

On quasi-amorphous sets

A set is said to be amorphous if it is infinite, but cannot be written as the disjoint union of two infinite sets. The possible structures which an amorphous set can carry were discussed in [5]. Here we study an analogous notion at the next level up, that is to say replacing finite/infinite by countable/uncountable, saying that a set is quasi-amorphous if it is uncountable, but is not the disjoint union of two uncountable sets, and every infinite subset has a countably infinite subset. We use the Fraenkel–Mostowski method to give many examples showing the diverse structures which can arise as quasi-amorphous sets, for instance carrying a projective geometry, or a linear ordering, or both; reconstruction results in the style of [1] are harder to come by in this case. Received: 8 April 1999 / Published online: 3 October 2001     

Index growth of hypersurfaces with constant mean curvature

In this paper we give the precise index growth for the embedded hypersurfaces of revolution with constant mean curvature (cmc) 1 in (Delaunay unduloids). When n=3, using the asymptotics result of Korevaar, Kusner and Solomon, we derive an explicit asymptotic index growth rate for finite topology cmc 1 surfaces with properly embedded ends. Similar results are obtained for hypersurfaces with cmc bigger than 1 in hyperbolic space. Received: 6 July 2000; in final form: 10 September 2000 / Published online: 25 June 2001     

Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

In this article we study surfaces in for which the unit normal makes a constant angle with the -direction. We give a complete classification for surfaces satisfying this simple geometric condition.