The Hartogs phenomenon on weakly pseudoconcave hypersurfaces

In 2002, Henkin and Michel proved a local Hartogs phenomenon for real analytic CR functions on real analytic weakly pseudoconcave CR manifolds. The aim of the present article is to remove the assumptions on real analyticity in the case of weakly pseudoconcave hypersurfaces ${M\subset\mathbb{C}^n}$ . If M is a graph of class ${\mathcal{C}^2}$ and n??? 3, a global theorem is proved for the extension of holomorphic germs along M. If the appearing domains have nicely shaped boundary, a Hartogs theorem even holds for continuous CR functions, where the difference to the case of holomorphic germs relies on the possible presence of lower-dimensional CR orbits. Levi flat hypersurfaces in ${\mathbb{C}^2}$ require a separate treatment. Here an affirmative answer is given to the question of Tomassini, whether 2-spheres bound 3-balls in M.     

Holomorphic Deformations of Real-Analytic CR Maps and Analytic Regularity of CR Mappings

Let \(M\subset {\mathbb {C}}^N\) and \(M'\subset {\mathbb {C}}^{N'}\) be real-analytic CR submanifolds, with M minimal. We provide a new sufficient condition, that happens to be also essentially necessary, for all sufficiently smooth CR maps \(h:U\rightarrow M'\) defined on a connected open subset of M and of rank larger than a prescribed integer r to be real-analytic on a dense open subset of U. This condition corresponds to the nonexistence of nontrivial holomorphic deformations of germs of real-analytic CR mappings whose rank is larger than r. As a consequence, we obtain several new results about analyticity of CR mappings that, at the same time, generalize and unify a number of previous existing ones.     

Formal and finite order equivalences

We show that two families of germs of real-analytic subsets in ${{\mathbb C}^{n}}$ are formally equivalent if and only if they are equivalent of any finite order. We further apply the same technique to obtain analogous statements for equivalences of real-analytic self-maps and vector fields under conjugations. On the other hand, we provide an example of two sets of germs of smooth curves that are equivalent of any finite order but not formally equivalent.     

Equisingularity of families of map germs between curves

Given a finite map germ f : (X, 0) → (Y, 0) between complex analytic reduced space curves, we look at invariants which control the topological triviality and the Whitney equisingularity in families of this type of map germs. In the case that (Y, 0) is smooth, the main invariant is the Milnor number of a function on a curve. We deduce some applications to the equisingularity of families of finitely determined map germs ${f : (\mathbb{C}^2, 0) \to (\mathbb{C}^2, 0)}$ and ${f : (\mathbb{C}^2, 0) \to (\mathbb{C}^3, 0)}$ .     

Estimates in Beurling-Helson type theorems: Multidimensional case

We consider the spaces A _p ( $\mathbb{T}^m $ ) of functions f on the m-dimensional torus $\mathbb{T}^m $ such that the sequence of Fourier coefficients $\hat f = \{ \hat f(k),k \in \mathbb{Z}^m \} $ belongs to l ^p (? ^m ), 1 ≤ p < 2. The norm on A _p ( $\mathbb{T}^m $ ) is defined by $\left\| f \right\|_{A_p (\mathbb{T}^m )} = \left\| {\hat f} \right\|_{l^p (\mathbb{Z}^m )} $ . We study the rate of growth of the norms $\left\| {e^{i\lambda \phi } } \right\|_{A_p (\mathbb{T}^m )} $ as |λ| → ∞, λ ∈ ?, for C ¹-smooth real functions φ on $\mathbb{T}^m $ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogs for the spaces A _p (? ^m ).     

New open-book decompositions in singularity theory

In this article, we study the topology of real analytic germs ${F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)}$ given by ${F(x,y,z)=\overline{xy}(x^p+y^q)+z^r}$ with ${p,q,r \in \mathbb{N}, p,q,r \geq 2}$ and (p, q)?=?1. Such a germ gives rise to a Milnor fibration ${\frac{F}{\mid F \mid}\colon \mathbb{S}^5\setminus L_F \to \mathbb{S}^1}$ . We describe the link L _F as a Seifert manifold and we show that in many cases the open-book decomposition of ${\mathbb{S}^5}$ given by the Milnor fibration of F cannot come from the Milnor fibration of a complex singularity in ${\mathbb{C}^3}$ .     

On maps between nonminimal hypersurfaces

We construct an analytic jet parametrization for nontrivial CR maps between real-analytic 1-nonminimal hypersurfaces in ${\mathbb{C}^2}$ . In particular, this gives a real-analytic structure on the set of all such maps, and shows that the automorphism groups of a 1-nonminimal hypersurface in ${\mathbb{C}^2}$ is a Lie group. Our results also hold for some classes of 1-nonminimal hypersurfaces in higher dimensions.     

Stationary holomorphic discs and finite jet determination problems

We construct a family of small analytic discs attached to Levi non-degenerate hypersurfaces in $\mathbb{C }^{n+1}$ , which is globally biholomorphically invariant. We then apply this technique to study unique determination problems along Levi non-degenerate hypersurfaces that are merely of class $\mathcal{C }^4$ . This method gives 2-jet determination results for germs of biholomorphisms, CR diffeomorphisms, as well as in the almost complex setting.     

Computing α-Invariants of Singular del Pezzo Surfaces

We prove a new local inequality for divisors on surfaces and utilize it to compute α-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$ , $\mathbb{A}_{2}$ , $\mathbb{A}_{3}$ , $\mathbb{A}_{4}$ , $\mathbb{A}_{5}$ , or $\mathbb{A}_{6}$ are Kähler-Einstein.     

Linearity and Second Fundamental Forms for Proper Holomorphic Maps from \mathbb{B}^{n+1} to \mathbb{B}^{4n-3}

For a holomorphic proper map F from the ball $\mathbb{B}^{n+1}$ into $\mathbb{B}^{N+1}$ that is C ³ smooth up to the boundary, the image $M=F(\partial\mathbb{B}^{n})$ is an immersed CR submanifold in the sphere $\partial \mathbb{B}^{N+1}$ on which some second fundamental forms II _M and $\mathit{II}^{CR}_{M}$ can be defined. It is shown that when 4??n+1<N+1??4n?3, F is linear fractional if and only if $\mathit{II}_{M} - \mathit{II}_{M}^{CR} \equiv 0$ .     

A class of subelliptic quasilinear equations

Suppose $\mathfrak {X} = \{X_1, X_2, \ldots,\,X_m\}$ is a system of real smooth vector fields on an open neighbourhood Ω of the closure of a bounded connected open set M in $\mathbb {R}^N$ satisfying the finite rank condition of Hörmander, namely the rank of the Lie algebra generated by $\mathfrak {X}$ under the usual bracket operation is a constant equal to N. We study the smoothness of solutions of a class of quasilinear equations of the form $$Q_{\mathfrak {X}}u = \sum _{j=1}^m X_j^*a_j(x, u, Xu) +b (x, u, Xu) = 0$$ where $a_j,\,b \in C^{\infty}(\Omega \times \mathbb {R} \times \mathbb {R}^m; \mathbb {R})$ . It is shown that if the matrix $\left({\frac {\partial a_j}{\partial \xi_i}}\right)$ is positive definite on $M \times \mathbb {R}^{m+1}$ then any weak solution $u \in \mathcal {C}^{2,\alpha}(M, \mathfrak {X})$ belongs to C ^∞(M).     

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities

We derive the inequality $$\int_\mathbb{R}M(|f'(x)|h(f(x))) dx\leq C(M,h)\int_\mathbb{R}M\left({\sqrt{|f''(x)\tau_h(f(x))|}\cdot h(f(x))}\right)dx$$ with a constant C(M, h) independent of f, where f belongs locally to the Sobolev space ${W^{2,1}(\mathbb{R})}$ and f′ has compact support. Here M is an arbitrary N-function satisfying certain assumptions, h is a given function and ${\tau_h(\cdot)}$ is its given transform independent of M. When M(λ) =  λ ^p and ${h \equiv 1}$ we retrieve the well-known inequality ${\int_\mathbb{R}|f'(x)|^{p}dx \leq (\sqrt{p - 1})^{p}\int_\mathbb{R}(\sqrt{|f''(x) f(x)|})^{p}dx}$ . We apply our inequality to obtain some generalizations of capacitary estimates and isoperimetric inequalities due to Maz’ya (1985).     

Ruled Weingarten Hypersurfaces in Hyperbolic Space {{\mathbb{H}}^{n+1}}

In this work we give a local classification of connected ruled Weingarten hypersurfaces M ⁿ , n ≥ 3 in the hyperbolic space ${{\mathbb{H}}^{n+1} \subset {\mathbb{L}}^{n+2}}$ .     

Localization in One-dimensional Quasi-periodic Nonlinear Systems

To investigate localization in one-dimensional quasi-periodic nonlinear systems, we consider the Schrödinger equation $${\rm i}\dot{q}_n+\epsilon(q_{n+1}+q_{n-1})+V(n\tilde{\alpha}+x)q_n+ |q_n|^2q_n=0,\quad n\in\mathbb{Z},$$ as a model, with V a nonconstant real-analytic function on ${\mathbb{R}/\mathbb{Z}}$ , and ${\tilde{\alpha}}$ satisfying a certain Diophantine condition. It is shown that, if ${\epsilon}$ is sufficiently small, then for a.e. ${x\in\mathbb{R}/\mathbb{Z}}$ , dynamical localization is maintained for “typical” solutions in a quasi-periodic time-dependent way.     

Permanent Versus Determinant over a Finite Field

Let $ \mathbb{F} $ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and for the whole matrix space M _n ( $ \mathbb{F} $ ). It is known that for n = 2, there are bijective linear maps Φ on $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, ?), where Φ is an arbitrary bijective map on matrices and ? : $ \mathbb{F} $ → $ \mathbb{F} $ is an arbitrary map such that per A = ?(det Φ(A)) for all matrices A from the spaces $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ), respectively. Moreover, for the space M _n ( $ \mathbb{F} $ ), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $ \mathbb{F} $ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.     

Parabolicity of maximal surfaces in Lorentzian product spaces

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form ${M^2 \times \mathbb {R}_1}$ , where M ² is a connected Riemannian surface with non-negative Gaussian curvature and ${M^2 \times \mathbb {R}_1}$ is endowed with the Lorentzian product metric ${{\langle , \rangle}={\langle , \rangle}_M-dt^2}$ . In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain ${\Omega \subseteq M}$ is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi–Bernstein result for entire maximal graphs in ${M^2 \times \mathbb {R}_1}$ .