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Register a new userLower Bound for the Simplicial Volume of Closed Manifolds Covered by $mathbb{H}^{2}timesmathbb{H}^{2}timesmathbb{H}^{2}$
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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 $big(mathbb{H}^2big)^n$.
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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.
We prove the positive energy conjecture for a class of asymptotically Horowitz-Myers metrics on $mathbb{R}^{2}timesmathbb{T}^{n-2}$. This generalizes the previous results of Barzegar-Chru{s}ciel-H{o}rzinger-Maliborski-Nguyen as well as the authors.
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 $-1.$
For the moduli space of unmarked convex $mathbb{RP}^2$ structures on the surface $S_{g,m}$ with negative Euler characteristic, we investigate the subsets of the moduli space defined by the notions like boundedness of projective invariants, area, Gromov hyperbolicity constant, quasisymmetricity constant etc. These subsets are comparable to each other. We show that the Goldman symplectic volume of the subset with certain projective invariants bounded above by $t$ and fixed boundary simple root lengths $mathbf{L}$ is bounded above by a positive polynomial of $(t,mathbf{L})$ and thus the volume of all the other subsets are finite. We show that the analog of Mumfords compactness theorem holds for the area bounded subset.
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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