Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds

The Dirichlet-to-Neumann (DN) map Λ_g: C^∞ (?M) → C^∞(?M) on a compact Riemannian manifold (M, g) with boundary is defined by Λ_gh = ?u/?v¦in{t6M}, where u is the solution to the Dirichlet problem Δu = 0, u¦?M = h and v is the unit normal to the boundary. If g^t = g + t? is a variation of the metric g by a symmetric tensor field ?, then Λ_g ^t = Λ_g + tΛ_? + o(t). We study the question: How do tensor fields ? look like for which Λ_? =0? A partial answer is obtained for a general manifold, and the complete answer is given in the two cases: For the Euclidean metric and in the 2D-case. The latter result is used for proving the deformation boundary rigidity of a simple 2-manifold.     

A gradient estimate in the Calabi–Yau theorem

We prove a C ¹-estimate for the complex Monge–Ampère equation on a compact Kähler manifold directly from the C ⁰-estimate, without using a C ²-estimate. This was earlier done only under additional assumption of non-negative bisectional curvature.     

An analog of Sard’s theorem for C 1-smooth functions of two variables

The main result of the article is Theorem 1. Let v: Ω → ? be a C¹-smooth function on a domain Ω → ?². Suppose that 0 ? Cl Int Dv(Ω). Then the measure of the image of the set of critical points equals zero.     

Hypersurfaces Levi-plates immergées dans les surfaces complexes de courbure positive

We show that there is no immersed compact Levi-flat hypersurface of class C¹ in the complex projective plane, if the foliation by holomorphic curves carries a harmonic current which is absolutely continuous with respect to the Lebesgue measure, with a density bounded from above and below. This is a corollary of a rigidity result for immersed compact Levi-flat hypersurfaces in complex surfaces of non-negative curvature.     

n-Widths,Faber Expansion,and Computation of Analytic Functions

LetBH^∞(Ω) be the space of analytic functionsfin the region Ω for which |f(z)| ≤ 1,z∈ Ω, and letKbe a compact subset of Ω. How can we compute the values of any functionf∈BH^∞(Ω) at an arbitrary pointz∈K? One of the approaches to this problem applies the results concerning then-widths and ϵ-entrophy of classBH^∞(Ω) in the metricC(K). In the case whenKhas a simply connected complement inC and Ω is a canonical neighbourhood ofK, the classical tools for approximation off∈BH^∞(Ω) inC(K) give the Faber series. This work is concerned with the following: the exact values of Kolmogorov and othern-widths of Hardy spacesH^p, then-widths and ϵ-entrophy of classBH^∞(Ω), the optimality of Faber approximations, and computing values of analytic functions with the help of Faber series.     

On the geometry of the reachability set of vector fields

We study the geometry of the reachability set of a family of vector fields on a C ^∞ manifold. We show that, for each real number T, the T-reachability set is a smooth submanifold of an orbit of codimension zero or one and that, on an arbitrary connected C ^∞ manifold of dimension greater than one, there exists a system of three vector fields such that each 0-reachability set coincides with the manifold itself.     

Properties of the <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth functions with nowhere dense gradient range

One of the main results of the present article is as follows Theorem. Let v: Ω → ? be a C¹-smooth function on a domain Ω ? ?². Suppose that Int?v(Ω) = ?. Then, for every point z ∈ Ω, there is a straight line L ? z such that ?v ≡ const on the connected component of the set L ? Ω containing z.Also, we prove that, under the conditions of the theorem, the range of the gradient ?v(Ω) is locally a curve and this curve has tangents in the weak sense and the direction of these tangents is a function of bounded variation.     

Geometric Anosov flows of dimension 5

We show that for a smooth Anosov flow on a closed five dimensional manifold, if it has C^∞ Anosov splitting and preserves a C^∞ pseudo-Riemannian metric, then up to a special time change and finite covers, it is C^∞ flow equivalent either to the suspension of a symplectic hyperbolic automorphism of $\text{T}{}^4$, or to the geodesic flow on a three dimensional hyperbolic manifold. To cite this article: Y. Fang, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A MANN ITERATIVE REGULARIZATION METHOD FOR ELLIPTIC CAUCHY PROBLEMS

We investigate the Cauchy problem for linear elliptic operators with C ^∞–coefficients at a regular set Ω ? R ², which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ ? ?Ω and our goal is to reconstruct the trace of the H ¹(Ω) solution of an elliptic equation at ?Ω/Γ. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.

     

Courbure scalaire prescrite sur une variété C∞ compacte de dimension 2

Let (V₂, g) be a C^∞ compact Riemannian manifold of negative scalar curvature of dimension 2, and let f be a C^∞ function defined on V₂. We intend to find a condition on f in order that f be the scalar curvature of a metric conformal to the initial metric g.     

Regularity for some scalar variational problems under general growth conditions

We consider integrals of the calculus of variations over a set Ω of ? ⁿ , and the related regularity result: are the minimizers smooth functions, say for example of classC ^∞(Ω)? Classically, the so-called natural growth conditions on the integrand have been the main sufficient assumptions for regularity. In recent years, motivated also by application, the interest in the study of this problem has increased under more general growth assumptions. In this paper, we propose some general growth conditions that guarantee regularity for a class of scalar variational problems.     

Derivations On a C∞-ring

The main purpose of this paper is to provide several results on objects lying between differential geometry and algebraic geometry such as C^∞-rings and derivations on a C^∞-ring. A C^∞-ring is defined as a set with operations by C^∞-functions on Euclidean spaces. A derivation on a C^∞-ring is defined in two method, an ?-derivation and a C^∞-derivation. The main result of this paper is to show that any ?-derivation is a C^∞-derivation for some classes of C^∞-rings (Theorems 3.1, 3.2).     

Right inverses for linear, constant coefficient partial differential operators on distributions over open half spaces

Results of Hörmander on evolution operators together with a characterization of the present authors [Ann. Inst. Fourier, Grenoble 40, 619–655 (1990)] are used to prove the following: Let P ∈ ?[z₁,...,z _n ] and denote by P _m its principal part. If P ? P_m is dominated by P _m then the following assertions for the partial differential operators P(D) and P _m(D) are equivalent for N ∈ S ^n?1:

1. P(D) and/or P_m D)admit a continuous linear right inverse on C ^∞(H ₊(N)).
2. P(D) admits a continuous linear right inverse on C ^∞ (? ⁿ ) and a fundamental solution E ∈ C ^∞(?ⁿ) satisfying Supp $E \subset \overline {H - (N)} $

where H ₊(N) := {x ∈ ? ⁿ :±(x,N) τ; 0}.     

Perturbation and Interpolation Theorems for the H ∞-Calculus with Applications to Differential Operators

We prove comparison theorems for the H ^∞-calculus that allow to transfer the property of having a bounded H ^∞-calculus from one sectorial operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ^∞-calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ^∞-calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for operators defined in interpolation scales such as the L _p -scale. We apply the results to give new proofs on the H ^∞-calculus for elliptic differential operators, including Schrödinger operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ^∞-calculus in certain Sobolev spaces and that the Stokes operator on bounded domains Ω with ?Ω ∈ C ^1,1 has a bounded H ^∞-calculus in the Helmholtz scale L _p,σ (Ω), p ∈ (1,∞).     

Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations

We present a very simple proof of the global existence of a C^∞ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has C^∞ dependence on initial data u₀ in the class of H^s divergence-free vector fields for s > 2.     

Isomorphisme de Dolbeault dans les variétés CR

Let M be a q-concave CR generic C^∞-smooth submanifold of real codimension k in an n-dimensional complex manifold X, then the truncated tangential Cauchy-Riemann complexes τ^≤q−1[ɛ^p,*](M), τ^{≥n−k−q+}2[ɛ^{p, *}](M) of C^∞-smooth forms and τ^≤q−1 [D′^p.*](M), τ^{≥n−k−q+}2[D′^p,*(M)] of currents on M are quasi-isomorphic for 0 ≤ p ≤ n.     

Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues

Suppose thatM ⁿ is a complete, noncompact, Riemannian manifold. If Δ denotes the Laplace operator ofM, one has associated Schrödinger operators ? Δ +V. Conditions onV are formulated, which ensures the essential self-adjointness of ? Δ +V. In particular, ifV ∈ Q_α,loc (M ⁿ), the local Stummel class, andV ≥ ? c outside of a compact set, then ? Δ +V is essentially self-adjoint on C ₀ ^∞ (M ⁿ). In addition, essential self-adjointness is proved for potentials which are strongly singular at a point. The absence of eigenvalues of ?Δ +V is also studied. This relies upon Rellich-type identities. The results on strongly singular potentials make use of a generalization of the classical uncertainty principle, inR ⁿ, to Riemannian manifolds with a pole.     

Unitary Equivalence, Similarity and Calculation of K0-Group

In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W＊（A＋B）′≈M2（C）, and two operators A and B in B1（Ω） are similar if and only if A′（AGB）/J≈M2（C）. Moreover, we obtain V（H^∞（Ω,μ）≈N and Ko（H^∞（Ω,μ）≈Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non-reducing measure on Ω.     

Exotic characteristic classes of quaternionic bundles

For aC ^∞ quaternionic vector bundle, the odd-dimensional real Chern classes vanish, and this allows for a construction of secondary (exotic) characteristic classes associated with a pair of quaternionic structures of a given complex vector bundle. This construction is then applied to obtain exotic characteristic classes associated with an automorphismβ of the holomorphic tangent bundle of a Kähler manifold. These results are the complex analoga of those given for the higher order Maslov classes in [V2].     

Uniqueness of Fokker-Planck equations for spin lattice systems (I): compact case

The main purpose of this paper is to prove the uniqueness of the Fokker-Planck equation for continuous lattice systems when the spin space is a compact, connected, C ^∞-Riemannian manifold M.