Electronic Edition Vol. 24, No. 2, 1998

Editors: G. Auchmuty (Houston), H. Brezis (Paris), S. S. Chern (Berkeley), J. Damon (Chapel Hill), K. Davidson (Waterloo), L. C. Evans (Berkeley), R. M. Hardt (Rice), J. A. Johnson (Houston), A. Lelek (Houston), J. Nagata (Osaka), B. H. Neumann (Canberra), G. Pisier (College Station and Paris), R. Scott (Houston), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)


Charalambos Charitos, Department of Mathematics, Agricultural University of Athens, 75 Iera Odos, Athens 11855 Greece, and Georgios Tsapogas, Department of Mathematics, University of the Agean, Karlovassi, Samos 83200 Greece.
Closed Geodesics on 2-Dimensional chi-Geometric Polyedra pp.185-196.
ABSTRACT. For 2-dimensional finite chi -geometric polyhedra of curvature K <= chi < 0 it is shown that the polygonal flow, applied to a closed curve, converges to a geodesic. Moreover, it is shown that there exists a finite number of closed geodesics with length smaller than a given positive B. As an application of the polygonal flow, a way of constructing closed, in particular simple, curves is given as well as a condition which implies that a curve is non-homotopic to a point.

Markov, Y., University of North Carolina, Chapel Hill NC 27599-3250 (markov@math.unc.edu), Tarasov, V., Osaka University, Toyonaka, Osaka 560, Japan (vt@math.sci.osaka-u.ac.jp), Varchenko, A., University of North Carolina, Chapel Hill NC 27599-3250 (av@math.unc.edu)
Determinant of a Hypergeometric Period Matrix pp. 197-220.
ABSTRACT. We consider a function U=e-f0.f1a1 ...fpap on a real affine space, here f0,..,fp are linear functions, a1,...,ap complex numbers. The zeros of the functions f1,..., fp form an arrangement of hyperplanes in the affine space. We study the period matrix of the hypergeometric integrals associated with the arrangement and the function U and compute its determinant as an alternating product of gamma functions and critical points of the functions f0,..., fp with respect to the arrangement.
We also give a determinant formula for Selberg type exponential integrals. In this case the arangements of hyperplanes is special and admits a symmetry group, the period matrix is decomposed into blocks corresponding to different representations of the symmetry group on the space of the hypergeometric integrals associated with the arrangement. We compute the determinant of the block corresponding to the trivial representation.

Groisser, David, University of Florida, Gainesville, FL 32611 (groisser@math.ufl.edu).
Totally geodesic boundaries of Yang-Mills moduli spaces, pp. 221-276.
ABSTRACT. Moduli spaces M of self-dual SU(2) connections (``instantons'') over a compact Riemannian 4-manifold (M,g) carry a natural L2 metric g, which is generally incomplete. For instantons of Pontryagin index 1 over a compact, simply connected, oriented, positive-definite base manifold, the completion M is Donaldson's compactification; in fact the boundary of the completion is an isometric copy of (M,4(pi)2 g) (Groisser and Parker, 1989). In this paper we show that the boundary is, furthermore, a totally geodesic submanifold of the completion. Along the way, we prove a regularity theorem: the continuous extension of g to the ``collar region'' of M is C 1,alpha (in the conventional scale/center coordinates) for small, positive alpha. The proofs rely on some new weighted Sobolev inequalities for concentrated instantons, in which the only dependence of the Sobolev constants on the connection is through the concentration parameter lambda. The exponent in the weighting function translates into the Holder exponent in the regularity theorem.

A. Bonome, R. Castro, E. García-Río, L. Hervella, R. Vázquez-Lorenzo Departamento de Xeometría e Topoloxía, Facultade de Matemáticas, Universidade de Santiago de Santiago de Compostela, 15706 Santiago deCompostela , Spain.
On the Paraholomorphic Sectional Curvature of Almost Para-Hermitian Manifolds, pp. 277-300.
ABSTRACT. The paraholomorphic sectional curvature of almost para-Hermitian manifolds is investigated. After discussing those manifolds with pointwise constant paraholomorphic sectional curvature, main attention is devoted to those being isotropic. Also, some boundedness conditions for the paraholomorphic sectional curvature are studied.

Tim Laberge Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115.
Supports of Quasi-Measures, pp. 301-312.
ABSTRACT. We define a useful notion of support for quasi-measures on compact Hausdorff spaces, investigate the consequences of this definition, and prove a decomposition theorem for quasi-measures. A consequence of the decomposition theorem is that any space carrying a fully supported 0--1 quasi-measure is connected and cannot be disconnected by the removal of a closed 0-dimensional subset.

Fulman, Igor, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen O, Denmark (ifulman@math.ku.dk).
Coordinatization for Fell Bundle Algebras, pp. 313-324.
ABSTRACT. Let Cr*(E) be the reduced C*-algebra generated by a Fell bundle E over an r-discrete principal groupoid. We show that each element in Cr*(E) is represented by a continuous section of E.
Also, the Coordinatization Theorem proved in this paper gives necessary and sufficient conditions for an abstract C*-algebra A to be isomorphic to Cr*(E) for some Fell bundle E.

Fernando Cobos, Depto. de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain and Tomas Schonbeck, Florida Atlantic University, Dept. of Mathematical Sciences, Boca Raton, Fl 33431, USA.
On a Theorem by Lions and Peetre About Interpolation Between a Banach Space and its Dual, pp. 325-344.
ABSTRACT. We show that if the duality between a Banach space A and its anti-dual A* is given by the inner product of a Hilbert space H, then (A,A*)1/2,2 = H = (A,A*)[1/2], provided A satisfies certain mild conditions. We do not assume A is reflexive. Applications are given to normed ideals of operators.

P. Celada, Dipartimento di Scienze Matematiche, UniversitàDegli Studi Trieste, P. Le Europa 1, I-34127 Trieste, Italia, and A. Cellina Scuola Internazionale Superiore di Studi Avanzati (SIISSA), Bia Beirut 4, I-34013 Trieste, Italia.
Existence and Non Existence of Solutions to a Variational Problem on a Square, pp. 345-375.
ABSTRACT. A non convex minimum problem on a square arising in shape optimization is studied. Conditions are discussed for the existence or the non existence of solutions.

E.M.E. Zayed Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt.
Short-Time Asymptotics of the Heat Kernel of the laplacian of a Bounded Domain with Robin Boundary Conditions, pp. 377-385.
ABSTRACT. The basic problem in this paper is that of determining some geometrical properties of a general bounded domain in two or three dimensions with a smooth boundary where smooth functions are entering the boundary conditions which are not strictly positive, from complete knowledge of the eigenvalues for the negative Laplacian, using the asymptotic expansions of the trace of the heat kernel for short-time t. Further results are obtained.