On the Teichmüller theorem and the heights theorem for quadratic differentials

By using the Marden-Strebel heights theorem for quadratic differentials, we provide a concrete method for finding the Teichmüller differential associated with the Teichmüller mapping between compact or finitely punctured Riemann surfaces.

     

An isometry theorem for quadratic differentials on Riemann surfaces of finite genus

Assume both and are Riemann surfaces which are subsets of compact Riemann surfaces and respectively, and that the set has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on and are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmüller space of onto the Teichmüller space of is induced by some quasiconformal map of onto . Consequently we can find an uncountable set of Riemann surfaces whose Teichmüller spaces are not biholomorphically equivalent.

     

A duality theorem for complex quadratic programming

A duality theory for complex quadratic programming over polyhedral cones is developed, following Dorn, by using linear duality theory.This research was partly supported by the National Science Foundation, Project No. GP-7550, and by the US Army Research Office, Durham, North Carolina, Contract No. DA-31-124-ARO-D-322. The authors are indebted to the referee for his helpful suggestions.     

Weierstrass preparation of quadratic differentials

Special cases or variants of the following play an important role in the asymptotic analysis of ordinary differential equations with turning points. Theorem.Let a(t, x) be a smooth complex-valued function germ at the origin in C × R ^m which is holomorphic in t. Suppose that a(t, 0) does not vanish identically. Then there is a smooth change of variable t = g(s, x), holomorphic in s, such that a(t, x)dt ² =P(s, x)ds ² where P is a monic polynomial in s.     

Moduli spaces of quadratic differentials

The cotangent bundle ofJ (g, n) is a union of complex analytic subvarieties, V(π), the level sets of the function “singularity pattern” of quadratic differentials. Each V(π) is endowed with a natural affine complex structure and volume element. The latter contracts to a real analytic volume element, Μ_π, on the unit hypersurface, V₁(π), for the Teichmüller metric. Μ_π is invariant under the pure mapping class group, γ(g, n), and a certain class of functions is proved to be L^p(Μ_π), 0 <p < 1, over the moduli space V₁(π)/γ (g, n). In particular, Μ_π(V₁(π)/γ(g, n)) < ∞, a statement which generalizes a theorem by H. Masur. Research supported by NSF-MCS-8219148 and NSF-DMS-8521620.     

Conformal invariants associated with quadratic differentials

We associate a functional of pairs of simply-connected regions D₂ ? D₁ to any quadratic differential on D₁ with specified singularities. This functional is conformally invariant, monotonic, and negative. Equality holds if and only if the inner domain is the outer domain minus trajectories of the quadratic differential. This generalizes the simply-connected case of results of Z. Nehari [20], who developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle for harmonic functions. Nehari’s method corresponds to the special case that the quadratic differential is of the form (?q)² for a singular harmonic function q on D₁.As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda [16].     

Meromorphic quadratic differentials with prescribed singularities

We study the singular flat structure associated to any meromorphic quadratic differential on a compact Riemann surface to prove an existence theorem as follows. There exists a meromorphic quadratic differential with given orders of the poles and zeros and orientability or non orientability of the horizontal foliation, iff these prescribed topological data are admissible according to the Gauss-Bonnet Theorem, the Residue Theorem and certain conditions arising from local orientability or non orientablity considerations. Some few exceptional cases remain excluded. Thus, we generalize two previous results. One due to Masur & Smillie, which assumes that poles are at most simple; and a second one due to Muciño-Raymundo, which assumes that the horizontal foliation is orientable.Partially supported by DGAPA-UNAM and CONACYT 28492-E.     

A limit theorem with weights for quadratic Dirichlet L-functions

Translated from Matematicheskie Zametki, Vol. 55, No. 2, pp. 128–129, February, 1994.     

A quantitative Oppenheim theorem for generic diagonal quadratic forms

We establish a quantitative version of Oppenheim’s conjecture for one-parameter families of ternary indefinite quadratic forms using an analytic number-theory approach. The statements come with power gains and in some cases are essentially optimal.     

A generalization of the inertia theorem for quadratic matrix polynomials

We show that the inertia of a quadratic matrix polynomial is determined in terms of the inertia of its coefficient matrices if the leading coefficient is Hermitian and nonsingular, the constant term is Hermitian, and the real part of the coefficient matrix of the first degree term is definite. In particular, we prove that the number of zero eigenvalues of such a matrix polynomial is the same as the number of zero eigenvalues of its constant term. We also give some new results for the case where the real part of the coefficient matrix of the first degree term is semidefinite.     

Existence of quadratic differentials with prescribed properties

The paper continues the author's studies of the question on the existence of quadratic differentials Q(z)dz² having given structure of trajectories and poles of high orders. It is shown that such differentials can be considered as the limits of sequences of quadratic differentials that have poles of second order with trajectories asymptotically similar to logarithmic spirals and realize extremal configurations in suitable families of nonoverlapping domains. It is established that there exist differentials Q(z)dz² of indicated form having given initial terms of the Laurent expansions in the vicinities of the poles of Q(z)dz² of order not smaller than three. Some discrepancies in an earlier paper are corrected. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 120–137.     

The norm theorem for semisingular quadratic forms

Let F be a field of characteristic 2. The aim of this paper is to give a complete proof of the norm theorem for singular F-quadratic forms which are not totally singular, i.e., we give necessary and sufficient conditions for which a normed irreducible polynomial of $F[x_1,\dots ,x_n]$ becomes a norm of such a quadratic form over the rational function field $F(x_1,\dots ,x_n)$. This completes partial results proved on this question in [8]. Combining the present work with the papers [1] and [7], we obtain the norm theorem for any type of quadratic forms in characteristic 2.