Real Analytic Case Research Articles

Direct proof of termination of the Kohn algorithm in the real-analytic case

In 1979 J.J. Kohn gave an indirect argument via the Diederich-Fornaess Theorem showing that finite D'Angelo type implies termination of the Kohn algorithm for a pseudoconvex domain with real-analytic boundary. We give here a direct argument for this same implication using the stratification coming from Catlin's notion of a boundary system as well as algebraic geometry on the ring of real-analytic functions. We also indicate how this argument could be used in order to compute an effective lower bound for the subelliptic gain in the $\bar\partial$-Neumann problem in terms of the D'Angelo type, the dimension of the space, and the level of forms provided that an effective Łojasiewicz inequality can be proven in the real-analytic case and slightly more information obtained about the behavior of the sheaves of multipliers in the Kohn algorithm.

Hidden toric symmetry and structural stability of singularities in integrable systems

The goal of the paper is to develop a systematic approach to the study of (perhaps degenerate) singularities of integrable systems and their structural stability. As the main tool, we use "hidden" system-preserving torus actions near singular orbits. We give sufficient conditions for the existence of such actions and show that they are persistent under integrable perturbations. We find toric symmetries for several infinite series of singularities and prove, as an application, structural stability of Kalashnikov's parabolic orbits with resonances in the real-analytic case. We also classify all Hamiltonian $k$-torus actions near a singular orbit on a symplectic manifold $M^{2n}$ (or on its complexification) and prove that the normal forms of these actions are persistent under small perturbations. As a by-product, we prove an equivariant version of the Vey theorem (1978) about local symplectic normal form of nondegenerate singularities.

On the Lyapunov instability in Newtonian dynamics

We prove Lyapunov instability for cases in which the local minimum of the potential energy is reached on a hypersurface of the configuration space. In contrast to the known results in this direction, which hold for potentials satisfying hypotheses on the first non-zero jet, this new result covers several real analytic cases that the previous do not.

On controlled invariance of regular distributions

This paper considers the problem of controlled invariance of involutive regular distribution, both for smooth and real analytic cases. After a review of some existing work, a precise formulation of the problem of local and global controlled invariance of involutive regular distributions for both affine control systems and affine distributions is introduced. A complete characterization for local controlled invariance of involutive regular distributions for affine control systems is presented. A geometric interpretation for this characterization is provided. A result on local controlled invariance for real analytic affine distribution is given. Then, we investigate conditions that allow passages from local controlled invariance to global controlled invariance, for both smooth and real analytic affine distributions. We clarify existing results in the literature. Finally, for manifolds with a symmetry Lie group action, the problem of global controlled invariance is considered.

On the Universal Unfolding of Vector Fields in One Variable: A Proof of Kostov\u2019s Theorem

In this note we present variants of Kostov’s theorem on a versal deformation of a parabolic point of a complex analytic 1-dimensional vector field. First we provide a self-contained proof of Kostov’s theorem, together with a proof that this versal deformation is indeed universal. We then generalize to the real analytic and formal cases, where we show universality, and to the {mathcal {C}}^infty case, where we show that only versality is possible.

Nonholonomic and constrained variational mechanics

Equations governing mechanical systems with nonholonomic constraints can be developed in two ways: (1) using the physical principles of Newtonian mechanics; (2) using a constrained variational principle. Generally, the two sets of resulting equations are not equivalent. While mechanics arises from the first of these methods, sub-Riemannian geometry is a special case of the second. Thus both sets of equations are of independent interest.The equations in both cases are carefully derived using a novel Sobolev analysis where infinite-dimensional Hilbert manifolds are replaced with infinite-dimensional Hilbert spaces for the purposes of analysis. A useful representation of these equations is given using the so-called constrained connection derived from the system's Riemannian metric, and the constraint distribution and its orthogonal complement. In the special case of sub-Riemannian geometry, some observations are made about the affine connection formulation of the equations for extremals.Using the affine connection formulation of the equations, the physical and variational equations are compared and conditions are given that characterise when all physical solutions arise as extremals in the variational formulation. The characterisation is complete in the real analytic case, while in the smooth case a locally constant rank assumption must be made. The main construction is that of the largest affine subbundle variety of a subbundle that is invariant under the flow of an affine vector field on the total space of a vector bundle.

Quasianalytic ultradifferentiability cannot be tested in lower dimensions

We show that, in contrast to the real analytic case, quasianalytic ultradifferentiability can never be tested in lower dimensions. Our results are based on a construction due to Jaffe.

Dimension preserving resolutions of singular Poisson structures

We give examples of Poisson structures that admit symplectic resolutions of the same dimension. We also give a simple condition under which proper in the smooth case or semi-connected symplectic resolutions in the real analytic and holomorphic case can not exist: open symplectic leaves have to be dense and the singular locus can not be of codimension one.

The Poisson embedding approach to the Calder\xf3n problem

We introduce a new approach to the anisotropic Calderón problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calderón type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result of Lassas et al. (Annales de l’ ENS 34(5):771–787, 2001) solving the Calderón problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.

Analytic Classification of Foliations Induced by Germs of Holomorphic Vector Fields in ( ℂ n , 0 ) $(\mathbb {C}^{n},0)$ with Non-isolated Singularities

We consider germs of holomorphic vector fields in $(\mathbb {C}^{n},0)$ , n ≥ 3, with non-isolated singularities. We assume that the set of singular points forms a submanifold of codimension 2, and the sum of the nonzero eigenvalues of the linearization of the germs at each singular point is zero. We give the orbital analytic classification of generic germs of such type. It happens that, unlike the formal classification (which is trivial), the analytic one has functional moduli. The same result is obtained in the real-analytic case (the smooth normalization was obtained earlier in Roussarie Asterisque. 1975;30:1–181).

On the Lower Bound of the Inner Radius of Nodal Domains

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions varphi _{lambda } on a closed Riemannian manifold (M, g) . In the real-analytic case, we present an improvement of the currently best-known bounds, due to Mangoubi (Commun Partial Differ Equ 33:1611–1621, 2008; Can Math Bull 51(2):249–260, 2008). Furthermore, using recent results of Hezari (P Am Math Soc, 2016, https://doi.org/10.1090/proc/13766; Anal PDE 11(4):855–871, 2018), we obtain log -type improvements in the case of negative curvature and improved bounds for (M, g) possessing an ergodic geodesic flow.

A Real-analytic Nonpolynomially Convex Isotropic Torus with no Attached Discs

AbstractWe showbymeans of an example in that Gromov’s theoremon the presence of attached holomorphic discs for compact Lagrangianmanifolds is not true in the subcritical real-analytic case, even in the absence of an obvious obstruction, i.e., polynomial convexity.

On damped second-order gradient systems

Using small deformations of the total energy, as introduced in [31], we establish that damped second order gradient systemsu″(t)+γu′(t)+∇G(u(t))=0, may be viewed as quasi-gradient systems. In order to study the asymptotic behavior of these systems, we prove that any (nontrivial) desingularizing function appearing in KL inequality satisfies φ(s)⩾cs whenever the original function is definable and C2. Variants to this result are given. These facts are used in turn to prove that a desingularizing function of the potential G also desingularizes the total energy and its deformed versions. Our approach brings forward several results interesting for their own sake: we provide an asymptotic alternative for quasi-gradient systems, either a trajectory converges, or its norm tends to infinity. The convergence rates are also analyzed by an original method based on a one-dimensional worst-case gradient system.We conclude by establishing the convergence of solutions of damped second order systems in various cases including the definable case. The real-analytic case is recovered and some results concerning convex functions are also derived.

Local and global liftings of analytic families of idempotents in Banach algebras

Generalizing results of our earlier paper, we investigate the following question. Let $\pi(\lambda) : A \to B$ be an analytic family of surjective homomorphisms between two Banach algebras, and $q(\lambda)$ an analytic family of idempotents in $B$. We want to find an analytic family $p(\lambda)$ of idempotents in $A$, lifting $q(\lambda)$, i.e., such that $\pi(\lambda)p(\lambda) = q(\lambda)$, under hypotheses of the type that the elements of $\text{Ker}\, \pi(\lambda)$ have small spectra. For spectra which do not disconnect $\Bbb C$ we obtain a local lifting theorem. For real analytic families of surjective $^*$-homomorphisms (for continuous involutions) and self-adjoint idempotents we obtain a local lifting theorem, for totally disconnected spectra. We obtain a global lifting theorem if the spectra of the elements in $\text{Ker}\, \pi(\lambda)$ are $\{0\}$, both in the analytic case, and, for $^*$-algebras (with continuous involutions) and self-adjoint idempotents, in the real analytic case. Here even an at most countably infinite set of mutually orthogonal analytic families of idempotents can be lifted to mutually orthogonal analytic families of idempotents. In the proofs, spectral theory is combined with complex analysis and general topology, and even a connection with potential theory is mentioned.

The Riemann extensions with cyclic parallel Ricci tensor

The property of being a D'Atri space (i.e., a Riemannian manifold with volume‐preserving geodesic symmetries) is equivalent, in the real analytic case, to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold satisfying the first odd Ledger condition L3 is said to be an L3‐space. This definition extends easily to the affine case. Here we investigate the torsion‐free affine manifolds and their Riemann extensions as concerns heredity of the condition L3. We also incorporate a short survey of the previous results in this direction, including also the topic of D'Atri spaces.

Tangent cones and regularity of real hypersurfaces

Abstract We characterize 𝒞 1 $\mathcal {C}^1$ embedded hypersurfaces of 𝐑 n $\mathbf {R}^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most m < 3 / 2 $m<3/2$ . It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is 𝒞 1 $\mathcal {C}^1$ . In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface X ⊂ 𝐑 n $X\subset \mathbf {R}^n$ is 𝒞 1 $\mathcal {C}^1$ . Furthermore, if X is real algebraic, strictly convex, and unbounded, then its projective closure is a 𝒞 1 $\mathcal {C}^1$ hypersurface as well, which shows that X is the graph of a function defined over an entire hyperplane. Finally we show that the last property is a special feature of real algebraic sets, in the sense that it does not hold in the real analytic category.

Remarks on global sub-Lorentzian geometry

This paper aims at being a starting point for the investigation of the global sub-Lorentzian (or more generally sub-semi-Riemannian) geometry, which is a subject completely not known. We present an adaptation to the fiber bundles setting (and state its elementary proof in the real analytic case) of the decomposition result for indefinite metrics. Using it and the result by Sussmann on global spanning of distributions, we state and prove some global facts on sub-Lorentzian manifolds. We also construct a global control system describing nonspacelike curves and study its controllability properties in case of compact manifolds.

Complexity reduction of C-Algorithm

The C-Algorithm introduced in [5] is designed to determine isochronous centers for Lienard-type differential systems, in the general real analytic case. However, it has a large complexity that prevents computations, even in the quartic polynomial case. The main result of this paper is an efficient algorithmic implementation of C-Algorithm, called ReCA (Reduced C-Algorithm). Moreover, an adapted version of it is proposed in the rational case. It is called RCA (Rational C-Algorithm) and is widely used in [1,2] to find many new examples of isochronous centers for the Liénard type equation.

Embedding into manifolds with torsion

We introduce a class of special geometries associated to the choice of a differential graded algebra contained in \({\Lambda^*\mathbb{R}^n}\). We generalize some known embedding results, that effectively characterize the real analytic Riemannian manifolds that can be realized as submanifolds of a Riemannian manifold with special holonomy, to this more general context. In particular, we consider the case of hypersurfaces inside nearly-Kähler and α-Einstein–Sasaki manifolds, proving that the corresponding evolution equations always admit a solution in the real analytic case.

Calabi-Yau cones from contact reduction

We consider a generalization of Einstein–Sasaki manifolds, which we characterize in terms both of spinors and differential forms, that in the real analytic case corresponds to contact manifolds whose symplectic cone is Calabi-Yau. We construct solvable examples in seven dimensions. Then, we consider circle actions that preserve the structure and determine conditions for the contact reduction to carry an induced structure of the same type. We apply this construction to obtain a new hypo-contact structure on S 2 × T 3.