Inverse Spectral Problem for Analytic {{(\mathbb{Z}/2 \mathbb{Z})^{n}}} -Symmetric Domains in {{\mathbb{R}^{n}}}

We prove that bounded real analytic domains in ${\mathbb{R}^{n}}$ , with the symmetries of an ellipsoid and with one axis length fixed, are determined by their Dirichlet or Neumann eigenvalues among other bounded real analytic domains with the same symmetries and axis length. Some non-degeneracy conditions are also imposed on the class of domains. It follows that bounded, convex analytic domains are determined by their spectra among other such domains. This seems to be the first positive result for the well-known Kac problem, “Can one hear the shape of a drum?”, in higher dimensions.     

New complete embedded minimal surfaces in $${{\mathbb {H} ^2\times \mathbb {R}}}$$

We construct three kinds of complete embedded minimal surfaces in \({\mathbb {H}^2\times \mathbb {R}}\) . The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These two are conjugate surfaces just as the helicoid and the catenoid are in \({\mathbb {R}^3}\) . The third one is a finite total curvature surface which is conformal to \({\mathbb {S}^2\setminus\{p_1,\ldots,p_k\}, k\geq3.}\)     

Boundedness of Singular Integrals on Multiparameter Weighted Hardy Spaces \text{{\textit{H}}}^\text{{\textit{p}}}_{\text{{\textit{w}}}}\ (\mathbb{R}^{\text{{\textit{n}}}}\times \mathbb{R}^{\text{{\textit{m}}}})

We apply the discrete version of Calderón??s identity and Littlewood?CPaley?CStein theory with weights to derive the $(H^p_w, H^p_w)$ and $(H^p_w, L^p_w) (0<p\le 1)$ boundedness for multiparameter singular integral operators. It turns out that even in the one-parameter case, our results substantially improve the known ones in the literature where w????A ₁ was needed. Our results in the multiparameter setting can be regarded as a natural extension of $L^p_w$ boundedness for p?>?1 for w????A _p to the case of weighted Hardy spaces $H^p_w$ for p????1, but under a weaker assumption that w belongs to the class of product A _???? weights with respect to rectangles in product spaces.     

Uniqueness of \mathbb{C}^{ * } - and \mathbb{C}_{{\text{ + }}} -actions on Gizatullin Surfaces

A Gizatullin surface is a normal affine surface V over $ \mathbb{C} $ , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with $ \mathbb{C}_{{\text{ + }}} $ -actions on V considered up to a “speed change”. Non-Gizatullin surfaces are known to admit at most one $ \mathbb{A}^{1} $ -fibration V → S up to an isomorphism of the base S. Moreover, an effective $ \mathbb{C}^{ * } $ -action on them, if it does exist, is unique up to conjugation and inversion t $ \mapsto $ t ^?1 of $ \mathbb{C}^{ * } $ . Obviously, uniqueness of $ \mathbb{C}^{ * } $ -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when $ \mathbb{A}^{{\text{1}}} $ -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base $ S \cong \mathbb{A}^{{\text{1}}} $ . We exhibit as well large subclasses of Gizatullin $ \mathbb{C}^{ * } $ -surfaces for which a $ \mathbb{C}^{ * } $ -action is essentially unique and for which there are at most two conjugacy classes of $ \mathbb{A}^{{\text{1}}} $ -fibrations over $ \mathbb{A}^{{\text{1}}} $ .     

Classification of rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^\mathbb{N} with degree 2

Rational proper holomorphic maps from the unit ball in ?² into the unit ball ? ^N with degree 2 are classified, up to automorphisms of balls.     

Separable forms of {{\mathbb{G}}_m} -ACTIONS ON {{\mathbb{A}}^3}

Let k be a field of characteristic zero. We consider k-forms of $ {\mathbb G} $ _m -actions on $ {\mathbb A} $ ³ and show that they are linearizable. In particular, $ {\mathbb G} $ _m -actions on $ {\mathbb A} $ ³ are linearizable, and k-forms of $ {\mathbb A} $ ³ that admit an effective action of an infinite reductive group are trivial.     

Uniqueness of minimal surfaces whose boundary is a horizontal graph and some Bernstein problems in {{\mathbb{H}^{2}\times \mathbb{R}}}

We deduce that a connected compact immersed minimal surface in ${{\mathbb{H}^{2}\times \mathbb{R}}}$ whose boundary has an injective horizontal projection on an admissible convex curve in ${\partial_\infty{\mathbb{H}^{2}\times \mathbb{R}}}$ , and satisfies an admissible bounded slope condition, is the Morrey’s solution of the Plateau problem and is a horizontal minimal graph. We prove that there is no entire horizontal minimal graph in ${{\mathbb{H}^{2}\times \mathbb{R}}}$ .     

Wandering Subspaces of the Bergman Space and the Dirichlet Space Over {{\mathbb{D}^{n}}}

Doubly commuting invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc \({\mathbb{D}^n}\) (with \({n \geq 2}\) ) are investigated. We show that for any non-empty subset \({\alpha=\{\alpha_1,\ldots,\alpha_k\}}\) of \({\{1,\ldots,n\}}\) and doubly commuting invariant subspace \({\mathcal{S}}\) of the Bergman space or the Dirichlet space over \({\mathbb{D}^n}\) , restriction of the multiplication operator tuple on \({\mathcal{S}, M_{\alpha}|_\mathcal{S}:=(M_{z_{\alpha_1}}|_\mathcal{S},\ldots, M_{z_{\alpha_k}}|_\mathcal{S})}\) , always possesses generating wandering subspace of the form $$\bigcap_{i=1}^k(\mathcal{S}\ominus z_{\alpha_i}\mathcal{S})$$ .     

Circle Geometries Modeled in Projective Lines over {{\mathbb{R}}^2} -rings

In this paper, we consider classical circle geometries and connect them with places of planar Cayley–Klein geometries. There are, in principle, only three types of $ {{\mathbb{R}}^2} $ -ring structures and, thus, only three types of corresponding circle geometries. Thus, each generalization to non-Euclidean planes turns out to be just another representation of the classical Euclidean cases. We believe that even the Euclidean cases of circle geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle geometries might also be of interest in themselves. For example, among the planar Cayley–Klein geometries, the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated similarly to the isotropic Möbius geometry by suitable projections of the Blaschke cylinder.     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

Rational holomorphic maps from \mathbb{B}^2 into \mathbb{B}^N with degree 2

Rational proper holomorphic maps from the unit ball in ?² into the unit ball ? ^N with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.     

Constant Angle Property and Canonical Principal Directions for Surfaces in {\mathbb{M}^{2}(c) \times {\mathbb{R}_{1}}}

In this paper, we study surfaces in Lorentzian product spaces ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ . We classify constant angle spacelike and timelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ . Moreover, complete classifications of spacelike surfaces in ${{\mathbb{S}^{2} \times \mathbb{R}_1}}$ and ${{\mathbb{H}^{2} \times \mathbb{R}_1}}$ and timelike surfaces in ${{\mathbb{M}^{2}(c) \times \mathbb{R}_1}}$ with a canonical principal direction are obtained. Finally, a new characterization of the catenoid of the 3rd kind is established, as the only minimal timelike surface with a canonical principal direction in Minkowski 3–space.     

Wiener-Hopf operators on L_{\omega}^{2}(\mathbb{R}^{+})

Let be a weighted space with weight . In this paper we show that for every Wiener-Hopf operator T on and for every a I, there exists a function such that

Explicit div-curl inequalities in bounded and unbounded domains of $${{\mathbb {R}}}^3$$

We give a constructive proof of some functional inequalities related to the div and curl operators in bounded and unbounded domains of \({{\mathbb {R}}}^3\). Our new innovation consists in giving explicit constants in several geometric configurations. These inequalities are of a first use in solving div-curl systems and vector potential problems arising in physics.