Joseph H. G. Fu

Theory of valuations on manifolds, III. Multiplicative structure in the general case

This is the third part of a series of articles where the theory of valuations on manifolds is constructed. In the second part of this series the notion of a smooth valuation on a manifold was introduced. The goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds, thus extending some of the affine constructions from the first authors 2004 paper and, from the first part of this series.

Criticality for the Gehring link problem

In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehrings problem to deal with links in a fixed link-homotopy class: we prove ropelength minimizers exist and introduce a theory of ropelength criticality. Our balance criterion is a set of necessary and sufficient conditions for criticality, based on a strengthened, infinite-dimensional version of the Kuhn--Tucker theorem. We use this to prove that every critical link is C^1 with finite total curvature. The balance criterion also allows us to explicitly describe critical configurations (and presumed minimizers) for many links including the Borromean rings. We also exhibit a surprising critical configuration for two clasped ropes: near their tips the curvature is unbounded and a small gap appears between the two components. These examples reveal the depth and richness hidden in Gehrings problem and our natural extension.

Integral geometry of complex space forms

We show how Alesker’s theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and complex Euclidean space. In particular, we compute the family of kinematic formulas for invariant valuations and invariant curvature measures in these spaces. In addition to new and more efficient framings of the tube formulas of Gray and the kinematic formulas of Shifrin, this approach yields a new formula expressing the volumes of the tubes about a totally real submanifold in terms of its intrinsic Riemannian structure. We also show by direct calculation that the Lipschitz-Killing valuations stabilize the subspace of invariant angular curvature measures, suggesting the possibility that a similar phenomenon holds for all Riemannian manifolds. We conclude with a number of open questions and conjectures.

STIEFEL-WHITNEY CLASSES AND THE CONORMAL CYCLE OF A SINGULAR VARIETY

A geometric construction of Sullivans Stiefel-Whitney homology classes of a real analytic variety X is given by means of the conormal cycle of an embedding of X in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. We also show that, for a complex analytic variety, these classes are the mod 2 reductions of the Chern-MacPherson classes. We present a new definition of the Stiefel-Whitney homology classes of a possibly singular real analytic variety X. The original definition, due to Sullivan [S], involves a triangulation of X; its geometric meaning is unclear. Our definition uses the conormal cycle of an embedding of X in a smooth variety. The conormal cycle of a subanalytic subset X of an analytic manifold M was defined by the first author [F4] using geometric measure theory. The conormal cycle is an integral current representing (up to sign) Kashiwaras characteristic cycle of the sheaf DMRX [F4, 4.7]. Our construction of the Stiefel-Whitney classes is based on the fundamental observation that the conormal cycle of a real analytic variety is antipodally symmetric mod 2. We use this observation to give a definition of Stiefel-Whitney classes which is parallel to the first authors definition of the Chern-Schwartz-MacPherson classes of a complex analytic variety [F5]. (This definition of Chern classes is related to earlier work of Brylinski-Dubson-Kashiwara [BDK] and Sabbah [Sa].) In fact we show that the Stiefel-Whitney homology classes of a complex analytic variety are the mod 2 reductions of the Chern classes. We prove that our Stiefel-Whitney classes satisfy axioms similar to the Deligne-Grothendieck axioms for Chern classes, and we prove a specialization formula for the Stiefel-Whitney classes of a family of varieties. We show that the Stiefel-Whitney classes of an affine real analytic variety X are represented by the polar cycles of X, introduced for simplicial spaces by Banchoff [B] and McCrory [Mc], and we give a new proof of the combinatorial formula for Stiefel-Whitney classes of manifolds. Our central result is a specialization formula for the conormal cycle (Theorem 3.7). We use this formula to prove the basic results of Kashiwara and Schapiras calculus of subanalytically constructible functions [Sc], [KS, 9.7], as well as the pushforward and specialization formulas for Stiefel-Whitney classes. Received by the editors October 2, 1995. 1991 Mathematics Subject Classification. Primary 14P25, 57R20; Secondary 14P15, 49Q15.

Algebraic Integral Geometry

Recent work of S. Alesker has catalyzed a flurry of progress in Blaschkean integral geometry and opened the prospect of further advances. By this term we understand the circle of ideas surrounding the kinematic formulas (Theorem 2.1.6 below), which express related fundamental integrals relating to the intersections of two subspaces K, L ⊂ ℝ n in general position in terms of certain “total curvatures” of K and L separately.

On Fine Differentiability Properties of Horizons and Applications to Riemannian Geometry

We study fine differentiability properties of horizons. We show that the set of end points of generators of an n-dimensional horizon H (which is included in an (n + 1)-dimensional space–time M) has vanishing n-dimensional Hausdorff measure. This is proved by showing that the set of end points of generators at which the horizon is differentiable has the same property. For 1 ≤ k ≤ n + 1, we show (using deep results of Alberti) that the set of points where the convex hull of the set of generators leaving the horizon has dimension k is “almost a C 2 manifold of dimension n + 1 − k”: it can be covered, up to a set of vanishing (n + 1 − k)-dimensional Hausdorff measure, by a countable number of C 2 manifolds. We use our Lorentzian geometry results to derive information about the fine differentiability properties of the distance function and the structure of cut loci in Riemannian geometry.

Symmetric criticality for tight knots

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength critical configurations for knots and links which are different from the ropelength minima for these knot and link types.

On Verdier's specialization formula for Chern classes*

Let M be a complex manifold of complex dimension m, and let X C M be a proper analytic subvariety. Let D C ~ be the unit disk, and f : X ~ D a proper analytic morphism. Put X~: = f l(s), s e D. We know that, for r > 0 sufficiently small, the inclusion Xo C f I(D,) is a homotopy equivalence. Thus for small nonzero s e D we may define the specialization map on homology a. :H,(X~)~H.(Xo) as the composition ~, H,(X~) , H , ( f I(D,)) ~ n,(Xo).

Integral geometry and valuations

Part I: New Structures on Valuations and Applications.- Translation invariant valuations on convex sets.- Valuations on manifolds.- Part II: Algebraic Integral Geometry.- Classical integral geometry.- Curvature measures and the normal cycle.- Integral geometry of euclidean spaces via Alesker theory.- Valuations and integral geometry on isotropic manifolds.- Hermitian integral geometry.

Dual Curvature Measures in Hermitian Integral Geometry

The local kinematic formulas on complex space forms induce the structure of a commutative algebra on the space CurvU(n)∗ of dual unitarily invariant curvature measures. Building on the recent results from integral geometry in complex space forms, we describe this algebra structure explicitly as a polynomial algebra. This is a short way to encode all local kinematic formulas. We then characterize the invariant valuations on complex space forms leaving the space of invariant angular curvature measures fixed.