Submanifolds of generalized $$(k,\mu )$$-space-forms

In this paper, we have studied submanifolds especially, totally umbilical submanifolds of generalized \((k,\mu )\)-space-forms. We have found a necessary and sufficient condition for such submanifolds to be either invariant or anti-invariant. It is also shown that every totally umbilical submanifold of a generalized \((k,\mu )\)-space-form is a pseudo quasi-Einstein manifold.     

Pointwise Pseudo-slant Warped Product Submanifolds in a Kähler Manifold

The purpose of this paper is to study the pointwise pseudo-slant warped product submanifolds of a Kähler manifold \(\widetilde{M}\). We derive the conditions of integrability and totally geodesic foliation for the distributions allied to the characterization of a pointwise pseudo-slant submanifolds of \(\widetilde{M}\). The necessary and sufficient conditions for isometrically immersed pointwise pseudo-slant submanifolds of \(\widetilde{M}\) to be a pointwise pseudo-slant warped product and a locally Riemannian product are obtained. Further, we classify pointwise pseudo-slant warped product submanifolds of \(\widetilde{M}\) by developing the sharp inequalities in terms of second fundamental form and wrapping function.     

Complete spacelike submanifolds with parallel mean curvature vector in a semi-Euclidean space

Our aim in this article is to study the geometry of n-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space \({\mathbb{R}^{n+p}_{q}}\) of index q, with \({1\leq q\leq p}\). Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of \({\mathbb{R}^{n+p}_{q}}\) be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in \({\mathbb{R}^{n+p}_{p}}\) and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz–Minkowski space \({\mathbb{R}^{n+1}_{1}}\).     

A class of non-compact homogeneous Lagrangian submanifolds in complex hyperbolic spaces

We study homogeneous Lagrangian submanifolds in complex hyperbolic spaces. We show there exists a correspondence between compact homogeneous Lagrangian submanifolds in \(\mathbb {C}H^{n}\) and the ones in \(\mathbb {C}^n\), or equivalently, in \(\mathbb {C}P^{n-1}\). Furthermore, we construct and classify non-compact homogeneous Lagrangian submanifolds in \(\mathbb {C}H^n\) obtained by the actions of connected closed subgroups of the solvable part of the Iwasawa decomposition.     

On Submanifolds with 2-Type Pseudo-Hyperbolic Gauss Map in Pseudo-Hyperbolic Space

In this paper, we study pseudo-Riemannian submanifolds of a pseudo-hyperbolic space \(\mathbb H^{m-1}_s (-1) \subset \mathbb E^m_{s+1}\) with 2-type pseudo-hyperbolic Gauss map. We give a characterization of proper pseudo-Riemannian hypersurfaces in \(\mathbb H^{n+1}_s (-1) \subset \mathbb E^{n+2}_{s+1}\) with non-zero constant mean curvature and 2-type pseudo-hyperbolic Gauss map. For \(n=2\), we prove classification theorems. In addition, we show that the hyperbolic Veronese surface is the only maximal surface fully lying in \(\mathbb H^4_2 (-1) \subset \mathbb H^{m-1}_2 (-1)\) with 2-type pseudo-hyperbolic Gauss map. Moreover, we prove that a flat totally umbilical pseudo-Riemannian hypersurface \(M^n_t\) of the pseudo-hyperbolic space \(\mathbb {H}^{n+1}_t(-1) \subset \mathbb E^{n+2}_{t+1}\) has biharmonic pseudo-hyperbolic Gauss map.     

On regular Stein neighborhoods of a union of two maximally totally real subspaces in $$\mathbb {C}^n$$

We construct regular Stein neighborhoods of a union of two maximally totally real subspaces \(M=(A+iI)\mathbb {R}^n\) and \(N=\mathbb {R}^n\) in \(\mathbb {C}^n\), provided that the eigenvalues of the real \(n \times n\) matrix A are sufficiently small. This result is applied to provide regular Stein neighborhoods of an immersed totally real n-manifold in a complex n-manifold, with only finitely many double points, and such that the union of the tangent spaces at each double point in some local coordinates coincides with \(M\cup N\), described above.     

Characterizations of Spacelike Submanifolds with Constant Scalar Curvature in the de Sitter Space

In this paper, we deal with complete spacelike submanifolds \(M^n\) immersed in the de Sitter space \(\mathbb S_p^{n+p}\) of index p with parallel normalized mean curvature vector and constant scalar curvature R. Imposing a suitable restriction on the values of R, we apply a maximum principle for the so-called Cheng–Yau operator L, which enables us to show that either such a submanifold must be totally umbilical or it holds a sharp estimate for the norm of its total umbilicity tensor, with equality if and only the submanifold is isometric to a hyperbolic cylinder of the ambient space. In particular, when \(n=2\) this provides a nice characterization of the totally umbilical spacelike surfaces of \(\mathbb {S}^{2+p}_p\) with codimension \(p\ge 2\). Furthermore, we also study the case in which these spacelike submanifold are L-parabolic.     

Parametrizing Shimura subvarieties of {\mathrm{A}_1} Shimura varieties and related geometric problems

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of \({X_{a, b} = (\mathbf{H}^2)^a\times (\mathbf{H}^3)^b}\). A special case describes all Shimura subvarieties of type \({\mathrm{A}_1}\) Shimura varieties. We produce, for any \({n\geq 1}\), examples of manifolds/Shimura varieties with precisely n commensurability classes of totally geodesic submanifolds/Shimura subvarieties. This is in stark contrast with the previously studied cases of arithmetic hyperbolic 3-manifolds and quaternionic Shimura surfaces, where the presence of one commensurability class of geodesic submanifolds implies the existence of infinitely many classes.     

Complements of convex topologies on products of finite totally ordered spaces

Let X be a finite product of finite totally ordered topological spaces. We show that in the lattice of topologies on X, every convex topology \(\tau \) on X has a convex complement \(\tau '\).     

Density and spectrum of minimal submanifolds in space forms

Let \(\varphi : M^m \rightarrow N^n\) be a minimal, proper immersion in an ambient space suitably close to a space form \(\mathbb {N}^n_k\) of curvature \(-k\le 0\). In this paper, we are interested in the relation between the density function \(\Theta (r)\) of M and the spectrum of its Laplace–Beltrami operator. In particular, we prove that if \(\Theta (r)\) has subexponential growth (when \(k<0\)) or sub-polynomial growth (\(k=0\)) along a sequence, then the spectrum of \(M^m\) is the same as that of the space form \(\mathbb {N}^m_k\). Notably, the result applies to Anderson’s (smooth) solutions of Plateau’s problem at infinity on the hyperbolic space, independently of their boundary regularity. We also give a simple condition on the second fundamental form that ensures M to have finite density. In particular, we show that minimal submanifolds with finite total curvature in the hyperbolic space also have finite density.     

<Emphasis Type="Italic">p</Emphasis>-Fundamental tone estimates of submanifolds with bounded mean curvature

In this paper, we estimate the p-fundamental tone of submanifolds in a Cartan–Hadamard manifold. First, we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension-one \(C^2\)-foliations of open subsets \(\Omega \) of Riemannian manifolds M and obtain lower bound estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of \(\Omega \).     

Warped Product Bi-slant Immersions in Kaehler Manifolds

A submanifold M of an almost Hermitian manifold \((\widetilde{M},g,J)\) is called slant, if for each point \(p\in M\) and \(0\ne X\in T_p M\), the angle between JX and \(T_p M\) is constant (see Chen in Bull Aust Math Soc 41:135–147, 1990). Later, Carriazo (in: Proceedings of the ICRAMS 2000, Kharagpur, 2000) defined the notion of bi-slant immersions as an extension of slant immersions. In this paper, we study warped product bi-slant submanifolds in Kaehler manifolds and provide some examples of warped product bi-slant submanifolds in some complex Euclidean spaces. Our main theorem states that every warped product bi-slant submanifold in a Kaehler manifold is either a Riemannian product or a warped product hemi-slant submanifold.     

Normal curvature of CR submanifolds of maximal CR dimension of the complex projective space

We prove that there do not exist CR submanifolds Mⁿ of maximal CR dimension of a complex projective space \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) with flat normal connection D of M, when the distinguished normal vector field is parallel with respect to D. If D is lift-flat, then there exists a totally geodesic complex projective subspace \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\) of \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) such that M is a real hypersurface of \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\).     

On homogeneous hypersurfaces in $${\mathbb {C}}^3$$

We consider a family \(M_t^n\), with \(n\geqslant 2\), \(t>1\), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in  \({\mathbb {C}}^n\) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of \(M_t^n\) in  \({\mathbb {C}}^n\) for \(n=3,7\). In our earlier article we showed that \(M_t^7\) is not embeddable in  \({\mathbb {C}}^7\) for every t and that \(M_t^3\) is embeddable in  \({\mathbb {C}}^3\) for all \(1<t<1+10^{-6}\). In the present paper, we improve on the latter result by showing that the embeddability of \(M_t^3\) in fact takes place for \(1<t<\sqrt{(2+\sqrt{2})/3}\). This is achieved by analyzing the explicit totally real embedding of the sphere \(S^3\) in \({\mathbb {C}}^3\) constructed by Ahern and Rudin. For \(t\geqslant {\sqrt{(2+\sqrt{2})/3}}\), the problem of the embeddability of \(M_t^3\) remains open.     

Failure of hypoellipticity for the canonical solution to $${\bar\partial_b}$$ on some nilpotent Lie groups

In this paper we show that Kohn’s solution to the \({\bar\partial_b}\) problem fails to be hypoelliptic on some class of high codimension submanifolds of \({\mathbb{C}^n}\). The examples presented here carry a Lie group structure which generalize the one-dimensional Heisenberg group.     

Coalescece ad Meetig Times o $$$$-Block Markov Chais

We consider finite-state, discrete-time, mixing Markov chains \((V,P)\), where \(V\) is the state space and \(P\) is the transition matrix. To each such chain \((V,P)\), we associate a sequence of chains \((V_n,P_n)\) by coding trajectories of \((V,P)\) according to their overlapping \(n\)-blocks. The chain \((V_n,P_n)\), called the \(n\)-block Markov chain associated with \((V,P)\), may be considered an alternate version of \((V,P)\) having memory of length \(n\). Along such a sequence of chains, we characterize the asymptotic behavior of coalescence times and meeting times as \(n\) tends to infinity. In particular, we define an algebraic quantity \(L(V,P)\) depending only on \((V,P)\), and we show that if the coalescence time on \((V_n,P_n)\) is denoted by \(C_n\), then the quantity \(\frac{1}{n} \log C_n\) converges in probability to \(L(V,P)\) with exponential rate. Furthermore, we fully characterize the relationship between \(L(V,P)\) and the entropy of \((V,P)\).     

Compact $$\lambda $$-translating Solitons with Boundary

A \(\lambda \)-translating soliton with density vector \(\mathbf {v}\) is a surface \(\varSigma \) in Euclidean space \(\mathbb {R}^3\) whose mean curvature H satisfies \(2H=2\lambda +\langle N,\mathbf {v}\rangle \), where N is the Gauss map of \(\varSigma \). In this article, we study the shape of a compact \(\lambda \)-translating soliton in terms of its boundary. If \(\varGamma \) is a given closed curve, we deduce under what conditions on \(\lambda \) there exists a compact \(\lambda \)-translating soliton \(\varSigma \) with boundary \(\varGamma \) and we provide estimates of the surface area depending on the height of \(\varSigma \). Finally, we study the shape of \(\varSigma \) related with the geometry of \(\varGamma \), in particular, we give conditions that assert that \(\varSigma \) inherits the symmetries of its boundary \(\varGamma \).     

Magnetic Trajectories in an Almost Contact Metric Manifold $${\mathbb{R}^{2N+1}}$$

In this paper we classify magnetic trajectories γ in \({{\mathbb{R}}^{2N+1}}\) endowed with a canonical quasi-Sasakian structure, corresponding to a magnetic field proportional to the fundamental 2-form. We prove that they are helices of order 5 and we show that there exists a totally geodesic \({{\mathbb{R}}^5}\) in \({\mathbb{R}^{2N+1}}\) such that γ lies in \({{\mathbb{R}}^5}\). Moreover, the quasi-Sasakian structure of \({{\mathbb{R}}^5}\) is that induced from the ambient manifold.     

Reducing Subspaces of Multiplication Operators on the Dirichlet Space

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product \({\phi}\) on the Dirichlet space D. We prove that any two distinct nontrivial minimal reducing subspaces of \({M_\phi}\) are orthogonal. When the order n of \({\phi}\) is 2 or 3, we show that \({M_\phi}\) is reducible on D if and only if \({\phi}\) is equivalent to \({z^n}\). When the order of \({\phi}\) is 4, we determine the reducing subspaces for \({M_\phi}\), and we see that in this case \({M_\phi}\) can be reducible on D when \({\phi}\) is not equivalent to \({z^4}\). The same phenomenon happens when the order n of \({\phi}\) is not a prime number. Furthermore, we show that \({M_\phi}\) is unitarily equivalent to \({M_{z^n} (n > 1)}\) on D if and only if \({\phi = az^n}\) for some unimodular constant a.