Real Analytic Set Research Articles

The Asymptotic Expansion of the Spacetime Metric at the Event Horizon

AbstractHawking’s local rigidity theorem, proven in the smooth setting by Alexakis-Ionescu-Klainerman, says that the event horizon of any stationary non-extremal black hole is a non-degenerate Killing horizon. In this paper, we prove that the full asymptotic expansion of any smooth vacuum metric at a non-degenerate Killing horizon is determined by the geometry of the horizon. This gives a new perspective on the black hole uniqueness conjecture. In spacetime dimension 4, we also prove an existence theorem: Given any non-degenerate horizon geometry, Einstein’s vacuum equations can be solved to infinite order at the horizon in a unique way (up to isometry). The latter is a gauge invariant version of Moncrief’s classical existence result, without any restriction on the topology of the horizon. In the real analytic setting, the asymptotic expansion is shown to converge and we get well-posedness of this characteristic Cauchy problem.

Quickly Computing Isotopy Type for Exponential Sums over Circuits (Extended Abstract)

Fewnomial Theory [Kho91] has established bounds on the number of connected components (a.k.a. pieces ) of a broad class of real analytic sets as a function of a particular kind of input complexity, e.g., the number of distinct exponent vectors among a generating set for the underlying ideal. Here, we pursue the algorithmic side: We show how to efficiently compute the exact isotopy type of certain (possibly singular) real zero sets, instead of just estimating their number of pieces. While we focus on the circuit case, our results form the foundation for an approach to the general case that we will pursue later.

The Maximum Locus of the Bloch Norm

Abstract For a Bloch function f in the unit ball in ℂ n , we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at most 2n − 2. This is not the case for n = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set Lf . In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.

On the Fukui–Kurdyka–Paunescu conjecture

In this paper, we prove the Fukui–Kurdyka–Paunescu conjecture, which says that sub-analytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (respectively, degree) of real and complex analytic (respectively, algebraic) sets. For instance, still in the real case, we prove a global version of the Fukui–Kurdyka–Paunescu conjecture. In the complex case, one of the results that we prove is the following: if$(X,0)\subset (\mathbb {C}^{n},0)$,$(Y,0)\subset (\mathbb {C}^{m},0)$are germs of analytic sets and$h\colon (X,0)\to (Y,0)$is a semi-bi-Lipschitz homeomorphism whose graph is a complex analytic set, then the germs$(X,0)$and$(Y,0)$have the same multiplicity. One of the results that we prove in the global case is the following: if$X\subset \mathbb {C}^{n}$,$Y\subset \mathbb {C}^{m}$are algebraic sets and$\phi \colon X\to Y$is a semi-algebraic semi-bi-Lipschitz homeomorphism such that the closure of its graph in$\mathbb {P}^{n+m}(\mathbb {C})$is an orientable homological cycle, then${\rm deg}(X)={\rm deg}(Y)$.

Multiplicity, regularity and blow-spherical equivalence of real analytic sets

This article is devoted to studying multiplicity and regularity of analytic sets. We present an equivalence for analytic sets, named blow-spherical equivalence, which generalizes differential equivalence and subanalytic bi-Lipschitz equivalence and, with this approach, we obtain several applications on analytic sets. On multiplicity, we present a generalization for Gau–Lipman’s Theorem about differential invariance of the multiplicity in the complex and real cases, and we show that the multiplicity \(\mathrm{mod}\,2\) is invariant under blow-spherical homeomorphisms in the case of real analytic curves and surfaces and also for a class of real analytic foliations and is invariant by (image) arc-analytic blow-spherical homeomorphisms in the case of real analytic hypersurfaces, generalizing some results proved by G. Valette. On regularity, we show that blow-spherical regularity of real analytic sets implies \(C^1\) smoothness only in the case of real analytic curves. We present also a complete classification of the germs of real analytic curves.

Differential invariance of the multiplicity of real and complex analytic sets

This paper is devoted to proving the differential invariance of the multiplicity of real and complex analytic sets. In particular, we prove the real version of Gau-Lipman's Theorem, i.e., it is proved that the multiplicity mod 2 of real analytic sets is a differential invariant. We prove also a generalization of Gau-Lipman's Theorem.

Formal versus analytic CR mappings

We discuss the convergence/divergence problem for formal holomorphic mappings sending real-analytic CR submanifolds into real-analytic sets both lying in complex Euclidean spaces of arbitrary dimension. In particular, we survey the recent developments on

Local topological algebraicity with algebraic coefficients of analytic sets or functions

We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.

Travelling waves over an arbitrary bathymetry: a local stability result

The problem of a travelling wave over an arbitrary quasi-flat bathymetry in a semi infinite channel is studied in the shallow-water formulation. It is shown how the streamfunction can be cast, in the vicinity of an elliptic equilibrium for the fluid flow, in the form of a nearly-integrable non-autonomous Hamiltonian with aperiodic time dependence. The proofs use the tools of perturbation theory in the real-analytic setting. The obtained Hamiltonian provides a natural example in the context of the aperiodically time-dependent Hamiltonian systems studied in Fortunati and Wiggins (2016). Some key properties of the system at hand, such as the stability, can be addressed as a consequence of this theory.

On dicritical singularities of Levi-flat sets

It is proved that dicritical singularities of real analytic Levi-flat sets coincide with the set of Segre degenerate points.

Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity

In this paper we show Whitney's fibering conjecture in the real and complex, local analytic and global algebraic cases.For a given germ of complex or real analytic set, we show the existence of a stratification satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivialization arc-wise analytic and we show that it can be constructed under the classical Zariski algebro-geometric equisingularity assumptions. Using a slightly stronger version of the Zariski equisingularity, we show the existence of Whitney's stratified fibration, satisfying the conditions (b) of Whitney and (w) of Verdier. Our construction is based on the Puiseux with parameter theorem and a generalization of Whitney's interpolation. For algebraic sets our construction gives a global stratification.We also present several applications of the arc-wise analytic trivialization, mainly to the stratification theory and the equisingularity of analytic set and function germs. In the real algebraic case, for an algebraic family of projective varieties, we show that the Zariski equisingularity implies local constancy of the associated weight filtration.

Local and global analysis of nodal sets

This is a survey for the JDG 50th Anniversary conference of recent results on nodal sets of eigenfunctions of the Laplacian on a compact Riemannian manifold. In part the techniques are `local', i.e. only assuming eigenfunctions are defined on small balls, and in part the techniques are `global', i.e. exploiting dynamics of the geodesic flow. The local part begins with a review of doubling indices and freqeuency functions as local measures of fast or slow growth of eigenfunctions. The pattern of boxes with maximal doubling indices plays a central role in the results of Logunov-Malinnokova, giving upper and lower bounds for hypersurface measures of nodal sets in the setting of general $C^{\infty}$ metrics. The proofs of both their polynomial upper bound and sharp lower bound are sketched. The survey continues with a global proof of the sharp upper bound for real analytic metrics (originally proved by Donnelly-Fefferman with local arguments), using analytic continuation to Grauert tubes. Then it reviews results of Toth-Zelditch giving sharp upper bounds on Hausdorff measures of intersections of nodal sets with real analytic submanifolds in the real analytic setting. Last, it goes over lower bounds of Jung-Zelditch on numbers of nodal domains in the case of $C^{\infty}$ surfaces of non-positive curvature and concave boundary or on negatively curved `real' Riemann surfaces, which are based on ergodic properties of the geodesic flow and eigenfunctions, and on estimates of restrictions of eigenfunctions to hypersurfaces. The last section details recent results on restrictions.

Tameness of Complex Dimension in a Real Analytic Set

AbstractGiven a real analytic set X in a complex manifold and a positive integer d, denote by Ad the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that Ad is a closed semianalytic subset of X.

Infinite type germs of real analytic pseudoconvex domains in ℂ3

In Lampert [On the boundary regularity of biholomorphic mappings, Contributions to several complex variables, Aspects Math. E9 (1986), pp. 193–215] and D'Angelo [Several Complex Variables and the Geometry of real Hypersurfaces, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1993, ISBN: 0-8493-8272-6] Lempert and D'Angelo showed that germs of real analytic sets in ℂ n of infinite type contain a complex curve. In this article we discuss a very special case of their result, germs of real analytic pseudoconvex domains in ℂ3. We reprove their theorem using a geometric construction which sheds light on the intricate structure of such boundaries in the presence of complex curves of high order tangency. The proof of Lempert and D'Angelo is somewhat more of an ideal theoretic nature.

On the holomorphic closure dimension of real analytic sets

Given a real analytic (or, more generally, semianalytic) set R in Cn (viewed as R2n), there is, for every p ∈ R, a unique smallest complex analytic germ Xp that contains the germ Rp. We call dimC Xp the holomorphic closure dimension of R at p. We show that the holomorphic closure dimension of an irreducible R is constant on the complement of a closed proper analytic subset of R, and we discuss the relationship between this dimension and the CR dimension of R.

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} . .

Real and complex analytic sets. The relevance of Segre varieties

Let X ⊂ C n be a closed real-analytic subset and put A := {z ∈ X |∃ A ⊂ X, germ of a complex-analytic set, z ∈ A, dimz A > 0} This article deals with the question of the structure of A.I nthe main result a natural proof is given for the fact, that A always is closed. As a main tool an interesting relation between complex analytic subsets of X of positive dimension and the Segre varieties of X is proved and exploited.

Critical points and level sets in exterior boundary problems

We study some geometrical properties of the critical set of the solutions to an exterior boundary problem in R n \Ω, where Ω is a bounded domain with C 2 connected boundary. We prove that this set can be nonempty (in fact, of codimension 3) even when Ω is contractible, thereby settling a question posed by Kawohl. We also obtain new sufficient geometric criteria for the absence of critical points in this problem and analyze the properties of the critical set for generic domains. The proofs rely on a combination of classical potential theory, transversality techniques and the geometry of real analytic sets.

On a global analytic Positivstellensatz

We consider several modified versions of the Positivstellensatz for global analytic functions that involve infinite sums of squares and/or positive semidefinite analytic functions. We obtain a general local-global criterion which localizes the obstruction to have such a global result. This criterion allows us to get completely satisfactory results for analytic curves, normal analytic surfaces and real coherent analytic sets whose connected components are all compact.

On the positive extension property and Hilbert's 17th problem for real analytic sets

In this work we study the Positive Extension property and Hilbert's 17th problem for real analytic germs and sets. A real analytic germ X0 of has the property if every positive semidefinite analytic function germ on X0 has a positive semidefinite analytic extension to ; analogously one states the property for a global real analytic set X in an open set Ω of ℝn. These properties are natural variations of Hilbert's 17th problem. Here, we prove that: (1) A real analytic germ has the property if and only if every positive semidefinite analytic function germ on X0 is a sum of squares of analytic function germs on X0; and (2) a global real analytic set X of dimension ≦ 2 and local embedding dimension ≦ 3 has the property if and only if it is coherent and all its germs have the property. If that is the case, every positive semidefinite analytic function on X is a sum of squares of analytic functions on X. Moreover, we classify the singularities with the property.