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In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al. [ABSTRACT FROM AUTHOR]

In this paper, we solve the asymptotic Plateau problem in hyperbolic space for constant \sigma _{n-1} curvature, i.e. the existence of a complete hypersurface in \mathbb {H}^{n+1} satisfying \sigma _{n-1}(\kappa)=\sigma \in (0,n) with a prescribed asymptotic boundary \Gamma. The key ingredient is the curvature estimates. Previously, this was only known for \sigma _0<\sigma

In this paper, we study a locally constrained mean curvature flow with free boundary in a hyperbolic ball. Under the flow, the enclosed volume is preserved and the area is decreasing. We prove the long time existence and smooth convergence for such flow under certain star-shaped condition. As an application, we give a flow proof of the isoperimetric problem for the star-shaped free boundary hypersurfaces in a hyperbolic ball. [ABSTRACT FROM AUTHOR]

In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia [Trans. Amer. Math. Soc. 372 (2019), pp. 6771–6803]. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzschild manifold. [ABSTRACT FROM AUTHOR]

Let G\subsetneq \mathbb {R}^n be an open set. It is shown by Hästö that G equipped with the \tilde {j}_G metric is Gromov hyperbolic. The purpose of this paper is to show that there is a natural quasisymmetric correspondence between the Gromov boundary of (G, \tilde {j}_G) and its Euclidean boundary \partial G. Both bounded and unbounded cases are in our considerations. [ABSTRACT FROM AUTHOR]

We prove an improved version of Poincaré-Hardy inequality in suitable subspaces of the Sobolev space on the hyperbolic space via Bessel pairs. As a consequence, we obtain a new Hardy type inequality with an improved constant (than the usual Hardy constant). Furthermore, we derive a new kind of improved Caffarelli-Kohn-Nirenberg inequality on the hyperbolic space. [ABSTRACT FROM AUTHOR]

A thrice-punctured sphere group is a non-elementary group generated by two parabolic isometries whose product is a parabolic isometry. We prove that the deformation space of a thrice-punctured sphere group acting on hyperbolic 4-space is 7-dimensional. Among them, there is a 5-dimensional parameter space of linked thrice-punctured sphere groups. In particular, there is a 1-parameter family of discrete linked thrice-punctured sphere groups such that the rotation angles of the two parabolic generators and the product of the generators are fixed. [ABSTRACT FROM AUTHOR]

We classify holomorphic isometric actions on complex space forms all of whose orbits are Lagrangian submanifolds, up to orbit equivalence. The only examples are Lagrangian affine subspace foliations of complex Euclidean spaces, and Lagrangian horocycle foliations of complex hyperbolic spaces. [ABSTRACT FROM AUTHOR]

In contrast to the fact that there are only finitely many maximal arithmetic reflection groups acting on the hyperbolic space \mathbb {H}^n, n\geq 2, we show that: one can produce infinitely many maximal quasi–arithmetic reflection groups acting on \mathbb {H}^2; they admit infinitely many different fields of definition; the degrees of their fields of definition are unbounded. However, for n\geq 14 an approach initially developed by Vinberg shows that there are still finitely many fields of definitions in the quasi–arithmetic case. [ABSTRACT FROM AUTHOR]

We prove an inequality bounding the renormalized area of a complete minimal surface in hyperbolic space in terms of the conformal length of its ideal boundary. [ABSTRACT FROM AUTHOR]

We give a new proof of the uniformization theorem of the leaves of a compact lamination by surfaces of hyperbolic conformal type. We use a laminated version of the Ricci flow to prove the existence of a laminated Riemannian metric (smooth on the leaves, transversally continuous) with leaves of constant Gaussian curvature equal to -1, which is conformally equivalent to the original metric. [ABSTRACT FROM AUTHOR]

The purpose of this article is to investigate a hyperbolic complex manifold M exhausted by a pseudoconvex domain \Omega in \mathbb {C}^n via an exhausting sequence \{f_j\colon \Omega \to M\} such that f_j^{-1}(a) converges to a boundary point \xi _0 \in \partial \Omega for some point a\in M. [ABSTRACT FROM AUTHOR]