The Hartogs phenomenon in hypersurfaces with constant signature

In this paper we study the -equation with zero Cauchy data along a hypersurface with constant signature. Applications to the solvability of the tangential Cauchy–Riemann equations for smooth forms with compact support and currents on the hypersurface are given. We also prove that the Hartogs phenomenon holds in weakly 2-convex–concave hypersurfaces with constant signature of Stein manifolds.     

The -equation and the Hartogs phenomenon on weakly q-pseudoconcave C R manifolds

We show that the Hartogs phenomenon holds in minimal, weakly 2-pseudoconcave generic C R submanifolds of a Stein manifold with trivial normal bundle. We also prove some results concerning the local and/or global solvability of the tangential Cauchy-Riemann equations for smooth forms and currents on weakly q-pseudoconcave C R manifolds.     

Regularity for weakly Dirac-harmonic maps to hypersurfaces

We prove that a weakly Dirac-harmonic map from a Riemann spin surface to a compact hypersurface is smooth. Supported by IMPRS “Mathematics in the Sciences” and the Klaus Tschira Foundation.     

Nonlinear degenerate curvature flows for weakly convex hypersurfaces

We study the evolution of closed, weakly convex hypersurfaces in in direction of their normal vector, where the speed equals a quotient of successive elementary symmetric polynomials of the principal curvatures. We show that there exists a solution for these weakly convex surfaces at least for some short time if the elementary symmetric polynomial in the denominator of the quotient is positive. The results for this nonlinear, degenerate flow are obtained by a cylindrically symmetric barrier construction.Received: 10 November 2003, Accepted: 5 April 2004, Published online: 16 July 2004     

Extensions of holomorphic maps through hypersurfaces and relations to the Hartogs extensions in infinite dimension

A generalization of Kwack's theorem to the infinite dimensional case is obtained. We consider a holomorphic map from into , where is a hypersurface in a complex Banach manifold and is a hyperbolic Banach space. Under various assumptions on , and we show that can be extended to a holomorphic map from into . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.

     

The nature of the bufferness phenomenon in weakly dissipative systems

We propose a mechanism for accumulating attractors in finite-dimensional weakly dissipative systems. The essence of this mechanism is that if a Hamiltonian or a conservative system with one and a half or more degrees of freedom is perturbed by small additional terms ensuring that it is dissipative, then under certain conditions, the number of its attractors appearing in small neighborhoods of different elliptic equilibriums or cycles of the nonperturbed system can increase without bound as the perturbations tend to zero. We consider meaningful examples from mechanics and radio physics: models of the bouncing ball dynamics, Fermi accelerations, the linear oscillator with impacts, and the self-excited oscillator with a discrete sequence of RLC circuits in the feedback circuit. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 447–466, March, 2006.     

The Bergman Kernel on Some Hartogs Domains

We obtain new explicit formulas for the Bergman kernel function on two families of Hartogs domains. To do so, we first compute the Bergman kernels on the slices of these Hartogs domains with some coordinates fixed, evaluate these kernel functions at certain points off the diagonal, and then apply a first order differential operator to them. We find, for example, explicit formulas for the kernel function on

$$\begin{aligned} \left\{ (z_1,z_2,w)\in \mathbb {C}^3:e^{|w|^2}|z_1|^2+|z_2|^2<1\right\} \end{aligned}$$

and on

$$\begin{aligned} \left\{ (z_1,z_2,w)\in \mathbb {C}^3:|z_1|^2+|z_2|^2+|w|^2<1+|z_2w|^2\;\mathrm{and} \;|w|<1\right\} . \end{aligned}$$

We use our formulas to determine the boundary behavior of the kernel function of these domains on the diagonal.

     

Balanced metrics on Hartogs domains

An n-dimensional strictly pseudoconvex Hartogs domain D _F can be equipped with a natural Kähler metric g _F . In this paper we prove that if m ₀ g _F is balanced for a given positive integer m ₀ then m ₀>n and (D _F ,g _F ) is holomorphically isometric to an open subset of the n-dimensional complex hyperbolic space.     

Hartogs Extension Theorems on Stein Spaces

We discuss various known generalizations of the classical Hartogs extension theorem on Stein spaces with arbitrary singularities and present an analytic proof based on \(\overline{\partial}\)-methods.     

The Coble hypersurfaces

Let A be an indecomposable principally polarized abelian variety of dimension g. Third order theta functions embed A in a projective space $\text{P}\text{(V}{}_3\text{)}$ of dimension 3^g?1, while second order theta functions embed the Kummer variety X=A/{±1} in a projective space $\text{P}\text{(V}{}_2\text{)}$ of dimension 2^g?1. Coble observed that for g=2 there is a unique cubic hypersurface in $\text{P}\text{(V}{}_3\text{)}$ that is singular along A, and for g=3 a unique quartic hypersurface in $\text{P}\text{(V}{}_2\text{)}$ singular along X. We explain these facts by a simple analysis of the representations of the corresponding Heisenberg group. To cite this article: A. Beauville, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Complete hyperbolicity of Hartogs domain

Let X be a complex space and an upper semicontinuous function on X. Consider the Hartogs domain (X) given by (X)={(z, w)X×C: |w| < e ^–(z) }. In this article, some necessary and sufficient conditions on the complete hyperbolicity of (X) are established. Mathematics Subject Classification (2000):32A10, 32C10, 32H20, 32A17     

A finite algorithm to maximize certain pseudoconcave functions on polytopes

This paper develops and proves an algorithm that finds the exact maximum of certain nonlinear functions on polytopes by performing a finite number of logical and arithmetic operations. Permissible objective functions need to be pseudoconcave and allow the closed-form solution of sets of equations , which are first order conditions associated with the unconstrained, but affinely transformed objective function. Examples are pseudoconcave quadratics and especially the homogeneous functioncx +m(xVx)^1/2,m < 0, V positive definite, for which sofar no finite algorithm existed.In distinction to most available methods, this algorithm uses the internal representation [6]|of the feasible set to selectively decompose it into simplices of varying dimensions; linear programming and a gradient criterion are used to select a sequence of these simplices, which contain a corresponding sequence of strictly increasing, relative and relatively interior maxima, the greatest of which is shown to be the global maximum on the feasible set. To find the interior maxima on these simplices in a finite way, calculus maximizations on the affine hulls of subsets of their vertices are necessary; thus the above requirement that be explicitly solvable.The paper presents a flow structure of the algorithm, its supporting theory, its decision-theoretic use, and an example, computed by an APL-version of the method.Research supported in part by The Canada Council, Ottawa. Earlier versions of the paper were presented at the European meeting of the Econometric Society in Budapest, September 1972 and at the meeting of the Econometric Society in San Francisco, December 1974.     

Divisors on hypersurfaces

Mathematische Zeitschrift -     

The density of rational points on non-singular hypersurfaces

LetF(x) =F[x₁,…,x_n]∈ℤ[x₁,…,x_n] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ ⁿ ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X ^n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan     

Zeros of Bergman kernels on some Hartogs domains

Explicit Bergman kernels are obtained on some Hartogs domains. For some special cases, zeros of the kernels are considered.     

A Vanishing Theorem on a Class of Hartogs Domain

In this paper, we consider the d-boundedness of the Bergman metric and a vanishing theorem of L²-cohomology on a class of Hartogs domain, whose base domain is the production of two irreducible bounded symmetric domains of the first type, by using the Bergman kernel function,invariant function, holomorphic automorphism group and so on.