Complex N-space Research Articles

On isometric Lagrangian immersions

This article uses Cartan-Kahler theory to show that a small neighborhood of a point in any surface with a Riemannian metric possesses an isometric Lagrangian immersion into the complex plane (or by the same argument, into any Kahler surface). In fact, such immersions depend on two functions of a single variable. On the other hand, explicit examples are given of Riemannian three-manifolds which admit no local isometric Lagrangian immersions into complex three-space. It is expected that isometric Lagrangian immersions of higher-dimensional Riemannian manifolds will almost always be unique. However, there is a plentiful supply of flat Lagrangian submanifolds of any complex n-space; we show that locally these depend on 1 2 n(n + 1) functions of a single variable.

On the Berezin-Toeplitz calculus

We consider the problem of composing Berezin-Toeplitz operators on the Hilbert space of Gaussian square-integrable entire functions on complex n-space, Cn. For several interesting algebras of functions on Cn, we have TWTp = T,o,p for all W,o in the algebra, where T. is the Berezin-Toeplitz operator associated with s and Woo is a twisted associative product on the algebra of functions. On the other hand, there is a C? function s for which Ti is bounded but TpTp # Tp for any 5.

A mean value theorem on bounded symmetric domains

LetΩ\Omegabe a Cartan domain of rankrrand genusppandBνB_\nu,ν>p−1\nu >p-1, the Berezin transform onΩ\Omega; the numberBνf(z)B_{\nu }f(z)can be interpreted as a certain invariant-mean-value of a functionffaround zz. We show that a Lebesgue integrable function satisfyingf=Bνf=Bν+1f=⋯=Bν+rff=B_\nu f=B_{\nu +1}f=\dots =B_{\nu +r}f,ν≥p\nu \ge p, must beM\mathcal {M}-harmonic. In a sense, this result is reminiscent of Delsarte’s two-radius mean-value theorem for ordinary harmonic functions on the complexnn-spaceCn\mathbf {C}^{n}, but with the role of radiusrrplayed by the quantity 1/ν1/\nu.

Flow of real hypersurfaces by the trace of the Levi form

Let F0 : N 2n−1 → C n , n ≥ 2, be a smooth immersion of a real (2n − 1)-dimensional hypersurface in complex n-space. In this paper we study the Cauchy-Riemann analogue of mean curvature flow [H] in Riemannian geometry: We deform the initial hypersurface N0 = F0(N 2n−1 ) in normal direction such that the speed at each point is given by the trace of the Levi form of the induced CR-structure on the hypersurface, with the defining one-form of norm one. For simplicity we assume in this paper that N 2n−1 is closed, i.e., compact without boundary. Our initial value problem for the flow along the trace of the Levi form then looks for a smooth family of immersions F : N 2n−1 ×[0 ,T ) → C n satisfying the system d

On M-harmonic Bloch functions and their Carleson measures

AbstractOn the setting of the unit ball of the complex n-space, some characterizations of M-harmonic Bloch functions are obtained. As an application, Carleson measures are characterized by means of Berezin type integrals of M-harmonic Bloch functions. As one may expect, these results carry over to M-harmonic little Bloch functions and vanishing Carleson measures.

M-harmonic functions with M-harmonic square

ℳ-harmonic functions with ℳ-harmonic square are proved to be either holomorphic or antiholomorphic in the unit ball of complex n-space under certain additional conditions. For example, if u and u2 are ℳ-harmonic in the unit ball of ℂ2 and if u is continuously differentiable up to the boundary then u is either holomorphic or antiholomorphic.

Fractional derivatives of Bloch functions, growth rate, and interpolation

Bloch functions on the ball are usually described by means of a restriction on the growth rate of ordinary derivatives of holomorphic functions. In this paper we first give a characterization of Bloch functions in terms of fractional derivatives. Then we show that the growth rate suggested by such a characterization is optimal in a certain sense. Also we prove a result concerning interpolating sequences for fractional derivatives of Bloch functions. 0. Introduction and Results Let B be the unit ball of the complex n-space C with norm |z| = 〈z, z〉 where 〈 , 〉 is the usual Hermitian inner product on C. A holomorphic function f on B is said to be a Bloch function if |∇f(z)|(1 − |z|2) is bounded on B where ∇f denotes the complex gradient of f . The space of Bloch functions endowed with norm ||f ||B = |f(0)|+ sup z∈B |∇f(z)|(1− |z|2) is called the Bloch space and denoted by B(B). If f ∈ B(B) satisfies the additional boundary vanishing condition |∇f(z)|(1 − |z|2) → 0 as |z| ↗ 1, we say f ∈ B0(B), the little Bloch space. In this paper we will investigate some properties of the Bloch space in terms of fractional derivatives. Let f be a function holomorphic on B with homogeneous expansion f = ∑∞ k=0 fk. Following [BB], we define the fractional derivative D f of order α > 0 as follows: D f(z) = ∞ ∑

On scalar curvature for totally real minimal submanifolds in CP n

Let CP~n be a complex projective n-space with the Fubini-Study metric of constantholomorphic sectional curvature c,and M be an n-dimensional compact totally real minimalsubmanifold in CP~n. It is known from refs. [1-3] that if the scalar curvature ρ≥n~2(n-2)c/2(2n-1) for M, then M is either totally geodesic in CP~n or n=2 and ρ=0, and M is a finiteRiemannian covering of the unique flat torus minimally imbedded in CP~2 with the parallel

Prescribing the Ricci curvature of Kähler asymptotically [formula omitted]n manifolds, n > 2

An analogue of Calabi's conjecture was posed on a class of complete noncompact Kähler manifolds [5], then solved on the simplest of them, the complex n-space with n > 2 [9]. Here we prove the conjecture in its full generality, by inverting an elliptic complex Monge-Ampère operator between suitable Fréchet spaces of smooth functions vanishing at infinity. A priori estimates benefit from recent simplifications of [2].

Pluriharmonic symbols of commuting Toeplitz operators

Our setting throughout the paper is the unit ball B n of the complex n-space cn; dimension n is fixed and thus we usually write B Bn unless otherwise specified. The Bergman space A2(B) is the closed subspace of L2(B) L2(B, V) consisting of holomorphic functions where V denotes the volume measure on B normalized to have total mass 1. For u L=(B), the Toeplitz operator Tu with symbol u is the bounded linear operator on A2(B) defined by Tu(f) P(uf)where P denotes the orthogonal projection of L2(B) onto A2(B). The projection P is the well-known Bergman projection which can be explicitly written as follows:

On the proof of Kuranishi’s embedding theorem

We prove a local holomorphic embedding theorem for a formally integrable, strictly pseudoconvex CR manifold M with dim M = 2n − 1 ≧ 7. This embedding is obtained as the limit of a sequence of approximate embeddings into complex n-space, which is constructed and shown to converge by the methods of Nash and Moser. The linearized problem is solved using the explicit integral operators constructed by Henkin. With estimates wich we have previously obtained for these operators, we show that if M is of class Cm, then it admits a Ck embedding provided 21 ≦ k, 6k + 5n − 2 ≦ m. Our argument is much shorter and simpler than previous arguments, which were based on the Neumann operator and carried out in the C∞ category.

On the local solution of the tangential Cauchy-Riemann equations

We study the solution operators P and homotopy formula introduced by G. M. Henkin for the tangential Cauchy-Riemann complex of a suitable small domain D on a strictly pseudoconvex real hypersurface in complex n-space. The main difficulties stem from the fact that P is an integral operator with a rather complicated kernel. For U ⊂⊂ D, we derive a Ck-norm estimate of the form ∥Pφ∥U, k ≦ K∥φ∥D, k, where the constant K blows up as U increases to D. We obtain careful control of the rate of this blow-up and of the dependence of K on the derivatives of the function defining the real hypersurface. Our estimates are sufficient for application to the local CR embedding problem.

Toeplitz operators and quantum mechanics

Toeplitz operators on the Segal-Bargmann spaces of Gaussian measure square-integrable entire functions on complex n-space C n are studied. The C ∗-algebra generated by the Weyl form of the canonical commutation relations consists precisely of the uniform limits of almost-periodic Toeplitz operators. The question of “which Toeplitz operators admit a symbol calculus modulo the compact operators” is raised and sufficient conditions are given for such a calculus. These conditions involve a notion of “slow oscillation at infinity.”

Zero-Free Regions for Exponential Sums

If the sum of the exponentials of the components of a complex n-vector P = (z1 . . . e zn) vanishes, then P is at least [I + o(l)JIn n from the diagonal of complex n-space, and this is essentially best possible. We shall establish an analogue in several complex variables of the fundamental fact that (1) exp z &0 for all complex z. Our contribution is one of formulation; the proof is quite simple. Define the distance d(A, P) between any points (2) A = (a,, I, an), P = (z,, . I Zn) in complex n-space by (3) d 2(A, P) = laj -Z 12. The diagonal of complex n-space is the set of complex n-tuples having all components identical. THEOREM. Let n > 2. If n (4) E exp zj = 0 j= 1 then P = (z, ... , zZn) has distance at least (5) dn = (1 + n)ln n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n=(1+'I from the diagonal of complex n-space. On the other hand, the sum of (4) vanishes at a point whose distance from the diagonal is at most (6) [ 1 + O(ln n)-2]ln n. REMARKS. The nonvanishing of the sum of (4) on the diagonal is simply (1). Since the point on the diagonal closest to P = (z1, . .. , zn) is easily shown to be Q = (q, q, .,q) where (7) q= (s zj)/n, Received by the editors June 2, 1980. 1980 Mathematics Subject Classification Primary 30C 15, 32A99, 33A 10.

An F. and M. Riesz Type Theorem for the Unit Ball in Complex N-Space

An F. and M. Riesz Type Theorem for the Unit Ball in Complex N-Space

The covering problem for the fields [formula omitted] and [formula omitted

Let k be a positive square free integer, N(−k) 1 2 the ring of algebraic integers in Q(−k) 1 2 and S the unit sphere in C n , complex n-space. If A 1,…, A n are n linearly independent points of C n then L = { u 1 Au + … + u n A n } with u r ∈ N(−k) 1 2 is called a k-lattice. The determinant of L is denoted by d( L). If L is a covering lattice for S, then θ(S, L) = V(S) d(L) is the covering density. L is called locally (absolutely) extreme if θ( S, L) is a local (absolute) minimum. In this paper we determine unique classes of extreme lattices for k = 1 and k = 3.

An F. and M. Riesz type theorem for the unit ball in complex 𝑁-space

Let M B {M_B} denote Lebesgue measure on the open unit ball, B B , in complex N N -space and let M ( B ) M(B) denote the space of Borel measures on B B . The volume Poisson kernel χ : B ¯ × B → ( 0 , ∞ ) \chi :\overline B \times B \to (0,\infty ) is defined and then we prove Theorem. If μ ∈ M ( B ) \mu \in M(B) and if μ ♯ ( w ) = ∫ B χ ( z , w ) d μ ( z ) {\mu ^\sharp }(w) = {\smallint _B}\chi (z,w)d\mu (z) is pluriharmonic in B B , then μ ♯ ∈ L ′ ( M B ) {\mu ^\sharp } \in L’({M_B}) and μ = μ ♯ ⋅ M B \mu = {\mu ^\sharp } \cdot {M_B} .

Elementary localization theorems for nonlinear eigenproblems

The minimax formula for linear eigenvalues of a linear operator is used to estimate the parameter values (λ) for which the self-adjoint operator L(λ) on Hilbert space to itself fails to have a bounded inverse. Such λ compose the “nonlinear spectrum” of L. The parameter spaces include regions in real or complex n-space. The localization theorems are used to demonstrate certain necessary conditions for stability of linear integro-partial-differential delay equations.

Bounded groups and norm-Hermitian matrices

An elementary proof is given that a bounded multiplicative group of complex (real) n× n nonsingular matrices is similar to a unitary (orthogonal) group. Given a norm on a complex n-space, it follows that there exists a nonsingular n× n matrix L (the Loewner-John matrix for the norm) such that LHL −1 is Hermitian for every norm-Hermitian H. Numerous applications of this result are given.

S-Algebras on Sets in C n

We give conditions which are necessary and sufficient for polynomial approximation of any continuous function on a compact subset of Cn. Let X be a compact set in Cn, complex n-space, P(X) the uniform closure of the polynomials on X, C(X) all continuous functions on X, m2n 2n-dimensional Lebesgue measure on Cn, and for any A in Cn let E(A)={z E Cn|Zi=Ai for some i}. A given set is a strong peak set if it is an intersection of peak sets and meets the boundary of each of them in a set which contains no nonempty perfect subsets. We say a Banach algebra A is an S-algebra if when x is in A and x, the Gelfand transform of x, vanishes at some p, then there exist xn in A such that xn vanish in (perhaps different) neighborhoods of p and j(xn-xI(--*0. For example, for any locally compact abelian group G, L'(G) is an S-algebra [6, p. 51]. The main question which motivates us here is: If A is a uniform algebra on a compact space X and A 'is an Salgebra, does A= C(X)? Our main result is the following. THEOREM. A necessary and sufficient condition that P(X)=C(X) is that (i) P(X) is an S-algebra, (ii) for almost all A E Cn with respect to m2', E(A) nX is a strong peak set, and (iii) each point of X is a peak point for P(X). We begin with some observations about uniform algebras which are S-algebras. LEMMA 1. Let A be a uniform algebra on a compact space X and suppose that A is an S-algebra. Then: (i) The maximal ideal space of A is X. (ii) A is normal. (iii) If each point of X is a peak point then A satisfies condition D [4, p. 86], i.e. iff E A and f (p)=O then there exist fn E A vanishing on neighborhoods of p such thatfInf-f. PROOF. (i) Let p be a homomorphism on A and y,u a representing measure for p with minimal closed support. If p, is not a point-mass then Presented to the Society, January 18, 1972 under the title A condition for polynomial approximation in Cn; received by the editors December 6, 1971 and, in revised form, October 18, 1972. AMS (MOS) subject classifications (1970). Primary 46J10; Secondary 41A10. ( American Mathematical Society 1973