Subanalytic Sets Research Articles

Levi-flat hypersurfaces with real analytic boundary

Let $X$ be a Stein manifold of dimension at least 3. Given a compact codimension 2 real analytic submanifold $M$ of $X$, that is the boundary of a compact Levi-flat hypersurface $H$, we study the regularity of $H$. Suppose that the CR singularities of $M$ are an $\mathcal{O}(X)$-convex set. For example, suppose $M$ has only finitely many CR singularities, which is a generic condition. Then $H$ must in fact be a real analytic submanifold. If $M$ is real algebraic, it follows that $H$ is real algebraic and in fact extends past $M$, even near CR singularities. To prove these results we provide two variations on a theorem of Malgrange, that a smooth submanifold contained in a real analytic subvariety of the same dimension is itself real analytic. We prove a similar theorem for submanifolds with boundary, and another one for subanalytic sets.

Algebraic approximation of germs of real analytic sets

Two subanalytic subsets of ℝ n are s-equivalent at a common point, say O, if the Hausdorff distance between their intersections with the sphere centered at O of radius r goes to zero faster than r s . In the present paper we investigate the existence of an algebraic representative in every s-equivalence class of subanalytic sets. First we prove that such a result holds for the zero-set V(f) of an analytic map f when the regular points of f are dense in V(f). Moreover we present some results concerning the algebraic approximation of the image of a real analytic map f under the hypothesis that f ―1 (O) = {O} . .

Harmonic measure and subanalytically tame measures

We introduce the notion of analytically and subanalytically tame mea- sures. These are measures which behave well in the globally subanalytic context; they preserve tameness: integrals of globally subanalytic functions with param- eters resp. analytic functions with parameters restricted to globally subanalytic compact sets are definable in an o-minimal structure. We consider the harmonic measure for a semianalytic bounded domain in the plane. We show that the har- monic measure for such a domain is analytically tame if the angles at singular boundary points are irrational multiples of . If the domain is a polygon and the angles at singular boundary points are rational or Diophantine irrational multiples of then the harmonic measure is subanalytically tame. 2000 Mathematics Subject Classification 03C64, 30C85, 32B20 (primary)

Radial index and Poincaré–Hopf index of 1-forms on semi-analytic sets

AbstractThe radial index of a 1-form on a singular set is a generalization of the classical Poincaré–Hopf index. We consider different classes of closed singular semi-analytic sets innthat contain 0 in their singular locus and we relate the radial index of a 1-form at 0 on these sets to Poincaré–Hopf indices at 0 of vector fields defined onn.

Bi-Lipschitz Sufficiency of Jets

We give some theorems of bi-Lipschitz or C1 sufficiency of jets which are expressed by means of transversality with respect to some strata of a stratification satisfying the (L) condition of T. Mostowski. This enables us to prove that the number of metric types of intersection of smooth transversals to a stratum of an (a) regular stratification of a subanalytic set is finite.

Densities of topological invariants of quasi-periodic subanalytic sets

We establish the existence of the densities of the Betti numbers and torsions of closed quasi-periodic subanalytic sets. In the course of the proof, we introduce the notion of a uniformly subanalytic set, which is useful when estimating the densities of topological invariants.

Distance functions and skeletal representations of rigid and non-rigid planar shapes

Shape skeletons are fundamental concepts for describing the shape of geometric objects, and have found a variety of applications in a number of areas where geometry plays an important role. Two types of skeletons commonly used in geometric computations are the straight skeleton of a (linear) polygon, and the medial axis of a bounded set of points in the k -dimensional Euclidean space. However, exact computation of these skeletons of even fairly simple planar shapes remains an open problem. In this paper we propose a novel approach to construct exact or approximate (continuous) distance functions and the associated skeletal representations (a skeleton and the corresponding radius function) for solid 2D semi-analytic sets that can be either rigid or undergoing topological deformations. Our approach relies on computing constructive representations of shapes with R-functions that operate on real-valued halfspaces as logic operations. We use our approximate distance functions to define a new type of skeleton, i.e, the C-skeleton, which is piecewise linear for polygonal domains, generalizes naturally to planar and spatial domains with curved boundaries, and has attractive properties. We also show that the exact distance functions allow us to compute the medial axis of any closed, bounded and regular planar domain. Importantly, our approach can generate the medial axis, the straight skeleton, and the C-skeleton of possibly deformable shapes within the same formulation, extends naturally to 3D, and can be used in a variety of applications such as skeleton-based shape editing and adaptive motion planning.

The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms

Let A ⊂ R n be a set-germ at 0 ∈ R n such that 0 ∈ A. We say that r ∈ S n−1 is a direction of A at 0 ∈ R n if there is a sequence of points {xi} ⊂ A\{0} tending to 0 ∈ R n such that x i k xik → r as i → ∞. Let D(A) denote the set of all directions of A at 0 ∈ R n . Let A, B ⊂ R n be subanalytic set-germs at 0 ∈ R n such that 0 ∈ A ∩ B. We study the problem of whether the dimension of the common direction set, dim(D(A) ∩ D(B)) is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of A and B are also subanalytic. In particular if two subanalytic set-germs are bi-Lipschitz equivalent their direction sets must have the same dimension. Let us recall the notion of blow-analyticity. Let f, g : (R n ,0) → (R,0) be analytic function-germs. We say that they are blow-analytically equivalent if there are real modifications � : (M, � −1 (0)) → (R n ,0), � ' : (M ' , � '−1 (0)) → (R n ,0) and an analytic isomorphism � : (M, � −1 (0)) → (M ' , � '−1 (0)) which induces a homeomorphism φ : (R n ,0) → (R n ,0), � ' ◦ � = φ ◦ � , such that f = g ◦ φ. A blow-analytic homeomorphism is such a φ, a homeomorphism induced by an analytic isomorphism via real modifications.

Separation of global semianalytic sets

Given global semianalytic sets $A$ and $B$, we define a minimal analytic set $N$ such that $\overline{A}\setminus N$ and $\overline{B}\setminus N$ can be separated by an analytic function. Our statement is very similar to the one proved by Bröcker

An elementary coordinate-dependent local resolution of singularities and applications

An elementary classical analysis resolution of singularities method is developed, extensively using explicit coordinate systems. The algorithm is designed to be applicable to subjects such as oscillatory integrals and critical integrability exponents. As one might expect, the trade-off for such an elementary method is a weaker theorem than Hironaka's work [H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, Ann. of Math. (2) 79 (1964) 109–203; H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero II, Ann. of Math. (2) 79 (1964) 205–326] or its subsequent simplications and extensions such as [E. Bierstone, P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (2) (1997) 207–302; S. Encinas, O. Villamayor, Good points and constructive resolution of singularities, Acta Math. 181 (1) (1998) 109–158; J. Kollar, Resolution of singularities—Seattle lectures, preprint; A.N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funct. Anal. Appl. 18 (3) (1976) 175–196]. Nonetheless the methods of this paper can be used to prove a variety of theorems of interest in analysis. As illustration, two consequences are given. First and most notably, a general theorem regarding the existence of critical integrability exponents are established. Secondly, a quick proof of a well-known inequality of Lojasiewicz [S. Lojasiewicz, Ensembles semi-analytiques, Inst. Hautes Études Sci., Bures-sur-Yvette, 1964] is given. The arguments here are substantially different from the general algorithms such as [H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, Ann. of Math. (2) 79 (1964) 109–203; H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero II, Ann. of Math. (2) 79 (1964) 205–326], or the elementary arguments of [E. Bierstone, P. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math. 67 (1988) 5–42] and [H. Sussman, Real analytic desingularization and subanalytic sets: an elementary approach, Trans. Amer. Math. Soc. 317 (2) (1990) 417–461]. The methods here have as antecedents the earlier work of the author [M. Greenblatt, A direct resolution of singularities for functions of two variables with applications to analysis, J. Anal. Math. 92 (2004) 233–257], Phong and Stein [D.H. Phong, E.M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997) 107–152], and Varchenko [A.N. Varchenko, Newton polyhedra and estimates of oscillatory integrals, Funct. Anal. Appl. 18 (3) (1976) 175–196].

Equivariant uniformization theorem for subanalytic sets

In this paper we prove an equivariant version of the uniformization theorem for closed subanalytic sets: Let G be a Lie group and let M be a proper real analytic G-manifold. Let X be a closed subanalytic G-invariant subset of M. We show that there exist a proper real analytic G-manifold N of the same dimension as X and a proper real analytic G-equivariant map \({\varphi{:} \ N \to M}\) such that \({\varphi(N) = X}\) .

Uniform Linear Bound in Chevalley’s Lemma

AbstractWe obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley’s lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function of a closed Nash (or formally Nash) subanalytic set.

Curve selection Lemma for semianalytic sets and conjugacy classes of finite order in Lie groups

Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group G, each connected component of the set E n (G) of all elements of order n in G is a conjugacy class in G. In particular, all conjugacy classes of finite order in G are closed. Some properties of connected components of E n (G) are also given.

On the homology of algebras of Whitney functions over subanalytic sets

In this article we study several homology theories of the algebra E ∞ (X) of Whitney functions over a subanalytic set X ⊂ R n with a view towards noncommutative geometry. Using a localization method going back to Teleman we prove a Hochschild-Kostant-Rosenberg type theorem for E ∞ (X), when X is a regular subset of R n having regularly situated diagonals. This includes the case of subanalytic X. We also compute the Hochschild cohomology of E ∞ (X) for a regular set with regularly situated diagonals and derive the cyclic and periodic cyclic theories. It is shown that the periodic cyclic homology coincides with the de Rham cohomology, thus generalizing a result of Feigin-Tsygan. Motivated by the algebraic de Rham theory of Grothendieck we finally prove that for subanalytic sets the de Rham cohomology of E ∞ (X) coincides with the singular cohomology. For the proof of this result we introduce the notion of a bimeromorphic subanalytic triangulation and show that every bounded subanalytic set admits such a triangulation. Contents

Parameterized stratification and piece number of D-semianalytic sets

We obtain results on the geometry of D-semianalytic and subanalytic sets over a complete, non-trivially valued non-Archimedean field K, which is not necessarily algebraically closed. Among the results are the Parameterized Smooth Stratification Theorem and several results concerning the dimension theory of D-semianalytic and subanalytic sets. Also, an extension of Bartenwerfer's definition of piece number for analytic K-varieties is provided for the D-semianalytic sets and the existence of a uniform bound for the piece number of the fibers of a D-semianalytic set is proved. There is a connection between the piece number and the complexity of a D-semianalytic set which is a subset of the affinoid line and therefore a simpler proof of the Complexity Theorem of Lipshitz and Robinson is made possible by these results. Finally we prove an analogue of a theorem by van den Dries, Haskell and Macpherson, which states that for each D-semianalytic X, there is a semialgebraic Y such that one dimensional fibers of X are among the one dimensional fibers of Y through an easy application of our earlier results.

Uniform Linear Bound in Chevalley's Lemma

We obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley's lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function for a closed Nash (or formally Nash) subanalytic set.

Capacity Density of Subanalytic Sets in Higher Dimension

We show that capacity density exists for subanalytic sets in any dimension. Moreover, we state connections between capacity density and volume density for subanalytic sets.

Approximation of subanalytic sets by normal cones

Let A be a closed semialgebraic subset of Euclidean space of codimension at least one, and containing the origin O as a non-isolated point. We prove that, for every real s ≥ 1, there exists an algebraic set V which approximates A to order s at O. The special case s = 1 generalizes the result of the authors that every semialgebraic cone of codimension at least one is the tangent cone of an algebraic set.

Tangent spaces and Gromov-Hausdorff limits of subanalytic spaces

It is shown that the Gromov-Hausdorff limit of a subanalytic 1-parameter family of compact connected sets (endowed with the inner metric) exists. If the family is semialgebraic, then the limit space can be identified with a semialgebraic set over some real closed field. Different notions of tangent cones (pointed Gromov-Hausdorff limits, blowups and Alexandrov cones) for a closed connected subanalytic set are studied and shown to be naturally isometric. It is shown that geodesics have well-defined Euclidean directions at each point.

On the connected components of a global semianalytic subset of an analytic surface

A global semianalytic subset of a real analytic manifold is a finite union of finite intersections of the solutions of equations and inequalities of real analytic functions on the manifold. Is a union of connected components of a global semianalytic set again global semianalytic? We consider a two-dimensional global semianalytic set such that the normalization of the Zariski closure of it is affine. We show that a union of connected components of it is again global semianalytic. We also give some partial results on connected components of global semianalytic subset of a three-dimensional analytic manifold.