Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

제6차 가정과 교육과정운영은 어떻게 할 것인가

On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables

We prove that if $M$ and $M'$ are algebraic hypersurfaces in $ C^ N$, i.e. both defined by the vanishing of real polynomials, then any sufficiently smooth CR mapping with Jacobian not identically zero extends holomorphically provided the hypersurfaces are holomorphically nondegenerate . Conversely, we prove that holomorphic nondegeneracy is necessary for this property of CR mappings to hold. For the case of unequal dimensions, we also prove that if $M$ is an algebraic hypersurface in $ C^N$ which does not contain any complex variety of positive codimension and $M'$ is the sphere in $ C^{N+1 }$ , then extendability holds for all CR mappings with certain minimal a priori regularity. 
Theorem A. Let $M$ and $M'$ be two algebraic hypersurfaces in $C^N$ and assume that $M$ is connected and holomorphically nondegenerate. If $H$ is a smooth CR mapping from $M$ to $M'$ with $ Jac H \not\equiv 0$, where $Jac H$ is the Jacobian determinant of $H$, then $H$ extends holomorphically in an open neighborhood of $M$ in $ C^N$. 
A recent example given by Ebenfelt shows that the conclusion of Theorem A need not hold if $M$ is real analytic, but not algebraic. 
Theorem B. Let $M$ be a connected real analytic hypersurface in $C^N$ which is holomorphically degenerate at some point $p_1$. Let $p_0 \in M$ and suppose there exists a germ at $p_0$ of a smooth CR function on $M$ which does not extend holomorphically to any full neighborhood of $p_0$ in $C^N$. Then there exists a germ at $p_0$ of a smooth CR diffeomorphism from $M$ into itself, fixing $p_0$, which does not extend holomorphically to any neighborhood of $p_0$ in $C^N$. 
Theorem C. Let $M\subset C^N$ be an algebraic hypersurface. Assume that there is no nontrivial complex analytic variety contained in $M$ through $p_0 \in M$, and let $m=m_{p_0}$ be the D'Angelo type. If $H: M \to S^{2N+1}\subset C^{N+1}$ is a CR map of class $C^m$, where $S^{2N+1}$ denotes the boundary of the unit ball in $C^{N+1 }$, then $H$ admits a holomorphic extension in a neighborhood of $p_0$.

Regularity of CR mappings between algebraic hypersurfaces

Théorèmes de finitude pour la cohomologie des espaces complexes

We study local holomorphic mappings sending a piece of a real hyperquadric in a complex space into a hyperquadric in another complex space of possibly larger dimension We show that these mappings possess strong super-rigidity properties when the hyperquadrics have positive signatures These results are applied in the context of holomorphic mappings between classical domains in complex projective spaces of different dimensions

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Super-rigidity for holomorphic mappings between hyperquadrics with positive signature

Mapping Bn into B2n-1

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2006/0013/0004/MRL-2006-0013-0004-a002.pdf

A new gap phenomenon for proper holomorphic mappings from ${\BB}^n$ into ${\BB}^{N}$

We consider local CR-immersions of a strictly pseudoconvex real hypersurface $M\subset\bC^{n+1}$, near a point $p\in M$, into the unit sphere $\mathbb S\subset\bC^{n+d+1}$ with $d>0$. Our main result is that if there is such an immersion $f\colon (M,p)\to \mathbb S$ and $d < n/2$, then $f$ is {\em rigid} in the sense that any other immersion of $(M,p)$ into $\mathbb S$ is of the form $\phi\circ f$, where $\phi$ is a biholomorphic automorphism of the unit ball $\mathbb B\subset\bC^{n+d+1}$. As an application of this result, we show that an isolated singularity of an irreducible analytic variety of codimension $d$ in $\bC^{n+d+1}$ is uniquely determined up to affine linear transformations by the local CR geometry at a point of its Milnor link.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2004/0012/0003/CAG-2004-0012-0003-a006.pdf

Rigidity of CR-immersions into spheres

We consider the class of Levi nondegenerate hypersurfaces M in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] that admit a local (CR transversal) embedding, near a point p e M , into a standard nondegenerate hyperquadric in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] with codimension k := N - n small compared to the CR dimension n of M . We show that, for hypersurfaces in this class, there is a normal form (which is closely related to the embedding) such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric. We also show that if the signature of M and that of the standard hyperquadric in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] are the same, then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric.

Xiaojun Huang

Papers

Super-rigidity for holomorphic mappings between hyperquadrics with positive signature

Mapping Bn into B2n-1

A new gap phenomenon for proper holomorphic mappings from ${\BB}^n$ into ${\BB}^{N}$

Rigidity of CR-immersions into spheres

The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics