On the Levi map of nondegenerate CR submanifolds
We investigate some of the properties of the Levi map and related objects for nondegenerate CR submanifolds in $\mathbb{C}^N$ of codimension d.
math.CV↗
arXiv subjects
Florian Bertrand
Publications and source records attributed to Florian Bertrand.
Generalized stationary discs attached to degenerate submanifolds in $\mathbb{C}^N$
We study the family of generalized stationary discs attached to a Levi degenerate submanifold M of codimension d in $\mathbb{C}^{n+d}$. We show, under suitable geometric assumptions on M, that this family forms a finite dimensional real submanifold of the Banach space of analytic discs.
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Group-Invariant CR maps of the sphere $\mathbb{S}^5$
We construct group-invariant CR maps from the unit sphere in $\mathbb{C}^3$ and provide sharp bounds for the gap termination in this setting.
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Construction of stationary discs for perturbations of decoupled submanifolds in $\mathbb{C}^4$
We construct generalized stationary discs to perturbations of decoupled real submanifolds of codimension $2$ in $\mathbb{C}^4$.
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New examples of CR submanifolds for the $2$-jet determination problem
This paper addresses two questions related to mapping problems. In the first part of the paper, we discuss some recent results regarding the $2$-jet determination for biholomorphisms between smooth (weakly) pseudoconvex hypersurfaces; in light of those, we formulate a general problem and illustrate it with examples. In the second part of the paper, we provide a new example of a real submanifold of codimension $7$ in $\mathbb{C}^{10}$ that is not strictly pseudoconvex and for which its germs of CR automorphisms are determined by their $2$-jets at a given point.
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A sphericity criterion for strictly pseudoconvex hyper surfaces in $\mathbb{C}^2$ via invariant curves
We prove that if every chain on a strictly pseudoconvex hypersurface $M$ in $\mathbb{C}^2$ coincides with the boundary of a stationary disc, then $M$ is locally spherical.
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On the envelope of Poisson functional on almost complex manifolds
We establish the plurisubharmonicity of the envelope of the Poisson functional on almost complex manifolds. That is, we generalize the corresponding result for complex manifolds and almost complex manifolds of complex dimension two.
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On the higher order Kobayashi pseudometric
We study the higher order Kobayashi pseudometric introduced by Yu. We first obtain estimates of this pseudometric in a special pseudoconvex domain in $\C^3$. We then study the structure of the higher order extremal discs and their connection with the standard extremal discs for the Kobayashi metric.
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The 1-jet determination of stationary discs attached to generic CR submanifolds
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.
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Explicit construction of stationary discs and its consequences for nondegenerate quadrics
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 $\mathbb{C}^N$.
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Nondefective stationary discs and 2-jet determination in higher codimension
We discuss the links between stationary discs, the defect of analytic discs, and 2-jet determination of CR automorphisms of generic nondegenerate real submanifolds of C^N of class C^4.
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Extremal discs and Segre varieties for real-analytic hypersurfaces in $\mathbb{C}^2$
We show that if the Segre varieties of a strictly pseudoconvex hypersurface in $\mathbb{C}^2$ are extremal discs for the Kobayashi metric, then that hypersurface has to be locally spherical. In particular, this gives yet another characterization of the unit sphere in terms of two important invariant families of objects coinciding.
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Riemann-Hilbert problems with constraints
This paper is devoted to Riemann-Hilbert problems with constraints. We obtain results characterizing the existence of solutions as well as the dimension of the solution space in terms of certain indices. As an application, we show how such results may be used to construct analytic discs attached to singular manifolds.
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Jet determination of smooth CR automorphisms and generalized stationary discs
We prove finite jet determination for (finitely) smooth CR diffeomorphisms of (finitely) smooth Levi degenerate hypersurfaces in $\mathbb{C}^{n+1}$ by constructing generalized stationary discs glued to such hypersurfaces.
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Stationary discs and finite jet determination for non-degenerate generic real submanifolds
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.
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Extremal and stationary discs for the Kobayashi $k$-metric
We generalize Lempert's and Poletsky's works on the description of extremal discs for the Kobayashi metric to a higher order setting.
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Local approximation of non-holomorphic discs in almost complex manifolds
We provide a local approximation result of non-holomorphic discs with small d-bar by pseudoholomorphic ones. As an application, we provide a certain gluing construction.
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Invariant holomorphic discs in some non-convex domains
We give a description of complex geodesics and we study the structure of stationary discs in some non-convex domains for which complex geodesics are not unique.
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