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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 $-1.$
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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 give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $ N times mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} leq kappa$. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces has positive fundamental tone.
We provide a method for fast and exact simulation of Gaussian random fields on spheres having isotropic covariance functions. The method proposed is then extended to Gaussian random fields defined over spheres cross time and having covariance functions that depend on geodesic distance in space and on temporal separation. The crux of the method is in the use of block circulant matrices obtained working on regular grids defined over longitude $times$ latitude.
Given $Hin [0,1)$ and given a $C^0$ exterior domain $Omega$ in a $H-$hypersphere of $mathbb{H}^3,$ the existence of hyperbolic Killing graphs of CMC $H$ defined in $overline{Omega}$ with boundary $ partial Omega $ included in the $H-$hypersphere is obtained.
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.
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