Estimates for the -Neumann problem and nonexistence of C 2 Levi-flat hypersurfaces in

Let be a pseudoconvex domain with C² boundary in , n 2. We prove that the -Neumann operator N exists for square-integrable forms on . Furthermore, there exists a number ₀>0 such that the operators and the Bergman projection are regular in the Sobolev space W ( ) for <₀. The -Neumann operator is used to construct -closed extension on for forms on the boundary b. This gives solvability for the tangential Cauchy-Riemann operators on the boundary. Using these results, we show that there exist no non-zero L²-holomorphic (p, 0)-forms on any domain with C² pseudoconcave boundary in with p > 0 and n 2. As a consequence, we prove the nonexistence of C² Levi-flat hypersurfaces in .This paper is a revision of our preprint (May 2003) formerly titled Estimates for the -Neumann problem and nonexistence of Levi-flat hypersurfaces in where the nonexistence of C², Levi-flat hypersurfaces is proved for >0.All three authors are partially supported by NSF grants.An erratum to this article can be found at     

On the logarithmic plurigenera of complements of plane curves

Let B be a (not necessarily irreducible) plane curve in ². In the present article, we prove that if and only if Moreover, we determine the curve B when and Mathematics Subject Classification (2000): 14R05, 14H50, 14J26     

On the solutions of the congruence

In this note, we show that if is a positive integer satisfying the congruence , then .

     

The cofinality of the saturated uncountable random graph

Assuming CH, let be the saturated random graph of cardinality ₁. In this paper we prove that it is consistent that and can be any two prescribed regular cardinals subject only to the requirement      

Metrics of positive curvature with conic singularities on the sphere

A simple proof is given of the necessary and sufficient condition on a triple of positive numbers for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles . The same condition is necessary and sufficient for the triple to be interior angles of a spherical triangular membrane.

     

Proofs of two conjectures of Gray involving the double suspension

In proving that the fiber of the double suspension has a classifying space, Gray constructed fibrations

and

He conjectured that is homotopic to the -power map on when is an odd prime. Harper proved this is true when looped once. We remove the loop when . Gray also conjectured that at odd primes  factors through a map

We show that this is true as well when .

     

An approach to the spectrum structure of Dirac operators by the local-compactness method

The purpose of this paper is to investigate the spectra of the Dirac operator . The local compactness of is shown under some assumption on . This method enables us to prove that if as , then and to give a significant sufficient condition that or has a purely discrete spectrum.

     

The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

The definite integral

is related to the Laplace transform of the digamma function

by when . Certain analytic expressions for in the complementary range, , are also provided.

     

On the evaluation of generalized Watson integrals

The triple integrals

and

where and are complex variables in suitably defined cut planes, were first evaluated by Watson in 1939 for the special cases and , respectively. In the present paper simple direct methods are used to prove that can be expressed in terms of squares of complete elliptic integrals of the first kind for general values of and . It is also shown that and are related by the transformation formula

where

Thus both of Watson's results for are contained within a single formula for .

     

Deterministic primality test for numbers of the form odd

We use a result of E. Lehmer in cubic residuacity to find an algorithm to determine primality of numbers of the form , odd, . The algorithm represents an improvement over the more general algorithm that determines primality of numbers of the form , , presented by Berrizbeitia and Berry (1999).

     

A proof of the Hoggatt-Bergum conjecture

It is proved that if and are positive integers such that the product of any two distinct elements of the set

increased by is a perfect square, then has to be . This is a generalization of the theorem of Baker and Davenport for .

     

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.

(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .

(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.

     

Convexity and the Exterior Inverse Problem of Potential Theory

Let and be bounded solid domains such that their associated volume potentials agree outside . Under the assumption that one of the domains is convex, it is deduced that .

     

On some dyadic models of the Euler equations

Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the Sobolev norm. It is shown that their model can be reduced to a dyadic model of the inviscid Burgers equation. The inviscid Burgers equation exhibits finite time blow-up in , for , but its dyadic restriction is even more singular, exhibiting blow-up for any . Friedlander and Pavlovic developed a closely related model for which they also prove finite time blow-up in . Some inconsistent assumptions in the construction of their model are outlined. Finite time blow-up in the norm, for any , is proven for a class of models that includes all those models. An alternative shell model of the Navier-Stokes equations is discussed.

     

Analysis of the -Neumann problem along a straight edge

We show there exists an L^p solution, for p (2,), to the -Neumann problem on an edge domain in ² for (0,1)-forms, and we explicitly compute the singularities, which are of complex logarithmic and arctangent type, along the edge, of the solution.Mathematics Subject Classification (2000): 32W05, 35B65     

On the biharmonic and harmonic indices of the Hopf map

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalize harmonic maps. We consider the Hopf map and modify it into a nonharmonic biharmonic map . We show to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawa's determination of its harmonic index and nullity.

     

Endpoint estimates for the circular maximal function

We consider the problem of endpoint estimates for the circular maximal function defined by

where is the normalized surface area measure on . Let be the closed triangle with vertices . We prove that for , there is a constant such that Furthermore, .

     

Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two

Let be the group of isotopy classes of orientation-preserving homeomorphisms of that preserve a Heegaard splitting of genus two. In this paper, we construct a tree in the barycentric subdivision of the disk complex of a handlebody of the splitting to obtain a finite presentation of .

     

A quadratic approximation to the Sendov radius near the unit circle

Define to be the set of complex polynomials of degree with all roots in the unit disk and at least one root at . For a polynomial , define to be the distance between and the closest root of the derivative . Finally, define . In this notation, a conjecture of Bl. Sendov claims that .

In this paper we investigate Sendov's conjecture near the unit circle, by computing constants and (depending only on ) such that for near . We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov's conjecture.

     

Approximation of holomorphic maps with a lower bound on the rank

Let be a closed polydisc or ball in , and let be a quasi-projective algebraic manifold which is Zariski locally equivalent to , or a complement of an algebraic subvariety of codimension in such a manifold. If is an integer satisfying , then every holomorphic map from a neighborhood of to with rank at every point of can be approximated uniformly on by entire maps with rank at every point of .