Valentin Burcea
We construct a family of analytic discs attached to a real submanifold M \subset $\mathbb{C}^{N+1}$ of codimension $2$ defined near a CR singularity.
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We prove that pseudo-holomorphic discs attached to a maximal totally real submanifold inherit their regularity from the regularity of the submanifold and of the almost complex structure. The proof is based on the computation of an explicit lower bound for the Kobayashi metric in almost complex manifolds, which also yields explicit estimates of Hölderian norms of such discs.
Let E be a generic real submanifold of an almost complex manifold. The geometry of Bishop discs attached to E is studied in terms of the Levi form of E.
Generalized stationary discs attached to degenerate submanifolds in $\mathbb{C}^N$
本文研究了退化子流形在复空间中的广义静止盘家族，证明其在适当条件下构成Banach空间中的有限维实子流形。
The existence of a nondefective stationary disc attached to a nondegenerate model quadric in C^N is a necessary condition to ensure the unique 1-jet determination of the lifts of a key family of stationary discs. In this paper, we give an elementary proof of the equivalence when the model quadric is strongly pseudoconvex, recovering a result of Tumanov. Our proof is based on the explicit expression of stationary discs, and opens up a conjecture for the unique 1-jet determination to hold when the model is not necessarily strongly pseudoconvex.
Let S be an arbitrary real surface, with or without boundary, contained in a hypersurface M of the complex euclidean space \C^2, with S and M of class C^{2, a}, where 0 < a < 1. If M is globally minimal, if S is totally real except at finitely many complex tangencies which are hyperbolic in the sense of E. Bishop and if the union of separatrices is a tree of curves without cycles, we show that every compact K of S is CR-, W- and L^p-removable (Theorem~1.3). We treat this seemingly global problem by means of purely local techniques, namely by means of families of small analytic discs partially …
Let $Ω$ be a smooth real analytic submanifold of a complex manifold $X$. We establish and study the link between the following 3 subjects: 1) topological properties of smooth families of attached analytic discs, the manifold $Ω$ admits, 2) lower bounds for dimensions of complex tangent spaces of $Ω$, 3) a generalization of the argument principle for smooth families of holomorphic mappings from the standard complex disc to $X$. In particular, we obtain characterization of complex manifolds and their boundaries in terms of attached analytic discs. The special case when $Ω$ is the graph, leads to…
We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further study the embeddability problem for real-analytic submanifolds of higher codimension and answer a question by Forstnerič.
We construct examples of $C^\infty$ smooth submanifolds in ${\Bbb C}^n$ and ${\Bbb R}^n$ of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianalytic functions. In the complex case, these submanifolds contain real $n$-dimensional tori or Euclidean spaces that are not pluripolar while the intersection with any complex analytic disk is polar.
We establish the Schwarz Reflection Principle for $J$-complex discs attached to a real analytic $J$-totally real submanifold of an almost complex manifold with real analytic $J$. We also prove the precise boundary regularity and derive the precise convergence in Gromov compactness theorem in $\calc^{k,α}$-classes.
We give an explicit construction of a key family of stationary discs attached to a nondegenerate model quadric in $\mathbb{C}^N$ and derive a necessary condition for which (each lift) of those stationary discs is uniquely determined by its $1$-jet at a given point via a local diffeomorphism. This unique $1$-jet determination is a crucial step to deduce $2$-jet determination for CR automorphisms of generic real submanifolds in $\mathbb{C}^N$.
We study boundary properties of plurisubharmonic functions near real submanifolds of almost complex manifolds.
In case M is Levi non-degenerate in the sense Tumanov, we construct stationary discs for $M$. If furthermore M satisfies an additional non-degeneracy condition, we apply the method of stationary discs to obtain 2-jet determination of CR automorphisms of M.
On a real analytic 5-dimensional CR-generic submanifold M^5 in C^4 of codimension 3, hence of CR dimension 1, which enjoys the generically satisfied nondegeneracy condition that Lie brackets up to length 3 of T^{1,0}M generate CTM, a canonical Cartan connection is constructed after reduction to a certain partially explicit e-structure of the concerned local biholomorphic equivalence problem. More advanced explorations of the incoming differential invariants due to the first and to the third authors already appeared in January 2014, hence the purpose is to show, while studying this specific Cla…
We prove that a germ of a holomorphic map $f$ between $C^n$ and $C^{n'}$ sending one real-algebraic submanifold $M\subset C^n$ into another $M'\subset C^{n'}$ is algebraic provided $M'$ contains no complex-analytic discs and $M$ is generic and minimal. We also propose an algorithm for finding complex-analytic discs in a real submanifold.
Construction of stationary discs for perturbations of decoupled submanifolds in $\mathbb{C}^4$
本文研究了 $\mathbb{C}^4$ 中余维数为 $2$ 的解耦实子流形的扰动问题，通过构造广义平稳圆盘来提供几何与分析工具。
We prove that generic homologically nontrivial $(2n-1)$-parameter family of analytic discs attached by their boundaries to a CR manifold $Ω$ in $\mathbb C^n, n \le 2$ tests CR functions: if a smooth function on $Ω$ extends analytically inside each analytic disc then it satisfies the tangential CR equations. In particular, we answer, in real analytic category, two open questions: on characterization of analytic functions in planar domains (the strip-problem), and on characterization of boundary values of holomorphic functions in domains in $\mathbb C^n$ (a conjecture of Globevnik and Stout). We…
We show --- with the means of a real-analytic example in $\mathbb{C}^3$ --- that Gromov's theorem on the presence of attached holomorphic discs for compact Lagrangian manifolds is not true in the isotropic (subcritical) case, even in the absence of an obvious obstruction, i.e, polynomial convexity.
Solutions of Pascali systems attached to convex boundaries
针对复平面内有界严格凸区域及指定内点，构造以该点为中心的 Pascali 型椭圆方程组连续解，实现单位圆盘到区域、单位圆周到边界的映射。
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