Upper bounds of small Eigenvalues of the Dirac operator and isometric immersions

We show that a topologically determined number of eigenvalues of the Dirac operatorD of a closed Riemannian spin manifoldM of even dimensionn can be bounded by the data of an isometric immersion ofM into the Euclidian spaceR ^N . From this we obtain similar bounds of the eigenvalues ofD in terms of the scalar curvature ofM ifM admits a minimal immersion intoS ^N or,ifM is complex, a holomorphic isometric immersion intoPC ^N .     

Self-accessibility of a set with respect to a multivalued field

LetF be a multivalued field on the manifoldM and letN be a submanifold ofM, possibly with boundary. We give a sufficient condition for the self-accessibility property of the pair (N, F), that is:N is contained in the interior of the attainable set fromN at a time smaller or equal toT, for everyT>0 (at the timeT, ifN is compact). To obtain such a condition, a modification of Petrov's implicit function theorem is proved (Ref. 1). Finally, a necessary condition for the self-accessibility property is given.This work was performed under the auspices of the Consiglio Nazionale delle Ricerche, Rome, Italy.     

Angle theorems for the Lagrangian mean curvature flow

We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. . Received: 11 April 2001 / Published online: 29 April 2002     

Isoperimetric inequalities on minimal submanifolds of space forms

For a domainU on a certaink-dimensional minimal submanifold ofS ⁿ orH ⁿ, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k ^k ω _k M (D) ^k-1 ≤Vol(∂D) ^k , where ω _k is the volume of the unit ball ofR ^k . Also, we prove that ifD is any domain on a minimal surface inS ₊ ⁿ (orH ⁿ, respectively), thenD satisfies an isoperimetric inequality2π A≤L ²+A² (2π A≤L²−A² respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH ⁿ, then(k−1) Vol(U)≤Vol(∂U). Supported in part by KME and GARC     

The lagrange intersections for (ℂP n , ℝP n )

Summary If (M, ω) is a compact symplectic manifold andL ⊂M a compact Lagrangian submanifold and if φ is a Hamiltonian diffeomorphism ofM then the V. Arnold conjecture states (possibly under additional conditions) that the number of intersection section points ofL and φ (L) can be estimated by #{Lϒφ (L)}≥ cuplength +1. We shall prove this conjecture for the special case (L, M)=(ℝP ⁿ , ℂP ⁿ ) with the standard symplectic structure.     

A minimax theorem for irreducible compact operators inL p-spaces

Let (X, Σ, μ) be a σ-finite measure space,T a compact irreducible (positive, linear) operator onL ^p (μ) (1≦p<+∞). It is shown that the spectral radiusr ofT is characterized by the minimax property {fx196-1} where ∑₀ denotes the ring of sets of finite measure and whereQ denotes the set of all, almost everywhere positive functions inL ^p. Moreover, ifr>0 then equality on either side is assumed ifff is the (essentially unique) positive eigenfunction ofT. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximatingT. Dedicated to H. G. Tillmann on his 60th birthday     

Geometry of the Normal Bundle of a Submanifold*

 We study the geometric behavior of the normal bundle T ^⊥ M of a submanifold M of a Riemannian manifold . We compute explicitely the second fundamental form of T ^⊥ M and look at the relation between the minimality of T ^⊥ M and M. Finally we show that the Maslov forms with respect to a suitable connection of the pair (T ^⊥ M, are null. Received March 14, 2001; in revised form February 11, 2002     

A matroid generalization of a result of Dirac

This paper generalizes a theorem of Dirac for graphs by proving that ifM is a 3-connected matroid, then, for all pairs {a,b} of distinct elements ofM and all cocircuitsC ^* ofM, there is a circuit that contains {a,b} and meetsC ^*. It is also shown that, although the converse of this result fails, the specified condition can be used to characterize 3-connected matroids.The author's research was partially supported by a grant from the National Security Agency.     

Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non-Kähler) manifolds and in twistor spaces Z ⁴ⁿ⁺² over quaternionic Kähler manifolds Q ⁴ⁿ are minimal. Moreover, we prove that any Lagrangian submanifold L in a nearly Kähler manifold M splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly Kähler part of M and the other factor is Lagrangian in the Kähler part of M. Using this splitting theorem, we then describe Lagrangian submanifolds in nearly Kähler manifolds of dimensions six, eight, and ten.     

On the total curvature and area growth of minimal surfaces in R n

LetM be an immersed complete minimal surface inR ⁿ. We show that the total curvature ofM is finite if and only ifM is of quadratic area growth and finite topological type.     

Cellular-indecomposable subnormal operators III

A bounded operatorT is called cellular-indecomposable ifL M {0} wheneverL andM are nonzero invariant subspaces forT. We prove that a cyclic subnormal operator is cellular-indecomposable if and only if it is quasi-similar to an analytic Toeplitz operator whose symbol is a weak-star generator ofH . This completes our previous work [5], [6].     

On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold

We investigate the existence of parallel sections in the normal bundle of a complex submanifold of a locally conformal Kaehler manifold with positive holomorphic bisectional curvature. Also, ifM is a quasi-Einstein generalized Hopf manifold then we show that any complex submanifoldM with a flat normal connection ofM is quasi-Einstein, too, provided thatM is tangent to the Lee field ofM. As an application of our results we study the geometry of the second fundamental form of a complex submanifold in the locally conformal Kaehler sphereQ _m (of a complex Hopf manifoldS ^2m+1 ×S ¹).     

An integral formula for the heat kernel of tubular neighborhoods of complete (connected) Riemannian manifolds

Let M be a complete connected Riemannian manifold and let N be a submanifold of M. Let v: E _v»N be the normal bundle of N and exp _v : E _v»M its exponential map.Let (exp _infv ^/sup-1 , M ₀) be the Fermi chart relative to the submanifold N. Then, by using the Fermi coordinates we obtain an integral formula for the Dirichlet heat kernel p _t ^m (-,-). That is, we obtain a probabilistic representation for the integral _N f(y)p _t ^M (x,y) dywhere f is any measurable function of compact support in M ₀. This representation involves a submanifold semi-classical Brownian Riemannian bridge process. Then applying the integral formula via a Riemannian submersion in [5], we obtain heat kernel formulae for the complex projective space cP ⁿ, the quaternionic projective space QP ⁿ and the Caley line CaP ¹. The case of the Caley plane CaP ² eludes us due to the lack of a submersion theorem.This work is part of a Ph.D. Thesis which was undertaken under Professor K. D. Elworthy, Mathematics Institute, Warwick University, Coventry CV47AL, England, Great Britain.     

Uniqueness Theorems for CR Functions

Let M be a CR manifold embedded in ?^s of arbitrary codimension. M is called generic if the complex hull of the tangent space in all points of M is the whole ?^s. M is minimal (in sense of Tumanov) in p ? M if there does not exist any CR submanifold of M passing through p with the same CR dimension as M but of smaller dimension. Let M be generic and minimal in some point p ? M and N be a generic submanifold of M passing through p. We prove that a continuous CR function on M vanishes identically in some neigbourhood of p if its restriction to N either vanishes in p faster then some function with non-integrable logarithm or it vanishes on a subset of N of positive measure.     

On a Theorem Due to Birkhoff

The manifold M being closed and connected, we prove that every submanifold of T*M that is Hamiltonianly isotopic to the zero-section and that is invariant by a Tonelli flow is a graph.     

Some properties of generating functions obtained by Amann-Conley-Zehnder reduction

Consider an initial Lagrangian submanifold Λ₀ ⊂ T* ℝ ⁿ that admits a global generating function and a Hamiltonian isotopy Φ _H ^t . Then, we provide a global generating function for the Lagrangian submanifold Λ _t = Φ _H ^t (Λ₀) realized by applying the so-called Amann-Conley-Zehnder reduction. When Λ₀ is the zero-section, we study in some detail the asymptotic behavior of such generating functions and give an approximation result. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

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     

The Arrow-Barankin-Blackwell theorem in a dual space setting

In 1953 Arrow, Barankin, and Blackwell proved that, ifR ⁿ is equipped with its natural ordering and ifF is a closed convex subset ofR ⁿ , then the set of points inF that can be supported by strictly positive linear functionals is dense in the set of all efficient (maximal) points ofF. Many generalizations of this density result to infinite-dimensional settings have been given. In this note, we consider the particular setting where the setF is contained in the topological dualY ^* of a partially ordered, nonreflexive normed spaceY, and the support functionals are restricted to be either nonnegative or strictly positive elements in the canonical embedding ofY inY ^*. Three alternative density results are obtained, two of which generalize a space-specific result due to Majumdar for the dual system (Y,Y ^*)=(L ¹,L ).This research was supported in part by funds provided by the Provident Chair of Excellence in Applied Mathematics at the University of Tennessee, Chattanooga, Tennessee.     

The projection theorem for spectral sets

LetG be a locally compact group with polynomial growth and symmetricL ¹-algebra andN a closed normal subgroup ofG. LetF be a closedG-invariant subset of Prim_* L ¹(N) andE={ker; with |N(k(F))=0}. We prove thatE is a spectral subset of Prim_* L ¹(G) ifF is spectral. Moreover we give the following application to the ideal theory ofL ¹(G). Suppose that, in addition,N is CCR andG/N is compact. Then all primary ideals inL ¹(G) are maximal, provided allG-orbits in Prim_* L ¹(N) are spectral.Dedicated to Professor Elmar Thoma on the occasion of his 60th birthday     

Analytic discs under sympletic transforms

By a holomorphic homogeneous symplectic transformation of T*X (for X = ?^N), we interchange the conormal bundle T _M ^* X to a higher codimensional submanifold M with the conormal bundle T _M ^* X to a hypersurface M of X. For an analytic disc A “attached” to M we are able to find a section A* ?T*X with π A* = A, attached to T _M ^* X, such that Ã:= πx(A*) is an analytic disc “attached” to M. By this procedure of “transferring” analytic discs, we get the higher codimensional version of our criteria of [5] on holomorphic extension of CR functions (with [5] being on its hand the main tool of the present proof). Thus, let W be a wedge of X with generic edge M and assume that there exists an analytic disc contained in M ∪ W, tangent to M at a boundary point z₀∈ ?A, and not contained in M in any neighborhood of z₀. Then germs of holomorphic functions on W at z₀ extend to a full neighborhood of z₀.