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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]

In this paper, we consider nonhomogeneous inverse Gauss curvature flows in hyperbolic space H3. We prove that if the initial hypersurface Σ0 has nonnegative scalar curvature, then the evolving surface Σt has nonnegative scalar curvature along the flow for t > 0. The solution of the flow exists for all time and becomes more and more umbilic as t→∞. [ABSTRACT FROM AUTHOR]

In this paper, we investigate universal inequalities for eigenvalues of a clamped plate problem on a bounded domain in an $ n$-dimensional Euclidean space, Payne, Pólya and Weinberger (1955), Hook (1990) and Chen and Qian (1990) studied universal inequalities for eigenvalues of the clamped plate problem. Recently, Cheng and Yang (2006) have derived the Yang-type universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the $ n$-dimensional hyperbolic space, although many mathematicians want to obtain a universal inequality for eigenvalues of the clamped plate problem, there are no results on universal inequalities for eigenvalues. The main reason that one could not derive a universal inequality is that one cannot find appropriate trial functions. In this paper, by constructing ``nice'' trial functions, we obtain a universal inequality for eigenvalues of the clamped plate problem on a bounded domain in the hyperbolic space. Furthermore, we can prove that if the first eigenvalue of the clamped plate problem tends to $ \tfrac{(n-1)^4}{16}$ $ \tfrac{(n-1)^4}{16}$ [ABSTRACT FROM AUTHOR]

This paper gives a new characterization of geodesic spheres in the hyperbolic space in terms of a "weighted" higher order mean curvature. Precisely, we show that a compact hypersurface Σn-1 embedded in ℍn with VHk being constant for some k = 1, · · ·, n - 1 is a centered geodesic sphere. Here Hk is the k-th normalized mean curvature of Σ induced from Hn and V = coshr, where r is a hyperbolic distance to a fixed point in ℍn. Moreover, this result can be generalized to a compact hypersurface Σ embedded in Hn with the ratio V (Hk/Hj) ≡ constant, 0 ≤ j < k ≤ n - 1 and Hj not vanishing on Σ. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider the mean curvature flow of surfaces in hyperbolic 3-space. First, we establish the isoperimetric inequality using the flow, provided the enclosed volume approaches zero at the final time. Second, we construct two singular examples of the flow. More precisely, there exists a torus which must develop a singularity in the flow before the volume it encloses decreases to zero. There also exists a topological sphere in the shape of dumbbell, which must develop a singularity in the flow before its area shrinks to zero. [ABSTRACT FROM AUTHOR]

Let X0 be a complete hyperbolic surface of infinite type that has a geodesic pants decomposition with cuff lengths bounded above. The length spectrum Teichmüller space Tls(X0) consists of homotopy classes of hyperbolic metrics on X0 such that the ratios of the corresponding simple closed geodesic for the hyperbolic metric on X0 and for the other hyperbolic metric are bounded from below away from 0 and from above away from ∞;. This paper studies earthquakes in the length spectrum Teichmüller space Tls(X0). We find a necessary condition and several sufficient conditions on the earthquake measure μ such that the corresponding earthquake Eμ describes a hyperbolic metric on X0 which is in the length spectrum Teichmüller space. Moreover, we give examples of earthquake paths t → Etμ, for t ≥ 0, such that Etμ ∈ Tls(X0) for 0 ≤ t < t0, Et0μ/∉ Tls(X0) and Etμ ∈ Tls(X0) for t > t0. [ABSTRACT FROM AUTHOR]

In this paper we investigate angle geometry in the universal Teichmüller space. We construct three examples of triangles bounded by geodesic segments such that the first example has the sum of the three inner angles less than n, the second example has the sum of the three angles equal to n, and the third example has the sum of the three angles greater than π. Our result gives a negative answer to a problem raised by Zhong Li and Yi Qi. Moreover, our results indicate that the universal Teichmüuller space presents all hyperbolic, Euclidean, and spherical phenomena in angle geometry. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to show that a complete hypersurface x : Mn → ℍn+1 immersed with constant mean curvature into the hyperbolic space ℍn+1 is totally umbilical provided that its Gauss mapping ν has some suitable behavior. In this setting, our first result requires that the image ν(M) lies in a totally umbilical spacelike hypersurface of the de Sitter space Sn+1 1 , while in our second one we suppose that Mn has scalar curvature bounded from below and that ν(M) is contained in the closure of a domain enclosed by a totally umbilical spacelike hypersurface of Sn+1 1 determined by some vector a of the Minkowski space Ln+2, with the tangential component of a with respect to Mn having Lebesgue integrable norm. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of multiple-layer solutions to the elliptic Allen-Cahn equation in hyperbolic space: ... here F is a nonnegative double-well potential with nondegenerate minima. We prove that for any collection of widely separated, nonintersecting hyperplanes in Hn, there is a solution to this equation which has a nodal set very close to this collection of hyperplanes. Unlike the corresponding problem in Rn, there are no constraints beyond the separation parameter. [ABSTRACT FROM AUTHOR]

In this paper, we will show that the limit of some quasilocal mass integrals of the coordinate spheres in an asymptotically hyperbolic (AH) manifold is the mass integral of the AH manifold. This is the analogue of the well-known result that the limit of the Brown-York mass of coordinate spheres is the ADM mass in an asymptotically flat manifold. [ABSTRACT FROM AUTHOR]

A k-involution is an involution with a fixed point set of codimension k. The conjugacy class of such an involution, denoted Sk, generates Möb(n)(the group of isometries of hyperbolic n-space) if k is odd and its orientation-preserving subgroup if k is even. In this paper, we supply effective lower and upper bounds for the Sk word length of Möb(n) if k is odd and the Sk word length of Möb+(n) if k is even. As a consequence, for a fixed codimension k, the length of Möb+(n) with respect to Sk, k even, grows linearly with n, with the same statement holding for Möb(n) in the odd case. Moreover, the percentage of involution conjugacy classes for which Möb+(n) has length two approaches zero as n approaches infinity. [ABSTRACT FROM AUTHOR]

Stimulated by a recent paper of J.-Ph. Anker and V. Pierfelice, we sharpen some dispersive estimates that arose in our previous work on nonlinear waves on 3D hyperbolic space. [ABSTRACT FROM AUTHOR]