arXiv subjects

Xianghong Gong

Publications and source records attributed to Xianghong Gong.

31 records · Page 2Linked to original sources

arXiv2014

Real submanifolds of maximum complex tangent space at a CR singular point

We study a germ of real analytic $n$-dimensional submanifold of ${\mathbf C}^n$ 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 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. If the singularity is {\it abelian}, we show, under some assumptions on the linear part of $σ$ at the singularity, that the real submanifold is holomorphically equivalent to an analytic normal form. 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 prove that, in general, there exists a complex submanifold of positive dimension in ${\mathbf C}^n$ that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a CR singularity of the {\it complex type}.

math.CV↗

arXiv2011

Dirichlet and Neumann problems for planar domains with parameter

Let $Γ(\cdot,λ)$ be smooth, i.e.\, $\mathcal C^\infty$, embeddings from $\barΩ$ onto $\bar{Ω^λ}$, where $Ω$ and $Ω^λ$ are bounded domains with smooth boundary in the complex plane and $λ$ varies in $I=[0,1]$. Suppose that $Γ$ is smooth on $\barΩ\times I$ and $f$ is a smooth function on $\partialΩ\times I$. Let $u(\cdot,λ)$ be the harmonic functions on $Ω^λ$ with boundary values $f(\cdot,λ)$. We show that $u(Γ(z,λ),λ)$ is smooth on $\barΩ\times I$. Our main result is proved for suitable Hölder spaces for the Dirichlet and Neumann problems with parameter. By observing that the regularity of solutions of the two problems with parameter is not local, we show the existence of smooth embeddings $Γ(\cdot,λ)$ from $\bar{\mathbb D}$, the closure of the unit disc, onto $\bar{Ω^λ}$ such that $Γ$ is smooth on $\bar{\mathbb D}\times I$ and real analytic at $(\sqrt{-1},0)\in\bar{\mathbb D}\times I$, but for every family of Riemann mappings $R(\cdot,λ)$ from $\bar{Ω^λ}$ onto $\bar{\mathbb D}$, the function $R(Γ(z,λ),λ)$ is not real analytic at $(\sqrt{-1},0)\in\bar{\mathbb D}\times I$.

math.CV↗

arXiv2010

Common boundary values of holomorphic functions for two-sided complex structures

Let $Ω_1,Ω_2$ be two disjoint open sets in $\mathbf C^n$ whose boundaries share a smooth real hypersurface $M$ as relatively open subsets. Assume that $Ω_i$ is equipped with a complex structure $J^i$ which is smooth up to $M$. Assume that the operator norm $\|J^2-J^1\|<2$ on $M$. Let $f$ be a continuous function on the union of $Ω_1,Ω_2, M$. If $f$ is holomorphic with respect to both structures in the open sets, then $f$ must be smooth on the union of $Ω_1$ with $M$. Although the result as stated is far more meaningful for integrable structures, our methods make it much more natural to deal with the general almost complex structures without the integrability condition. The result is therefore proved in the framework of almost complex structures.

math.CV↗

arXiv2009

Regularity for The CR Vector Bundle Problem I

We give a new solution to the local integrability problem for CR vector bundles over strictly pseudoconvex real hypersurfaces of dimension seven or greater. It is based on a KAM rapid convergence argument and avoids the previous more difficult Nash-Moser methods. The solution is sharp as to Hölder continuity.

math.CV↗

arXiv2009

Regularity for the CR vector bundle problem II

We derive a $\mathcal C^{k+\yt}$ Hölder estimate for $Pϕ$, where $P$ is either of the two solution operators in Henkin's local homotopy formula for $\bar\partial_b$ on a strongly pseudoconvex real hypersurface $M$ in $\mathbf C^{n}$, $ϕ$ is a $(0,q)$-form of class $\mathcal C^{k}$ on $M$, and $k\geq0$ is an integer. We also derive a $\mathcal C^{a}$ estimate for $Pϕ$, when $ϕ$ is of class $\mathcal C^{a}$ and $a\geq0$ is a real number. These estimates require that $M$ be of class $\mathcal C^{k+{5/2}}$, or $\mathcal C^{a+2}$, respectively. The explicit bounds for the constants occurring in these estimates also considerably improve previously known such results. These estimates are then applied to the integrability problem for CR vector bundles to gain improved regularity. They also constitute a major ingredient in a forthcoming work of the authors on the local CR embedding problem.

math.CV↗

arXiv2009

Regularity in the local CR embedding problem

We consider a formally integrable, strictly pseudoconvex CR manifold $M$ of hypersurface type, of dimension $2n-1\geq7$. Local CR, i.e. holomorphic, embeddings of $M$ are known to exist from the works of Kuranishi and Akahori. We address the problem of regularity of the embedding in standard Hölder spaces $C^{a}(M)$, $a\in\mathbf{R}$. If the structure of $M$ is of class $C^{m}$, $m\in\mathbf{Z}$, $4\leq m\leq\infty$, we construct a local CR embedding near each point of $M$. This embedding is of class $C^{a}$, for every $a$, $0\leq a < m+(1/2)$. Our method is based on Henkin's local homotopy formula for the embedded case, some very precise estimates for the solution operators in it, and a substantial modification of a previous Nash-Moser argument due to the second author.

math.CV↗

arXiv2007

Differential inequalities of continuous functions and removing singularities of Rado type for J-holomorphic maps

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable singularity theorem of Rado type for J-holomorphic maps. Let $Ω$ be an open subset in $\mathbf C$ and let $E$ be a closed polar subset of $Ω$. Let $u$ be a continuous map from $Ω$ into an almost complex manifold $(M,J)$ with $J$ of class $C^1$. We show that if $u$ is J-holomorphic on $Ω\setminus E$ then it is J-holomorphic on $Ω$.

math.CV↗

arXiv2003

Existence of divergent Birkhoff normal forms of Hamiltonian functions

We show that for $n \geq 2$ there exist real analytic Hamiltonian systems on $\mathbf{R}^{2n}$ with non-resonant eigenvalues at a singular point, of which the Birkhoff normal form itself is divergent. The proof of the result is achieved by analyzing how the small-divisors enter the normal form.

math.DS↗

arXiv1996

On totally real spheres in complex space

We shall prove that there are totally real and real analytic embeddings of $S^k$ in $\cc^n$ which are not biholomorphically equivalent if $k\geq 5$ and $n=k+2[\frac{k-1}{4}]$. We also show that a smooth manifold $M$ admits a totally real immersion in $\cc^n$ with a trivial complex normal bundle if and only if the complexified tangent bundle of $M$ is trivial. The latter is proved by applying Gromov's weak homotopy equivalence principle for totally real immersions to Hirsch's transversal fields theory.

math.CV↗

arXiv1995

Fixed points of elliptic reversible transformations with integrals

We show that for a certain family of integrable reversible transformations, the curves of periodic points of a general transformation cross the level curves of its integrals. This leads to the divergence of the normal form for a general reversible transformation with integrals. We also study the integrable holomorphic reversible transformations coming from real analytic surfaces in C^2 with non-degenerate complex tangents. We show the existence of real analytic surfaces with hyperbolic complex tangents, which are contained in a real hyperplane, but cannot be transformed into the Moser-Webster normal form through any holomorphic transformation.

math.CV↗

arXiv1995

Divergence of the normalization for real Lagrangian surfaces near complex tangents

We study real Lagrangian analytic surfaces in C^2 with a non-degenerate complex tangent. Webster proved that all such surfaces can be transformed into a quadratic surface by formal symplectic transformations of C^2. We show that there is a certain dense set of real Lagrangian surfaces which cannot be transformed into the quadratic surface by any holomorphic (convergent) transformation of C^2. The divergence is contributed by the parabolic character of a pair of involutions generated by the real Lagrangian surfaces.

math.CV↗

arXiv1995

Integrable analytic vector fields with a nilpotent linear part

We study the normalization of integrable analytic vector fields with a nilpotent linear part. We prove that such an analytic vector field can be transformed into a certain form by convergent transformations when it has a non-singular formal integral. In particular, we show that a formally linearizable analytic vector field with a nilpotent linear part is linearizable by convergent transformations. We then prove that there are smoothly linearizable parabolic analytic transformations which cannot be embedded into the flow of any analytic vector field with a nilpotent linear part.

math.CV↗

arXiv1995

Unimodular invariants of totally real tori in C^n

We study the global invariants of real analytic manifolds in the complex space with respect to the group of holomorphic unimodular transformations. We consider only totally real manifolds which admits a certain fibration over the circle. We find a complete set of invariants for totally real tori in C^n which are close to the standard torus. The invariants are obtained by an analogous classification of complex-valued analytic $n$-forms on the standard torus. We also study the realization of certain exact complex-valued analytic $n$-forms on the standard torus through non-critical totally real embeddings.

math.CV↗