Isolated complex singularities and their CR links Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper, we discuss some recent studies on the complex structure of an isolated normal singularity by using the information from its link. We also give some open problems to be further pursued.     

CR extension from hypersurfaces of higher type

We prove extension of CR functions from a hypersurface M of CN in presence of the so-called sector property. If M has finite type in the Bloom-Graham sense, then our result is already contained in [C. Rea, Prolongement holomorphe des fonctions CR, conditions suffisantes, C. R. Acad. Sci. Paris 297 (1983) 163-166] by Rea. We think however, that the argument of our proof carries an expressive geometric meaning and deserves interest on its own right. Also, our method applies in some case to hypersurfaces of infinite type; note that for these, the classical methods fail. CR extension is treated by many authors mainly in two frames: extension in directions of iterated of commutators of CR vector fields (cf., for instance, [A. Boggess, J. Pitts, CR extension near a point of higher type, Duke Math. J. 52 (1) (1985) 67-102; A. Boggess, J.C. Polking, Holomorphic extension of CR functions, Duke Math. J. 49 (1982) 757-784. [4]; M.S. Baouendi, L. Rothschild, Normal forms for generic manifolds and holomorphic extension of CR functions, J. Differential Geom. 25 (1987) 431-467. [1]]); extension through minimality towards unprecised directions [A.E. Tumanov, Extension of CR-functions into a wedge, Mat. Sb. 181 (7) (1990) 951-964. [6]; A.E. Tumanov, Analytic discs and the extendibility of CR functions, in: Integral Geometry, Radon Transforms and Complex Analysis, Venice, 1996, in: Lecture Notes in Math., vol. 1684, Springer, Berlin, 1998, pp. 123-141].     

Explicit construction of graph invariant for strongly pseudoconvex compact 3-dimensional rational CR manifolds

Let X be a strongly pseudoconvex compact 3-dimensional CR manifolds which bounds a complex variety with isolated singularities in some C^N. The weighted dual graph of the exceptional set of the minimal good resolution of V is a CR invariant of X; in case X has a tranversal holomorphic S¹ action, we show that it is a complete topological invariant of except for two special cases. When X is a rational CR manifolds, we give explicit algebraic algorithms to compute the graph invariant in terms of the ring structure of _k=0 m^k/m^k+1, where m is the maximal ideal of each singularity. An example is computed explicitly to demonstrate how the algorithms work.     

Extension and approximation of CR functions on tube manifolds

A complete generalization of the classical Bochner theorem for infinite tubes is given.

     

Smooth Real Hypersurfaces in ℂn with a Non-compact Isotropy Group of CR Transformations

Let M ²ⁿ⁺¹ be an (abstract) smooth, real hypersurface of ⁿ⁺¹ , strongly pseudoconvex, and let us assume that there exists a non-compact CR transformation f fixing a point p M ²ⁿ⁺¹ . We prove that there exists an f-contractible neighborhoodU of p (i.e. the closure of any orbit f ⁿ (u) with u U contains p). This gives an elementary proof of the following recent result by R. Schoen: a strongly pseudoconvex smooth real hypersurface in ²ⁿ⁺¹ , admitting a non-compact transformation fixing a point, is globally equivalent to S ²ⁿ⁺¹ or to S ²ⁿ⁺¹ with a point deleted.     

Explicit construction of a Chern–Moser connection for CR manifolds of codimension two

In the present paper we suggest an explicit construction of a Cartan connection for an elliptic or hyperbolic CR manifold M of dimension six and codimension two, i.e. a pair , consisting of a principal bundle over M and of a Cartan connection form over P, satisfying the following property: the (local) CR transformations are in one to one correspondence with the (local) automorphisms for which . For any , this construction determines an explicit monomorphism of the stability subalgebra Lie (Aut(M)_x) into the Lie algebra of the structure group H of P. Mathematics Subject Classification (2000) Primary 32V05, Secondary 53C15, 53A55     

The zero curvature equation for rigid CR-manifolds

In this paper, we present the general analytic solution to the zero curvature equation for rigid three-dimensional CR-manifolds. The solutions are uniquely determined by one function and four real parameters.     

On the Schwarz symmetry principle in a model case

In this article, we prove that smooth CR diffeomorphisms between two real analytic holomorphically nondegenerate hypersurfaces, one of which is rigid and polynomial, extend to be locally biholomorphic. It turns out that the result can be generalized to not totally degenerate mappings, in the sense of Baouendi and Rothschild.

     

Global Homotopy Formulas on <Emphasis Type="Italic">q</Emphasis>-concave <Emphasis Type="Italic">CR</Emphasis> Manifolds for Small Degrees

Using functional analysis, we derive from local homotopy formulas for the tangential Cauchy-Riemann operator a global homotopy formula for compact CR manifolds without loss of regularity.      

Groups acting transitively on compact CR manifolds of hypersurface type

Let be a compact homogeneous manifold with acting effectively and with a -invariant CR structure of hypersurface type; then any maximal compact subgroup acts transitively on .

     

Maximal L 2 and Pointwise Hölder Estimates for □ b on CR Manifolds of Class C 2

In this paper we prove the optimal pointwise Hölder estimates for the Kohn's Laplacian □ _b for CR manifolds of class C ². We first derive optimal L ² theory for solutions of □ _b directly without using pseudo-differential operators. Then we use the scaling method to obtain pointwise estimates in non-istotropic Campanato spaces based on the maximal L ² theory.     

Rigidity of Noncompact Finsler Manifolds

For a Euclidean space or a Minkowski space, we change the metric in a compact subset and show that the resulting Finsler manifold is isometric to the original standard space under certain conditions. We assume that the mean tangent curvature vanishes and the metric satisfies some curvature conditions or have no conjugate points.     

Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds

We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton–Jacobi equations defined on Riemannian manifolds.     

On Ricci eigenvalues of locally homogeneous Riemannian 3-manifolds

We find the necessary and sufficient conditions for three constants ₁, ₂, ₃ ³ to be the principal Ricci curvatures of some 3-dimensional locally homogeneous Riemannian space.The first author was supported by the grant GAR 201/93/0469; the second author was supported by the grant SFS, Project #0401.     

Boundary Hölder and estimates for local solutions of the tangential Cauchy-Riemann equation

We study the local solvability of the tangential Cauchy-Riemann equation on an open neighborhood of a point when is a generic -concave manifold of real codimension in , where . Our method is to first derive a homotopy formula for in when is the intersection of with a strongly pseudoconvex domain. The homotopy formula gives a local solution operator for any -closed form on without shrinking. We obtain Hölder and estimates up to the boundary for the solution operator. RÉSUMÉ. Nous étudions la résolubilité locale de l'opérateur de Cauchy- Riemann tangentiel sur un voisinage d'un point d'une sous-variété générique -concave de codimension quelconque de . Nous construisons une formule d'homotopie pour le sur , lorsque est l'intersection de et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un opérateur de résolution pour toute forme -fermée sur . Nous en déduisons des estimations et des estimations hölderiennes jusqu'au bord pour la solution de l'équation de Cauchy-Riemann tangentielle sur .

     

Geometric properties of almost contact metric 3-submersions

Summary In this paper, we discuss some geometric properties of three types of Riemannian submersions whose total space is an almost contact metric manifold with 3-structure. The study is focused on the transference of structures.     

The compact support principle for differential inequalities with gradient terms

We give some sufficient conditions to guarantee the validity of the Compact Support Principle for solutions of some singular elliptic differential inequalities on complete, non-compact manifolds. Our results apply in particular to the p-Laplace and the Mean Curvature operators. We also show that our results are sharp in many important cases.