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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]