A survey on normal forms of real submanifolds with CR singularity
We survey results dating back from the seminal works of Bishop and Moser-Webster as well as more recent advances.
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arXiv subjects
Xianghong Gong
Publications and source records attributed to Xianghong Gong.
Integrability of Koszul connections on complex vector bundles over domains in ${\mathbf C}^n$
We study invertible matrix solutions $A$ to the equation $A^{-1}\overline\partial A=ω^{(0,1)}$ on a small open subset $U$ of the closure $\overline M$ of a domain $M\subset{\mathbf C}^n$, where $ω^{(0,1)}$ is a matrix of $(0,1)$ forms on $\overline M$ satisfying the formal integrable condition $\overline{\partial}ω^{(0,1)}=ω^{(0,1)}\wedgeω^{(0,1)}$. For a $C^2$ domain $M$ that is either strongly pseudoconvex or has at least $3$ negative Levi eigenvalues at a boundary point contained in $U$, we obtain existence and sharp regularity of the solutions.
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A homotopy formula for $a_q$ domains in complex manifolds
We construct a global homotopy formula for $a_q$ domains in a complex manifold. The homotopy operators in the formula will gain $1/2$ derivative in Hölder-Zygmund spaces $Λ^{r}$ when the boundaries of the domains are in $Λ^{r+3}$ with $r>0$.
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Global Newlander-Nirenberg theorem on domains with finite smooth boundary in complex manifolds
Let $M$ be a relatively compact $C^2$ domain in a complex manifold $\mathcal M$ of dimension $n$. Assume that $H^{1}(M,Θ)=0$ where $Θ$ is the sheaf of germs of holomorphic tangent fields of $M$. Suppose that the Levi-form of the boundary of $M$ has at least 3 negative eigenvalues or at least $n-1$ positive eigenvalues pointwise. We first construct a homotopy formula for $Θ$-valued $(0,1)$-forms on $\overline M$. We then apply a Nash-Moser iteration scheme to show that if a formally integrable almost complex structure of the Hölder-Zygmund class $Λ^r$ on $\overline M$ is sufficiently close to the complex structure on $ M$ in the Hölder-Zygmund norm $Λ^{r_0}(\overline M)$ for some $r_0>5/2$, then there is a diffeomorphism $F$ from $\overline M$ into $\mathcal M$ that transforms the almost complex structure into the complex structure on $F(M)$, where $F \in Λ^s(M)$ for all $s<r+1/2$.
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On regularity of $\overline\partial$-solutions on $a_q$ domains with $C^2$ boundary in complex manifolds
We study regularity of solutions $u$ to $\overline\partial u=f$ on a relatively compact $C^2$ domain $D$ in a complex manifold of dimension $n$, where $f$ is a $(0,q)$ form. Assume that there are either $(q+1)$ negative or $(n-q)$ positive Levi eigenvalues at each point of boundary $\partial D$. Under the necessary condition that a locally $L^2$ solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain $1/2$ derivative when $q=1$ and $f$ is in the Hölder-Zygmund space $Λ^r(\overline D)$ with $r>1$. For $q>1$, the same regularity for the solutions is achieved when $\partial D$ is either sufficiently smooth or of $(n-q)$ positive Levi eigenvalues everywhere on $\partial D$.
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Smooth equivalence of families of strongly pseudoconvex domains
We establish a smoothness result for families of biholomorphisms between smooth families of strongly pseudoconvex domains, each with trivial biholomorphism group. This is accomplished by considering the Riemannian geometry of their Bergman metrics and proving a result about the smoothness of families of isometries between smooth families of Riemannian manifolds.
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A structure theorem for neighborhoods of compact complex manifolds
We construct an injective map from the set of holomorphic equivalence classes of neighborhoods $M$ of a compact complex manifold $C$ into ${\mathbb C}^m$ for some $m<\infty$ when $(TM)|_C$ is fixed and the normal bundle of $C$ in $M$ is either weakly negative or $2$-positive.
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On neighborhoods of embedded complex tori
The goal of the article is to show that an n-dimensional complex torus embedded in a complex manifold of dimensional n+d, with a split tangent bundle, has neighborhood biholomorphic a neighborhood of the zero section in its normal bundle, provided the latter has (locally constant) Hermitian transition functions and satisfies a non-resonant Diophantine condition.
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Regularity of a $\bar\partial$-solution operator for strongly $\mathbf C$-linearly convex domains with minimal smoothness
We prove regularity of solutions of the $\bar\partial$-problem in the Hölder-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains which is of independent interest, and on techniques that were recently developed by the first-named author to prove estimates for the $\bar\partial$-problem on strongly pseudoconvex domains of class $C^2$.
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Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations
We consider an embedded $n$-dimensional compact complex manifold in $n+d$ dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert's formal principle program. We will give conditions ensuring that a neighborhood of $C_n$ in $M_{n+d}$ is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold's result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in $M_{n+d }$ having $C_n$ as a compact leaf, extending Ueda's theory to the high codimension case. Both problems appear as a kind linearization problem involving small divisors condition arising from solutions to their cohomological equations.
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Global Newlander-Nirenberg theorem for domains with $C^2$ boundary
The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander-Nirenberg theorem under the presence of a $C^2$ strictly pseudoconvex boundary. When a given formally integrable complex structure $X$ is defined on the closure of a bounded strictly pseudoconvex domain with $C^2$ boundary $D\subset \mathbb{C}^n$, we show the existence of global holomorphic coordinate systems defined on $\overline{D}$ that transform $X$ into the standard complex structure provided that $X$ is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small $C^2$ perturbation of $\partial D$. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a $C^2$ real hypersurface $M\subset \mathbb{C}^n$, we prove the existence of local one-sided holomorphic coordinate systems provided that $M$ is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.
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Hölder estimates for homotopy operators on strictly pseudoconvex domains with $C^2$ boundary
We derive a new homotopy formula for a strictly pseudoconvex domain of $C^2$ boundary in ${\mathbf C}^n$ by using a method of Lieb and Range and obtain estimates in Lipschitz spaces for the homotopy operators. For $r>1$ and $q>0$, we obtain a $Λ_{r+{1}/{2}}$ solution $u$ to $\overline\partial u=f$ for $\overline\partial$-closed $(0,q)$ forms $f$ of class $Λ_{r}$ on the domain. We apply the estimates to obtain boundary regularities of $\mathcal D$-solutions for a domain in the Levi-flat Euclidean space.
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A Frobenius-Nirenberg theorem with parameter
The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander-Nirenberg theorem with parameter. The first extends the Newlander-Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster's proof for the Newlander-Nirenberg theorem. The second concerns a version of Nirenberg's complex Frobenius theorem and its proof yields a result with a mild loss of regularity.
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The $\overline{\partial}$-equation on variable strictly pseudoconvex domains
We investigate regularity properties of the $\overline{\partial}$-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous family of pseudoconvex domains. The boundary regularity and the regularity in the parameter are also obtained for smoothly bounded strongly pseudoconvex domains.
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Smooth equivalence of deformations of domains in complex euclidean spaces
We prove that two smooth families of 2-connected domains in $\cc$ are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct, for $m \geq 3$, two smooth families of smoothly bounded $m$-connected domains in $\cc$, and for $n\geq2$, two families of strictly pseudoconvex domains in $\cc^n$, that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains.
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Real submanifolds of maximum complex tangent space at a CR singular point, II
We study a germ of real analytic n-dimensional submanifold of $C^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under the condition that its complexification admits the maximum number of deck transformations, we first classify holomorphically its quadratic CR singularity. We then study its transformation to a normal form under the action of local (possibly formal) biholomorphisms at the singularity. We first conjugate formally its associated reversible map $σ$ to suitable normal forms and show that all these normal forms can be divergent. We then construct a unique formal normal form under a non degeneracy condition.
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Real submanifolds of maximum complex tangent space at a CR singular point, I
We study a germ of real analytic n-dimensional submanifold of C n that has a complex tangent space of maximal dimension at a CR singularity. Under some assumptions , we show its equivalence to a normal form under a local biholomorphism at the singularity. We also show that if a real submanifold is formally equivalent to a quadric, it is actually holomorphically equivalent to it, if a small divisors condition is satisfied. Finally, we investigate the existence of a complex submanifold of positive dimension in C n that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a possibly non-isolated CR singularity.
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Normal forms for CR singular codimension two Levi-flat submanifolds
Real-analytic Levi-flat codimension two CR singular submanifolds are a natural generalization to ${\mathbb{C}}^m$, $m > 2$, of Bishop surfaces in ${\mathbb{C}}^2$. Such submanifolds for example arise as zero sets of mixed-holomorphic equations with one variable antiholomorphic. We classify the codimension two Levi-flat CR singular quadrics, and we notice that new types of submanifolds arise in dimension 3 or greater. In fact, the nondegenerate submanifolds, i.e. higher order purturbations of $z_m=\bar{z}_1z_2+\bar{z}_1^2$, have no analogue in dimension 2. We prove that the Levi-foliation extends through the singularity in the real-analytic nondegenerate case. Furthermore, we prove that the quadric is a (convergent) normal form for a natural large class of such submanifolds, and we compute its automorphism group. In general, we find a formal normal form in ${\mathbb{C}}^3$ in the nondegenerate case that shows infinitely many formal invariants.
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