Determination of formal CR mappings by a finite jet

We prove that if M is a connected real-analytic holomorphically nondegenerate hypersurface in Cn+1, then for any point p∈M there exists an integer k such that any two germs at p of local biholomorphic mappings that send M into itself and whose k-jets agree at p are identical.The above is a special case of a more general theorem stated for formal hypersurfaces that gives a finite jet determination result for the class of formal mappings whose Jacobian determinant does not vanish identically.     

On the pseudohermitian sectional curvature of a strictly pseudoconvex CR manifold

We show that the pseudohermitian sectional curvature Hθ(σ) of a contact form θ on a strictly pseudoconvex CR manifold M measures the difference between the lengths of a circle in a plane tangent at a point of M and its projection on M by the exponential map associated to the Tanaka-Webster connection of (M,θ). Any Sasakian manifold (M,θ) whose pseudohermitian sectional curvature Kθ(σ) is a point function is shown to be Tanaka-Webster flat, and hence a Sasakian space form of φ-sectional curvature c=−3. We show that the Lie algebra i(M,θ) of all infinitesimal pseudohermitian transformations on a strictly pseudoconvex CR manifold M of CR dimension n has dimension ?²(n+1) and if dimRi(M,θ)=²(n+1) then Hθ(σ)= constant.     

Partial rigidity of degenerate CR embeddings into spheres

In this paper, we study degenerate CR embeddings f$f$ of a strictly pseudoconvex hypersurface M⊂Cⁿ⁺¹$M\subset {\mathbb{C}}^{n+1}$ into a sphere S$\mathbb{S}$ in a higher dimensional complex space C^N+1${\mathbb{C}}^{N+1}$. The degeneracy of the mapping f$f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings f$f$ into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained in a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank d$d$ of the second fundamental form and all of its covariant derivatives is <n$<n$ (here, n$n$ is the CR dimension of M$M$), then f(M)$f\left(M\right)$ is contained in a complex plane of dimension n+d+1$n+d+1$. The converse of this statement is also true, as is easy to see. When the total rank d$d$ exceeds n$n$, it is no longer true, in general, that f(M)$f\left(M\right)$ is contained in a complex plane of dimension n+d+1$n+d+1$, as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension n$n$, then partial rigidity may still persist, but there is a “defect” k$k$ that arises from the ranks exceeding n$n$ such that f(M)$f\left(M\right)$ is only contained in a complex plane of dimension n+d+k+1$n+d+k+1$. Moreover, this defect occurs in general, as is illustrated by examples.     

Lichnerowicz-Obata theorem for Kohn Laplacian on the real ellipsoid

We give the sharp lower bound for Ricci curvature on the real ellipsoid in Cⁿ⁺¹, and prove the Lichnerowicz-Obata theorem for Kohn Laplacian.     

Finite jet determination of CR embeddings

We prove finite jet determination results for smooth CR embeddings, which are of constant degeneracy, using the method of complete systems. As an application, we obtain a reflection principle for mappings between a Levinondegenerate hypersurface in ℂ ^N and a Levinondegenerate hypersurface in ℂ ^N+1.We also give an independent proof of the reflection principle for mappings between strictly pseudoconvex hypersurfaces in any codimension due to Forstneric [14].     

An algebraic formula for the intersection number of a polynomial immersion

An algorithm is presented for computing the topological degree for a large class of polynomial mappings. As an application there is given an effective algebraic formula for the intersection number of a polynomial immersion M→R^2m, where M is an m-dimensional algebraic manifold.     

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.     

Holomorphic extension of CR functions from quadratic cones

It is proved that CR functions on a quadratic cone M in , n > 1, admit one-sided holomorphic extension if and only if M does not have two-sided support, a geometric condition on M which generalizes minimality in the sense of Tumanov. A biholomorphic classification of quadratic cones in is also given.     

Monge-Ampère equations and moduli spaces of manifolds of circular type

A (bounded) manifold of circular type is a complex manifold M of dimension n admitting a (bounded) exhaustive real function u, defined on M minus a point xo, so that: (a) it is a smooth solution on M?{xo} to the Monge-Ampère equation n(ddcu)=0; (b) xo is a singular point for u of logarithmic type and eu extends smoothly on the blow up of M at xo; (c) ddc(eu)>0 at any point of M?{xo}. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of Cn.A set of modular parameters for bounded manifolds of circular type is considered. In particular, for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form. It is also shown that the class of normalizing maps for an n-dimensional manifold M is a new holomorphic invariant with the following property: it is parameterized by the points of a finite dimensional real manifold of dimension n² when M is a (non-convex) circular domain while it is of dimension n²+2n when M is a strictly linearly convex domain. New characterizations of the circular domains and of the unit ball are also obtained.     

Extension of C R Structures on three dimensional compact pseudoconvex C R manifolds

Let M be a smooth compact orientable pseudoconvex C R manifold of real dimension three and assume that there is a smooth function λ which is strictly subharmonic in any direction where the Levi-form vanishes on M. Then we extend the given C R structure on M to an integrable almost complex structure on the concave side of M. As an application, if M is a non-compact pseudoconvex C R manifold of real dimension three, we prove that the given C R structure on M can be locally extended to an integrable almost complex structure on the concave side of M. Partially supported by Korea research foundation Grant (KRF-2001-015-DP0016) and by R14-2002-044-01000-0 from KRF     

Stabilizers and orbits of smooth functions

Let be a smooth function such that f(0)=0. We give a condition J(id) on f when for arbitrary preserving orientation diffeomorphism such that ?(0)=0 the function ?○f is right equivalent to f, i.e. there exists a diffeomorphism such that ?○f=f○h at 0∈Rm. The requirement is that f belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated, simple, and many other singularities.We also globalize this result as follows. Let M be a smooth compact manifold, a surjective smooth function, DM the group of diffeomorphisms of M, and the group of diffeomorphisms of R that have compact support and leave [0,1] invariant. There are two natural right and left-right actions of DM and on C^∞(M,R). Let SM(f), SMR(f), OM(f), and OMR(f) be the corresponding stabilizers and orbits of f with respect to these actions. We prove that if f satisfies J(id) at each critical point and has additional mild properties, then the following homotopy equivalences hold: SM(f)≈SMR(f) and OM(f)≈OMR(f). Similar results are obtained for smooth mappings M→S¹.     

Remarks on the generic rank of a CR mapping

We study germs of smooth CR mappings between embedded real hypersurfaces in complex spaces of the same dimension. In particular, we are interested in the generic rank of such mappings. IfH:M →M′ is a CR map between two hypersurfacesM andM′, we prove that ifM′ does not contain any germ of a holomorphic manifold then eitherH is constant or the generic rank ofH is odd. We also prove that if there is no formal holomorphic vector field tangent toM, then eitherH is constant or genericallyH is a local diffeomorphism. It follows, as a special case, that ifM andM′ are of D-finite type (in the sense of D’Angelo) thenH is either constant or is generically a local diffeomorphism. Supported by NSF Grant DMS 8901268.     

Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics

Using Green’s hyperplane restriction theorem, we prove that the rank of a Hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a rigidity theorem for CR mappings between hyperquadrics in the spirit of the results of Baouendi–Huang and Baouendi–Ebenfelt–Huang. Given a real-analytic CR mapping of a hyperquadric (not equivalent to a sphere) to another hyperquadric $Q(A,B)$ , either the image of the mapping is contained in a complex affine subspace, or $A$ is bounded by a constant depending only on $B$ . Finally, we prove a stability result about existence of nontrivial CR mappings of hyperquadrics. That is, as long as both $A$ and $B$ are sufficiently large and comparable, then there exist CR mappings whose image is not contained in a hyperplane. The rigidity result also extends when mapping to hyperquadrics in infinite dimensional Hilbert-space.     

On the analyticity of CR mappings between nonminimal hypersurfaces

Let be a connected real-analytic hypersurface containing a connected complex hypersurface , and let be a smooth CR mapping sending M into another real-analytic hypersurface . In this paper, we prove that if f does not collapse E to a point and does not collapse M into the image of E, and if the Levi form of M vanishes to first order along E, then f is real-analytic in a neighborhood of E. In general, the corresponding statement is false if the Levi form of M vanishes to second order or higher, in view of an example due to the author. We also show analogous results in higher dimensions provided that the target M' satisfies a certain nondegeneracy condition. The main ingredient in the proof, which seems to be of independent interest, is the prolongation of the system defining a CR mapping sending M into M' to a Pfaffian system on M with singularities along E. The nature of the singularity is described by the order of vanishing of the Levi form along E. Received: 12 February 2001 / Published online: 18 January 2002     

Generalized Dirichlet to Neumann operator on invariant differential forms and equivariant cohomology

In recent work, Belishev and Sharafutdinov show that the generalized Dirichlet to Neumann (DN) operator Λ on a compact Riemannian manifold M with boundary ∂M determines de Rham cohomology groups of M. In this paper, we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and the corresponding vector field XM on M, Witten defines an inhomogeneous coboundary operator dXM=d+ιXM on invariant forms on M. The main purpose is to adapt Belishev-Sharafutdinov?s boundary data to invariant forms in terms of the operator dXM in order to investigate to what extent the equivariant topology of a manifold is determined by the corresponding variant of the DN map. We define an operator ΛXM on invariant forms on the boundary which we call the XM-DN map and using this we recover the XM-cohomology groups from the generalized boundary data (∂M,ΛXM). This shows that for a Zariski-open subset of the Lie algebra, ΛXM determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the ring structure of XM-cohomology groups from ΛXM. These results explain to what extent the equivariant topology of the manifold in question is determined by ΛXM.     

Weak compactness and fixed point property for affine mappings

It is shown that a closed convex bounded subset of a Banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings. The class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly Lipschitzian with some constant M>1 in the case of c₀, the class of affine mappings which are uniformly Lipschitzian with some constant in the case of quasi-reflexive James’ space J and the class of nonexpansive affine mappings in the case of L-embedded spaces.     

Invariant holomorphic mappings

Given a real-analytic hypersurface invariant under a finite unitary group, we construct an invariant holomorphic mapping to a hyperquadric, and prove the basic properties of this mapping. When the hypersurface is the unit sphere, the groups are cyclic, and the quotient is a Lens space, we prove that the coefficients of this mapping must be square roots of integers. For the Lens spacesL(p, p - 1) we evaluate these integers by some combinatorial reasoning. We indicate how these calculations bear on a conjecture about the multiplicity of proper mappings between balls in different dimensions.