Analytic Hypersurface Research Articles

Advances in the Parameter Space Concept towards Picometer Precise Crystal Structure Refinement—A Resolution Study

The Parameter Space Concept (PSC) is an alternative approach to solving and refining (partial) crystal structures from very few pre-chosen X-ray or neutron diffraction amplitudes without the use of Fourier inversion. PSC interprets those amplitudes as piecewise analytic hyper-surfaces, so-called isosurfaces, in the Parameter Space, which is spanned by the spatial coordinates of all atoms of interest. The intersections of all isosurfaces constitute the (possibly degenerate) structure solution. The present feasibility study investigates the La and Sr split position of the potential high-temperature super-conductor (La0.5Sr1.5)MnO4, I4/mmm, with a postulated total displacement between La and Sr of a few pm by theoretical amplitudes of pre-selected 00l reflections (l=2,4,…,20). The revision of 15-year-old results with state-of-the-art computing equipment enhances the former simplified model by varying the scattering power ratio fLa/fSr, as exploitable by means of resonant scattering contrast at synchrotron facilities, and irrevocably reveals one of the two originally proposed solutions as being a “blurred” pseudo-solution. Finally, studying the resolution limits of PSC as a function of intensity errors by means of Monte-Carlo simulations shows both that the split can only be resolved for sufficiently low errors and, particularly for the resonant scattering contrast, a theoretical precision down to ±0.19 pm can be achieved for this specific structural problem.

Jet scheme and simple isolated hypersurface singularities

Let m≥2 be a positive integer. Let (H(f),o) be the analytic germ of hypersurface attached to a weighted-homogeneous polynomial f∈C[x1,…,xm]. In this article, we mainly characterize the simpleness of any isolated analytic quasi-homogeneous complex hypersurface singularity (H,o) in Cm in terms of the associated jet schemes. We also compute the dimension of the jet scheme Ln(H)o, formed by the n-jets centered at o, for every positive integer n>>0, which is the crucial technical part of our simpleness criterion.

On Real Analytic Levi-Flat Hypersurfaces Associated with Milnor Fibrations

On Real Analytic Levi-Flat Hypersurfaces Associated with Milnor Fibrations

Realization of arbitrary Lie algebras by automorphisms of $\mathrm{CR}$ manifolds and symmetries of differential equations

For any finite-dimensional real Lie algebra $\mathfrak{h}$, we construct a germ of a real analytic hypersurface in complex space such that its Lie algebra of infinitesimal holomorphic automorphisms is isomorphic to $\mathfrak{h}$. For any $\mathfrak{h}$, we also construct a system of partial differential equations whose Lie algebra of symmetries is isomorphic to the complexification of the algebra $\mathfrak{h}$.

Recursive formula for covariant derivatives and geometric classification of quotient modules

Let H be a vector-valued holomorphic reproducing kernel function space and H0n be the subspace of functions vanishing to order n on an analytic hyper-surface. We determine unitary equivalence of the quotient space H⊖H0n by covariant derivatives of the curvature on the vector bundle associated to the reproducing kernel of H. The proof is a combination of some geometric and combinatorial results. In particular, we show that on a holomorphic Hermitian vector bundle, matrix representations for covariant derivatives of its curvature satisfy a recursive formula, which is of independent interest.

On the full regularity of the free boundary for minima of Alt–Caffarelli functionals in Orlicz spaces

In this paper, we discuss two issues about the full regularity of the free boundary for overdetermined Bernoulli-type problems in Orlicz spaces. First, we show that in dimension \(n = 2\) there are no singular points on the free boundary \(F(u) := \partial \{ u > 0 \} \cap \Omega\) of minimizers of the Alt-Caffarelli functional
 \(J_G(u) := \int_{\Omega} \left( G(\vert \nabla u \vert) + \lambda \chi_{\{ u > 0 \}} \right) dx\)
 for suitable N-functions \(G\). Next, we prove as a consequence of our main results that there exist a critical dimension \(5 \leq n_0 \leq 7\) and a universal constant \(\varepsilon_0 \in (0,1)\) such that if \(G(t)\) is "\(\varepsilon_0\)-close" of \(t^2\), then for \(2 \leq n < n_0\), \(F(u)\) is a real analytic hypersurface.

On Boundedness of Maximal Operators Associated with Hypersurfaces

In this paper, we obtain a criterion of boundedness of maximal operators associated with smooth hypersurfaces. Also, we compute the exact value of the boundedness index of such operators associated with arbitrary convex analytic hypersurfaces in the case where the Varchenko height of the hypersurface is greater than two. We obtain the exact value of the boundedness index for degenerate smooth hypersurfaces, i.e., for hypersurfaces satisfying the assumptions of the classical Hartman–Nirenberg theorem. The obtained results justify the Stein–Iosevich–Sawyer conjecture for arbitrary convex analytic hypersurfaces as well as for smooth degenerate hypersurfaces. Also, we discuss related problems of the theory of oscillatory integrals.

Chow's theorem for real analytic Levi-flat hypersurfaces

In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space Pn, n≥2. More specifically, we prove that a real analytic Levi-flat hypersurface M⊂Pn, with singular set of real dimension at most 2n−4 and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in Pn. As a consequence, M is a semialgebraic set. We also prove that a Levi foliation on Pn — a singular real analytic foliation whose leaves are immersed complex manifolds of codimension one — satisfying similar conditions — singular set of real dimension at most 2n−4 and all leaves algebraic — is defined by the level sets of a rational function.

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.

On Levi flat hypersurfaces with transversely affine foliation

We prove the non-existence of real analytic Levi flat hypersurface whose complement is 1-convex and Levi foliation is transversely affine in a compact Kähler surface.

Generic properties of geodesic flows on analytic hypersurfaces of Euclidean space

<p style='text-indent:20px;'>Consider the geodesic flow on a real-analytic closed hypersurface <inline-formula><tex-math id="M1">\begin{document}$ M $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, equipped with the induced metric. How commonly can we expect such flows to have a transverse homoclinic orbit? In this paper, we give the following two partial answers to this question:</p><p style='text-indent:20px;'>● If <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> is a real-analytic closed hypersurface in <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> (with <inline-formula><tex-math id="M5">\begin{document}$ n \geq 3 $\end{document}</tex-math></inline-formula>) on which the geodesic flow with respect to the induced metric has a nonhyperbolic periodic orbit, then <inline-formula><tex-math id="M6">\begin{document}$ C^{\omega} $\end{document}</tex-math></inline-formula>-generically the geodesic flow on <inline-formula><tex-math id="M7">\begin{document}$ M $\end{document}</tex-math></inline-formula> with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit; and</p><p style='text-indent:20px;'>● There is a <inline-formula><tex-math id="M8">\begin{document}$ C^{\omega} $\end{document}</tex-math></inline-formula>-open and dense set of real-analytic, closed, and strictly convex surfaces <inline-formula><tex-math id="M9">\begin{document}$ M $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M10">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula> on which the geodesic flow with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit.</p><p style='text-indent:20px;'>These are among the first perturbation-theoretic results for real-analytic geodesic flows.</p>

Multiplicity, regularity and Lipschitz geometry of real analytic hypersurfaces

Multiplicity, regularity and Lipschitz geometry of real analytic hypersurfaces

A List of Integral Representations for the Diagonal of Power Series of a Rational Function

In this paper we present integral representations for the diagonals of power series. Such representations are obtained by lowering the multiplicity of integration for the previously known integral representation. The procedure for reducing the order of integration is carried out in the framework of the Leray theory of multidimensional residues. The concept of the amoeba of a complex analytic hypersurface plays a special role in the construction of new integral representations

On local hulls of Levi-flat hypersurfaces

We discuss local polynomial convexity of real analytic Levi-flat hypersurfaces in [Formula: see text], [Formula: see text], near singular points.

Hardy spaces for a class of singular domains

We set a framework for the study of Hardy spaces inherited by complements of analytic hypersurfaces in domains with a prior Hardy space structure. The inherited structure is a filtration, various aspects of which are studied in specific settings. For punctured planar domains, we prove a generalization of a famous rigidity lemma of Kerzman and Stein. A stabilization phenomenon is observed for egg domains. Finally, using proper holomorphic maps, we derive a filtration of Hardy spaces for certain power-generalized Hartogs triangles, although these domains fall outside the scope of the original framework.

ON ESTIMATES FOR THE TRANSFORMATION FOURIER WITH DAMPED FACTOR

In this paper we consider estimates for the Fourier transform measures, concentrated on analytic hypersurfaces containing the of damping factor. The paper presents the solution of the problem S.D.Soggi and I.M. Stein about the optimal decay of the transformation Fourier measures with a damping factor for any analytic surfaces in three-dimensional Euclidean space.

Nodal intersections and geometric control

We prove that the number of nodal points on an $\mathcal{S}$-good real analytic curve $\mathcal{C}$ of a sequence $\mathcal{S}$ of Laplace eigenfunctions $\varphi_j$ of eigenvalue $-\lambda^2_j$ of a real analytic Riemannian manifold $(M, g)$ is bounded above by $A_{g , \mathcal{C}} \lambda_j$. Moreover, we prove that the codimension-two Hausdorff measure $\mathcal{H}^{m-2} (\mathcal{N}_{\varphi \lambda} \cap H)$ of nodal intersections with a connected, irreducible real analytic hypersurface $H \subset M$ is $\leq A_{g, H} \lambda_j$. The $\mathcal{S}$-goodness condition is that the sequence of normalized logarithms $\frac{1}{\lambda_j} \operatorname{log} {\lvert \varphi_j \rvert}^2$ does not tend to $-\infty$ uniformly on $\mathcal{C}$, resp. $H$. We further show that a hypersurface satisfying a geometric control condition is $\mathcal{S}$-good for a density one subsequence of eigenfunctions. This gives a partial answer to a question of Bourgain–Rudnick about hypersurfaces on which a sequence of eigenfunctions can vanish. The partial answer characterizes hypersurfaces on which a positive density sequence can vanish or just have $L^2$ norms tending to zero.

Pull-back of singular Levi-flat hypersurfaces

We study singular real analytic Levi-flat subsets invariant by singular holomorphic foliations in complex projective spaces. We give sufficient conditions for a real analytic Levi-flat subset to be the pull-back of a semianalytic Levi-flat hypersurface in a complex projective surface under a rational map or to be the pull-back of a real algebraic curve under a meromorphic function. In particular, we give an application to the case of a singular real analytic Levi-flat hypersurface. Our results improve previous ones due to Lebl and Bretas -- Fernandez-Perez -- Mol.

Vanishing Hachtroudi Curvature and Local Equivalence to the Heisenberg Pseudosphere

To any completely integrable second-order system of real or complex partial differential equations: $$\begin{aligned} y_{x^{k_1}x^{k_2}} = F_{k_1,k_2} \left( x^1,\dots ,x^n,\,y,\,y_{x^1},\ldots ,y_{x^n} \right) \end{aligned}$$ with $$1 \leqslant k_1,\, k_2 \leqslant n$$ and with $$F_{ k_1, k_2} = F_{ k_2, k_1}$$ in $$\underline{n \geqslant 2}$$ independent variables $$(x^1, \ldots , x^n)$$ and in one dependent variable y, Mohsen Hachtroudi associated in 1937 a normal projective (Cartan) connection, and he computed its curvature. By means of a natural transfer of jet polynomials to the associated submanifold of solutions, what the vanishing of the Hachtroudi curvature gives can be precisely translated to characterize when both families of Segre varieties and of conjugate Segre varieties associated to a Levi nondegenerate real analytic hypersurface M in $$\mathbb {C}^n$$ ( $$n \geqslant 3$$ ) can be straightened to be affine complex (conjugate) lines. In continuation to a previous paper devoted to the quite distinct $$\mathbb {C}^2$$ -case, this then characterizes in an effective way those hypersurfaces of $$\mathbb {C}^{n+1}$$ in higher complex dimension $$n + 1\geqslant 3$$ that are locally biholomorphic to a piece of the $$(2n + 1)$$ -dimensional Heisenberg quadric, without any special assumption on their defining equations.

Analytic deformations of pencils and integrable one-forms having a first integral

We consider integrable analytic deformations of codimension one holomorphic foliations near an initially singular point. Such deformations are of two possible types. The first type is given by an analytic family [Formula: see text] of integrable one-forms [Formula: see text] defined in a neighborhood [Formula: see text] of the initial singular point, and parametrized by the disc [Formula: see text]. The initial foliation is defined by [Formula: see text]. The second type, more restrictive, is given by an integrable holomorphic one-form [Formula: see text] defined in the product [Formula: see text]. Then, the initial foliation is defined by the slice restriction [Formula: see text]. In the first part of this work, we study the case where the starting foliation has a holomorphic first integral, i.e. it is given by [Formula: see text] for some germ of holomorphic function [Formula: see text] at the origin [Formula: see text]. We assume that the germ [Formula: see text] is irreducible and that the typical fiber of [Formula: see text] is simply-connected. This is the case if outside of a dimension [Formula: see text] analytic subset [Formula: see text], the analytic hypersurface [Formula: see text] has only normal crossings singularities. We then prove that, if cod sing [Formula: see text] then the (germ of the) developing foliation given by [Formula: see text] also exhibits a holomorphic first integral. For the general case, i.e. cod sing [Formula: see text], we obtain a dimension two normal form for the developing foliation. In the second part of the paper, we consider analytic deformations [Formula: see text], of a local pencil [Formula: see text], for [Formula: see text]. For dimension [Formula: see text] we consider [Formula: see text]. For dimension [Formula: see text] we assume some generic geometric conditions on [Formula: see text] and [Formula: see text]. In both cases, we prove: (i) in the case of an analytic deformation there is a multiform formal first integral of type [Formula: see text] with some properties; (ii) in the case of an integrable deformation there is a meromorphic first integration of the form [Formula: see text] with some additional properties, provided that for [Formula: see text] the axes remain invariant for the foliations [Formula: see text].