CR Hypersurfaces with a Contracting Automorphism

Let M be a real analytic CR hypersurface in ℂ ⁿ⁺¹ admitting no varieties of positive dimension. We show first that every contracting local CR automorphism of M is linearizable. As a consequence, we show that such M admitting a contracting local CR automorphism is holomorphically equivalent to a weighted homogeneous hypersurface. Finally, we apply these results to prove that a bounded domain in ℂ ⁿ⁺¹ with a real analytic boundary admitting an automorphism contracting at a boundary point must admit a Lie subgroup of real dimension at least two in its automorphism group. Research of the first named author is partially supported by The Grant R01-2005-000-10771-0 of The Korea Science and Engineering Foundation.     

Free CR distributions

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n ² are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n ²)-dimensional submanifolds in for all n > 1. In particular, we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry.     

On a CR Transversality Problem Through the Approach of the Chern–Moser Theory

In this paper, we give a geometric condition for a CR map, which sends a CR non-umbilical Levi non-degenerate hypersurface in ? ⁿ⁺¹ into the hyperquadric in ? ⁿ⁺² with the same signature, to be CR transversal.     

Characterization of isolated homogeneous hypersurface singularities in ℂ4

Let V be a hypersurface with an isolated singularity at the origin in ? ⁿ⁺¹. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface signularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau’s theorem remains true for singularities with geometric genus equal to zero.     

Infinitesimal CR automorphisms of rigid hypersurfaces in C2

In this paper, we describe the space of infinitesimal CR automorphisms of a rigid, real analytic, real hypersurface in C². We use these results to obtain a geometric characterization of the homogeneous hypersurfaces. Here, a hypersurface is called homogeneous if it is equivalent to one given by an equation of the formIm(w) =p wherep is a homogeneous polynomial inz and \(\bar z\) . This gives an answer in dimension 2 to a problem posed by Linda Rothschild. We give another answer, in terms of a normal form for the defining function, in our paper “A normal form for rigid hypersurfaces in C².”     

Inhomogeneous Calderón reproducing formula on spaces of homogeneous type

In this paaper we use the Calderón-Zygmund operator theory to prove an inhomogenous Calderón reproducing formula on spaces of homogeneous type with finite or infinite measures. Our formula is new even for classical spaces of homogeneous type such as the surface of the unit ball and then-torus inR ⁿ, compact Lie groups,C ^∞-compact Riemannian manifolds, and the boundary of any bounded Lipschitz domain inR ⁿ.     

Foliations by complex curves and the geometry of real surfaces of finite type

We show that if the Levi form of a smooth CR manifold is de-generate in every conormal direction, then on a dense open set, the manifold is foliated by complex curves. As a consequence we show that every real analytic manifold of finite D'Angelo type can be stratified so that each stratum locally is contained in a Levi nondegenerate hypersurface. Received in final form: 11 June 2001 / Published online: 28 February 2002     

Qregularity and an extension of the Evans-Griffiths criterion to vector bundles on quadrics

Here we define the concept of Qregularity for coherent sheaves on a smooth quadric hypersurface Q_n⊂Pⁿ⁺¹. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on Q_n with the Castelnuovo-Mumford regularity of their extension by zero in Pⁿ⁺¹. We also classify the coherent sheaves with Qregularity −∞. We use our notion of Qregularity in order to prove an extension of the Evans-Griffiths criterion to vector bundles on quadrics. In particular, we get a new and simple proof of Knörrer’s characterization of ACM bundles.     

On covariant functions and distributions under the action of a compact group

Let G be a compact subgroup of GLn(R) acting linearly on a finite dimensional complex vector space E. B. Malgrange has shown that the space C^∞G(Rn,E) of C^∞ and G-covariant functions is a finite module over the ring C^∞G(Rn) of C^∞ and G-invariant functions. First, we generalize this result for the Schwartz space SG(Rn,E) of G-covariant functions. Secondly, we prove that any G-covariant distribution can be decomposed into a sum of G-invariant distributions multiplied with a fixed family of G-covariant polynomials. This gives a generalization of an Oksak result proved in [4].     

Global solvability for first order real linear partial differential operators

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C^∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C^∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C^∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C^∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.     

Characterization of isolated homogeneous hypersurface singularities in ?4

Let V be a hypersurface with an isolated singularity at the origin in ℂ ⁿ⁺¹. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface signularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau’s theorem remains true for singularities with geometric genus equal to zero. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Asymptotics of convex sets in Euclidean and hyperbolic spaces

We study convex sets C of finite (but non-zero) volume in Hn and En. We show that the intersection C_∞ of any such set with the ideal boundary of Hn has Minkowski (and thus Hausdorff) dimension of at most (n−1)/2, and this bound is sharp, at least in some dimensions n. We also show a sharp bound when C_∞ is a smooth submanifold of _∂∞Hn. In the hyperbolic case, we show that for any k?(n−1)/2 there is a bounded section S of C through any prescribed point p, and we show an upper bound on the radius of the ball centered at p containing such a section. We show similar bounds for sections through the origin of a convex body in En, and give asymptotic estimates as 1?k?n.     

First nonzero eigenvalue of a pseudo-umbilical hypersurface in the unit sphere

S. Deshmukh has obtained interesting results for first nonzero eigenvalue of a minimal hypersurface in the unit sphere. In the present article, we generalize these results to pseudoumbilical hypersurface and prove: What conditions are satisfied by the first nonzero eigenvalue λ ₁ of the Laplacian operator on a compact immersed pseudo-umbilical hypersurface M in the unit sphere S ⁿ⁺¹. We also show that a compact immersed pseudo-umbilical hypersurface of the unit sphere S ⁿ⁺¹ with λ ₁ = n is either isometric to the sphere S ⁿ or for this hypersurface an inequaluity is fulfilled in which sectional curvatures of the hypersuface M participate.     

Finite jet determination of CR mappings

We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M⊂CN, N?2, that is essentially finite and of finite type at each of its points, for every point p∈M there exists an integer ?p, depending upper-semicontinuously on p, such that for every smooth generic submanifold M^′⊂CN of the same dimension as M, if are two germs of smooth finite CR mappings with the same ?p jet at p, then necessarily for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in CN of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Ω,Ω^′⊂CN are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary ∂Ω, such that if are two proper holomorphic mappings extending smoothly up to ∂Ω near some point p∈∂Ω and agreeing up to order k at p, then necessarily H₁=H₂.     

Domains with Hyperbolic Orbit Accumulation Boundary Points

Let Ω be a smoothly bounded pseudoconvex domain in ℂ ⁿ satisfying the condition R. Suppose that its Bergman kernel extends to [`(W)]×[`(W)]\overline{\Omega}\times\overline{\Omega} minus the boundary diagonal set as a locally bounded function. In this paper we show that for each hyperbolic orbit accumulation boundary point p, there exists a contraction f∈Aut(Ω) at p. As an application, we show that Ω admits a hyperbolic orbit accumulation boundary point if and only if it is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial and that Ω is of finite D’Angelo type.     

Hartogs-Bochner type theorem in projective space

We prove the following Hartogs-Bochner type theorem: Let M be a connected C² hypersurface of P_n(C) (n≥2) which divides P_n(C) in two connected open sets Ω₁ and Ω₂. Suppose that M has at most one open CR orbit. Then there exists i∈{1,2} such that C¹ CR functions defined on M extends holomorphically to Ω _i . Supported by the TMR network.     

Proper holomorphic maps on an irreducible bounded symmetric domain of classical type

In this paper it is extended a result by W. Rudin on proper holomorphic maps from the open ball ofC ⁿ intoC ⁿ. I show that, forn>1, every proper map from an irreducible bounded symmetric domain of classical type intoC ⁿ, can be obtained, module biholomorphic maps, via a finite reflection group.     

The Hamiltonian dynamics over real hypersurfaces in Kaehler manifolds

Let X be a Kaehler manifold with complex dimension n. Let ωX be its Kaehler form. Let M be a strongly pseudo convex real hypersurface in X. For this hypersurface, the deformation theory of CR structures is successfully developed. And we find that H¹(M,T^′) (the T^′-valued Kohn-Rossi cohomology) is the Zariski tangent space of the versal family. In this paper, the geometrical meaning of H¹(M,O) is studied, and we propose to study displacements of the real hypersurface, which preserves the type of the differential form, ωX, over CR structures, on M, infinitesimally.     

Nonnegative functions as squares or sums of squares

We prove that, for n?4, there are C^∞ nonnegative functions f of n variables (and even flat ones for n?5) which are not a finite sum of squares of C² functions. For n=1, where a decomposition in a sum of two squares is always possible, we investigate the possibility of writing f=g². We prove that, in general, one cannot require a better regularity than g∈C¹. Assuming that f vanishes at all its local minima, we prove that it is possible to get g∈C² but that one cannot require any additional regularity.     

Straffe Untermannigfaltigkeiten in konvexen Hyperflächen

It will be proved that a tight substantial embedding ofS ^m×Sⁿ intoE ^m+n+2 whose image lies in a strictly convex hypersurface is projectively equivalent to the productC ₁×C ₂E ^m+1×E ^m+1=E ^m+n+2 of two convex hypersurfacesC ₁ undC ₂.