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Register a new userLagrangian surfaces in $mathbb H^2 times mathbb H^2$
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The Riemannian product of two hyperbolic planes of constant Gaussian curvature -1 has a natural Kahler structure. In fact, it can be identified with the complex hyperbolic quadric of complex dimension two. In this paper we study Lagrangian surfaces in this manifold. We present several examples and classify the totally umbilical and totally geodesic Lagrangian surfaces, the Lagrangian surfaces with parallel second fundamental form, the minimal Lagrangian surfaces with constant Gaussian curvature and the complete minimal Lagrangian surfaces satisfying a bounding condition on an important function that can be defined on any Lagrangian surface in this particular ambient space.
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It is known that for $Omega subset mathbb{R}^{2}$ an unbounded convex domain and $H>0$, there exists a graph $Gsubset mathbb{R}^{3}$ of constant mean curvature $H$ over $Omega $ with $partial G=$ $partial Omega $ if and only if $Omega $ is included in a strip of width $1/H$. In this paper we obtain results in $mathbb{H}^{2}times mathbb{R}$ in the same direction: given $Hin left( 0,1/2right) $, if $Omega $ is included in a region of $mathbb{ H}^{2}times left{ 0right} $ bounded by two equidistant hypercycles $ell(H)$ apart, we show that, if the geodesic curvature of $partial Omega $ is bounded from below by $-1,$ then there is an $H$-graph $G$ over $Omega $ with $partial G=partial Omega$. We also present more refined existence results involving the curvature of $partialOmega,$ which can also be less than $-1.$
We estimate the upper bound for the $ell^{infty}$-norm of the volume form on $mathbb{H}^2timesmathbb{H}^2timesmathbb{H}^2$ seen as a class in $H_{c}^{6}(mathrm{PSL}_{2}mathbb{R}timesmathrm{PSL}_{2}mathbb{R}timesmathrm{PSL}_{2}mathbb{R};mathbb{R})$. This gives the lower bound for the simplicial volume of closed Riemennian manifolds covered by $mathbb{H}^{2}timesmathbb{H}^{2}timesmathbb{H}^{2}$. The proof of these facts yields an algorithm to compute the lower bound of closed Riemannian manifolds covered by $big(mathbb{H}^2big)^n$.
In this article we complete the classification of the umbilical submanifolds of a Riemannian product of space forms, addressing the case of a conformally flat product $mathbb{H}^ktimes mathbb{S}^{n-k+1}$, which has not been covered in previous works on the subject. We show that there exists precisely a $p$-parameter family of congruence classes of umbilical submanifolds of $mathbb{H}^ktimes mathbb{S}^{n-k+1}$ with substantial codimension~$p$, which we prove to be at most $mbox{min},{k+1, n-k+2}$. We study more carefully the cases of codimensions one and two and exhibit, respectively, a one-parameter family and a two-parameter family (together with three extra one-parameter families) that contain precisely one representative of each congruence class of such submanifolds. In particular, this yields another proof of the classification of all (congruence classes of) umbilical submanifolds of $mathbb{S}^ntimes mathbb{R}$, and provides a similar classification for the case of $mathbb{H}^ntimes mathbb{R}$. We determine all possible topological types, actually, diffeomorphism types, of a complete umbilical submanifold of $mathbb{H}^ktimes mathbb{S}^{n-k+1}$. We also show that umbilical submanifolds of the product model of $mathbb{H}^ktimes mathbb{S}^{n-k+1}$ can be regarded as rotational submanifolds in a suitable sense, and explicitly describe their profile curves when $k=n$. As a consequence of our investigations, we prove that every conformal diffeomorphism of $mathbb{H}^ktimes mathbb{S}^{n-k+1}$ onto itself is an isometry.
We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $mathbb{R}^{3}_{raisepunct{.}}$ We also show that any minimal hypersurface immersed with bounded curvature in $Mtimes R_+$ equals some $Mtimes {s}$ provided $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with $text{Ric}_M geq 0$ and whose sectional curvatures are bounded from above. For $H$-surfaces we prove that a stochastically complete surface $M$ can not be in the mean convex side of a $H$-surface $N$ embedded in $R^3$ with bounded curvature if $sup vert H_{_M}vert < H$, or ${rm dist}(M,N)=0$ when $sup vert H_{_M}vert = H$. Finally, a maximum principle at infinity is shown assuming $M$ has non-empty boundary.
This survey aims to give a brief introduction to operator theory in the Hardy space over the bidisc $H^2(mathbb D^2)$. As an important component of multivariable operator theory, the theory in $H^2(mathbb D^2)$ focuses primarily on two pairs of commuting operators that are naturally associated with invariant subspaces (or submodules) in $H^2(mathbb D^2)$. Connection between operator-theoretic properties of the pairs and the structure of the invariant subspaces is the main subject. The theory in $H^2(mathbb D^2)$ is motivated by and still tightly related to several other influential theories, namely Nagy-Foias theory on operator models, Andos dilation theorem of commuting operator pairs, Rudins function theory on $H^2(mathbb D^n)$, and Douglas-Paulsens framework of Hilbert modules. Due to the simplicity of the setting, a great supply of examples in particular, the operator theory in $H^2(mathbb D^2)$ has seen remarkable growth in the past two decades. This survey is far from a full account of this development but rather a glimpse from the authors perspective. Its goal is to show an organized structure of this theory, to bring together some results and references and to inspire curiosity on new researchers.
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