Electronic Edition Vol. 28, No. 2, 2002

Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson (Houston), J. Nagata (Osaka), B. H. Neumann (Canberra), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)

Houston Journal of Mathematics


It gives us distinct pleasure to dedicate this issue to Professor Shiing-Shen Chern. Prof. Chern had been an editor of the Houston Journal of Mathematics from 1992 to 2000. His association with the Department of Mathematics involved yearly visits and stimulating  lectures that began in 1988. He has collaborated with members of the department in publications and in organizing conferences. He is currently a Distinguished Visiting Professor Emeritus of the University of Houston. Professor Chern is also Professor Emeritus of Mathematics at the University of California at Berkeley, and Director Emeritus of  the Mathematical Sciences Research Institute (M.S.R.I.). He is the founding father of three  renowned institutes: the Institute of Mathematics of the Academia Sinica, the M.S.R.I., and the Nankai Institute of Mathematics.

Professor Chern has rather broad but impeccable tastes in Mathematics. We have taken the liberty of singling out some areas in which he has made profound contributions: real and complex differential geometry, several complex variables, and Finsler geometry. The collection of papers that we have assembled in this issue comes from a  small cross-section of mathematicians in whose careers Professor Chern has played a unique role. We are delighted, but not  surprised, that the authors in question submitted their manuscripts with much enthusiasm. We hope this special issue will become a source of inspiration for many years to come.

Editors of the special issue for S.S. Chern, Volume 28, No. 2

David Bao 
Shanju Ji
Klaus Kaiser (Managing Editor)
Min Ru

Picture of Professor Chern

Subscribers might wish to order an additional copy of this issue Vol. 28(2)  which is also available to individuals. 
The price is  $25, plus $5 for shipping and handling. 


Robert L. Bryant, Duke University Mathematics Department, P.O. Box 90320 Durham, NC 27708-0320 (bryant@math.duke.edu) .
Some Remarks on Finsler Manifolds with Constant Flag Curvature, pp. 221-262.
ABSTRACT. Abstract: This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature.
The first remark is that there is a canonical Kähler structure on the space of geodesics of such a manifold.
The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in CPn.
The third remark is that there is a description of the Finsler metrics of constant curvature on S2 in terms of a Riemannian metric and 1-form on the space of its geodesics. In particular, this allows one to use any (Riemannian) Zoll metric of positive Gauss curvature on S2 to construct a global Finsler metric of constant positive curvature on S2.
The fourth remark concerns the generality of the space of (local) Finsler metrics of constant positive flag curvature in dimension n+1>2. It is shown that such metrics depend on n(n+1) arbitrary functions of n+1 variables and that such metrics naturally correspond to certain torsion-free S1·GL(n,R)-structures on 2n-manifolds. As a by-product, it is found that these groups do occur as the holonomy of torsion-free affine connections in dimension 2n, a hitherto unsuspected phenomenon.

Patrick Foulon, Institut de Recherche Mathèmatique Avancèe, Universitè Louis Pasteur Strasbourg, France.
Curvature and Global Rigidity in Finsler Manifolds, pp. 263-292.
ABSTRACT. We present some strong global rigidity results for reversible Finsler manifolds. Following É Cartan's definition (1926), a locally symmetric Finsler metric is one whose curvature is parallel. These spaces strictly contain the spaces such that the geodesic reflections are local isometries and also constant curvature manifolds. In the case of negative curvature, we prove that the locally symmetric Finsler metrics on compact manifolds are Riemannian and this, therefore, extends A. Zadeh's rigidity result. Our approach uses dynamical properties of the flag curvature. We also give a full generalization of the Ossermann Sarnak minoration of the metric entropy of the geodesic flow. In positive curvature, we just announce some partial results and remarks concerning Finsler metrics of curvature +1 on the 2-sphere. We show that in the reversible case the geodesic flow is conjugate to the standard one. We also observe that a condition of integral geometry (of Radon type) forces such a metric to be Riemannian. This indicates a deep link with (exotic) projective structures

Daniel S. Freed, Department of Mathematics, University of Texas, Austin, TX 78712 (dafr@math.utexas.edu).
Classical Chern-Simons Theory, Part 2, pp. 293-310.
ABSTRACT. We first recall the construction of Chern-Weil and Chern-Simons forms, and then their synthesis in Cheeger-Simons differential characters. Then given a family of connections parametrized by a manifold T, we integrate these invariants to obtain differential forms and differential characters on T. There are special constructions of geometric representatives in low degrees, but the general case requires a suitable "cochain theory" for differential characters. This is provided by recent work of Hopkins-Singer. It applies in particular to the classical three-dimensional topological field theory known as Chern-Simons theory.

Peter B. Gilkey, Mathematics Department, University of Oregon, Eugene Or 97403 USA (gilkey@darkwing.uoregon.edu) and Tan Zhan, Department of Mathematics and Statistics, Murray State University, Murray, KY 42071-0009 (tan.zhang@murraystate.edu).
Algebraic Curvature Tensors for Indefinite Metrics whose Skew-Symmetric Curvature Operator has Constant Jordan Normal Form, pp. 311-328.
ABSTRACT. We classify the connected pseudo-Riemannian manifolds of signature (p,q) with q≥5 so that at each point of M the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least one point of M.

Mark L. Green, Department of Mathematics, UCLA, Los Angeles, CA 9009 (mlg@math.ucla.edu) and Phillip A. Griffiths, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540 (pg@ias.edu).
Abel's Differential Equations, pp. 329-351.
ABSTRACT. Abel's differential equations govern the rational motion of algebraic cycles on an algebraic variety. In this paper, we will discuss Abel's differential equations, first from a historical and classical perspective. Then we will discuss their modern form, which is definitely non-classical in that arithmetic considerations enter in an essential way. Finally, we will discuss the integration of Abel's differential equations, which may be accomplished only by assuming an important conjecture of Bloch/Beilinson.

Wolf Kohn, Vladimir Brayman, Hynomics Corporation,Kirkland, WA 98033-7921 (wk@hynomics.com) (vbrayman@hynomics.com) and Anil Nerode, Department of Mathematics, Cornell University, Ithaca, New York 14853 (anil@math.cornell.edu).
Control Synthesis in Hybrid Systems with Finsler Dynamics, pp. 353-375.
ABSTRACT. This paper is concerned with a symbolic-based synthesis of feedback control policies for hybrid and continuous dynamic systems. A key step in our synthesis procedure is a new method to solve a dynamic optimization problem in which the continuous dynamics generates trajectories on a smooth manifold in which a Finsler metric has been defined. The proposed method can be generalized for finding explicitly the control laws for a wide variety of problems by "Finslerizing" their formulation. The feedback control law generated is of the form of the Cartan connection. The coefficients of this control law (generalized Christoffel symbols) can be determined analytically. This paper illustrates the computation by establishing a direct connection between optimality in a Finsler domain (geodesic trajectories) and Dynamic Programming.

Bernard Shiffman, Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA (shiffman@math.jhu.edu) and Mikhail Zaidenberg, Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d'Hères Cédex, France (zaidenbe@ujf-grenoble.fr) .
Constructing Low Degree Hyperbolic Surfaces in P 3 , pp. 377-388.
ABSTRACT. We describe a new method of constructing Kobayashi-hyperbolic surfaces in complex projective 3-space based on deforming surfaces with a ``hyperbolic non-percolation'' property. We use this method to show that general small deformations of certain singular abelian surfaces of degree 8 are hyperbolic. We also show that a union of 15 planes in general position in projective 3-space admits hyperbolic deformations.

Yum-Tong Siu, Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138} (siu@math.harvard.edu).
A New Bound for the Effective Matsusaka Big Theorem , pp. 389- 409.
ABSTRACT. Matsusaka's Big Theorem gives the very ampleness of a multiple of an ample line bundle over an n-dimensional compact complex manifold with the factor depending ineffectively on the top Chern class of the line bundle and the product of the second top Chern class of the line bundle and the canonical class. An earlier result of the author gives an effective bound on the factor which is of the order of the sum of the absolute values of the above two Chern numbers raised to the power 4n. Demailly reduced the power to 3n by reducing the twisting required for the existence of nontrivial global holomorphic sections for anticanonical sheaves of subvarieties which occur in the verification of the numerical effectiveness of the sum of the anticanonical line bundle and an effective multiple of the ample line bundle. Twisted sections of anticanonical sheaves are needed to offset the addition of the canonical sheaf in vanishing theorems. We introduce here a technique to get a new bound with power 2n. The technique avoids the use of sections of twisted anticanonical sheaves of subvarieties by transferring the use of vanishing theorems on subvarieties of the ambient space to the ambient space itself and is more in line with techniques for Fujita conjecture type results.

Gang Tian, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 (tian@math.mit.edu).
Extremal Metrics and Geometric Stability, pp. 411-432.
ABSTRACT. This paper grew out of my lectures at Nankai Institute as well as a few other conferences in the last few years. The purpose of this paper is to describe some of my works on extremal Kähler metrics in the last fifteen years in a more unified way.
At the beginning of the 90's, the author developed a method of relating certain stability of underlying manifolds to Kähler-Einstein metrics (cf. [Ti4], [Ti2]). An necessary and new condition was derived in terms of the stability for a Kähler manifold to admit Kähler-Einstein metrics with positive scalar curvature. It was clear then that similar results should also hold for general extremal Kähler metrics. Extremal Kähler metrics were introduced by Calabi [Ca]. Extremal Kähler metrics are critical points of the K-energy introduced by T. Mabuchi. Most extremal metrics are Kähler metrics of constant scalar curvature. It was conjectured by the author before that the existence of Kähler metrics with constant scalar curvature is equivalent to the properness of the K-energy. This has been verified for the case of Kähler-Einstein metrics ([Ti2]).
We will explain how extremal metrics are related to the stability of the underlying manifolds and compare it with the standard picture from symplectic geometry. We will outline the proof of the Calabi's conjecture for complex surfaces. We will also list a few problems and indicate the difficulties in solving them.
[Ca] : Calabi, E.: Extremal Kähler metrics. Seminar on Diff. Geom., Ann. of math. Stud., 102, Princeton Univ. Press, 1982.
[Ti2] : Tian, G: Kähler-Einstein metrics with positive scalar curvature. Invent. Math., 130 (1997), 1-39.
[Ti4]: Tian, G: The K-energy on Hypersurfaces and Stability. Communications in Geometry and Analysis, 2 (1994), 239-265.

S. M. Webster , University of Chicago (webster@math.uchicago.edu).
A remark on the Chern-Moser tensor, pp. 433-435.
ABSTRACT. We compute the fourth order Chern-Moser tensor for real hypersurfaces of revolution in complex Euclidean space.