Hyperbolic Space Form Research Articles

Geometric estimates on weighted p-fundamental tone and applications to the first eigenvalue of submanifolds with bounded mean curvature

ABSTRACT This paper generalizes to the context of smooth metric measure spaces and submanifolds with negative sectional curvatures some well-known geometric estimates on the p-fundamental tone by using vector fields satisfying a positive divergence condition. Choosing the vector field to be the gradient of an appropriately chosen distance function yields generalised McKean estimates whilst other choices of vector fields yield new geometric estimates generalising certain results of Lima et al. (Nonlinear Anal. 2010;72:771–781). We also obtain a lower bound on the spectrum of the weighted p-Laplacian on a complete noncompact smooth metric space with the underlying space being a submanifold with bounded mean curvature in the hyperbolic space form of constant negative sectional curvature generalising results of Du and Mao (J Math Anal Appl. 2017;456:787–795).

On effective existence of symmetric differentials of complex hyperbolic space forms

For a noncompact complex hyperbolic space form of finite volume $$X=\mathbb {B}^n/\varGamma $$ , we consider the problem of producing symmetric differentials vanishing at infinity on the Mumford compactification $$\overline{X}$$ of X similar to the case of producing cusp forms on hyperbolic Riemann surfaces. We introduce a natural geometric measurement which measures the size of the infinity $$\overline{X}-X$$ called canonical radius of a cusp of $$\varGamma $$ . The main result in the article is that there is a constant $$r^*=r^*(n)$$ depending only on the dimension, so that if the canonical radii of all cusps of $$\varGamma $$ are larger than $$r^*$$ , then there exist symmetric differentials of $$\overline{X}$$ vanishing at infinity. As a corollary, we show that the cotangent bundle $$T_{\overline{X}}$$ is ample modulo the infinity if moreover the injectivity radius in the interior of $$\overline{X}$$ is larger than some constant $$d^*=d^*(n)$$ which depends only on the dimension.

Curvature free rigidity for higher rank three-manifolds

We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.

Hopf hypersurfaces in spaces of oriented geodesics

A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector. We consider particular Hopf hypersurfaces in the space of oriented geodesics of a non-flat space form of dimension greater than 2. For spherical and hyperbolic space forms, the oriented geodesic space admits a canonical Kaehler-Einstein and para-Kaehler-Einstein structure, respectively, so that a natural notion of a Hopf hypersurface exists. The particular hypersurfaces considered are formed by the oriented geodesics that are tangent to a given convex hypersurface in the underlying space form. We prove that a tangent hypersurface is Hopf in the space of oriented geodesics with respect to this canonical (para-)Kaehler structure iff the underlying convex hypersurface is totally umbilic and non-flat. In the case of 3 dimensional space forms, however, there exists a second canonical complex structure which can also be used to define Hopf hypersurfaces. We prove that in this dimension, the tangent hypersurface of a convex hypersurface in the space form is always Hopf with respect to this second complex structure.

Einstein manifolds with finite Lp-norm of the Weyl curvature

Let (Mn,g)(n≥4) be an n-dimensional complete Einstein manifold. Denote by W the Weyl curvature tensor of M. We prove that (Mn,g) is isometric to a spherical space form if (Mn,g) has positive scalar curvature and unit volume, and the Lp(p≥n2)-norm of W is pinched in [0,C), where C is an explicit positive constant depending only on n, p and S, which improves the isolation theorems given by [24,14,17].This paper also states that W goes to zero uniformly at infinity if for p≥n2, the Lp-norm of W of M with non-positive scalar curvature and positive Yamabe constant is finite. Assume that M has negative scalar curvature and the Lα-norm of W is finite. As application, we prove that M is a hyperbolic space form if the Lp-norm of W is sufficiently small, which generalizes an Ln2-norm of W pinching theorem in [19].

Functions with finite Dirichlet sum of order $p$ and quasi-monomorphisms of infinite graphs

In this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of p-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension n and it is not p-parabolic for p > n - 1, then it admits a lot of p-harmonic functions with finite Dirichlet sum of order p.

The Inradius of a Hyperbolic Truncated $$n$$ n -Simplex

Hyperbolic truncated simplices are polyhedra bounded by at most $$2n+2$$ 2 n + 2 hyperplanes in hyperbolic $$n$$ n -space. They provide important models in the context of hyperbolic space forms of small volume. In this work, we derive an explicit formula for their inradius by algebraic means and by using the concept of reduced Gram matrix. As an illustration, we discuss implications for some polyhedra related to small volume arithmetic orientable hyperbolic orbifolds.

SOME HYPERBOLIC SPACE FORMS WITH FEW GENERATED FUNDAMENTAL GROUPS

We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane re ections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.

Mean curvature flow of higher codimension in hyperbolic spaces

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a pinching condition in a hyperbolic space form to a round point in finite time.

Functions with finite Dirichlet sum of orderpand quasi-monomorphisms of infinite graphs

AbstractIn this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space ofp-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimensionnand it is notp-parabolic forp > n -1, then it admits a lot ofp-harmonic functions with finite Dirichlet sum of orderp.

Spectral theory on 3-dimensional hyperbolic space and Hermitian modular forms

Abstract We study some arithmetics of Hermitian modular forms of degree two by applying the spectral theory on 3-dimensional hyperbolic space. This paper presents three main results: (1) a 3-dimensional analogue of Katok–Sarnak's correspondence, (2) an analytic proof of a Hermitian analogue of the Saito–Kurokawa lift by means of a converse theorem, (3) an explicit formula for the Fourier coefficients of a certain Hermitian Eisenstein series.

The second variation of the Ricci expander entropy

The critical points of the W + functional introduced by M. Feldman, T. Ilmanen and L. Ni are the expanding Ricci solitons, which are special solutions of the Ricci flow. On compact manifolds, expanding solitons coincide with Einstein metrics. In this paper, we compute the first and second variations of the entropy functional of the W + functional, and briefly discuss the linear stability of compact hyperbolic space forms.

On complete submanifolds with bounded mean curvature

We consider M n , n ≥ 3 , an n -dimensional complete and connected submanifold of a space form M n + p ( c ) , whose mean curvature H does not vanish and is bounded with a parallel normalized mean curvature vector. We prove that if S ≤ n 2 H 2 n − 1 + 2 c , where S denotes the squared norm of the second fundamental form of M n , then the codimension of isometric immersion reduces to 1. This result generalizes the case where the mean curvature vector is parallel or mean curvature is constant. If M n is a compact submanifold of hyperbolic space form with constant mean curvature H ≠ 0 , then M n is a geodesic sphere. When M n is a submanifold of the unit sphere with constant mean curvature H ≠ 0 , then either M n is a great or small sphere in S n + 1 ( 1 ) or M n is a product of spheres S l ( r ) × S m ( s ) .

Almost quarter-pinched Kähler metrics and Chern numbers

We prove that compact K\"ahler manifolds whose sectional curvatures are close to 1/4-pinched have ratios of Chern numbers close to the corresponding ratios of a complex hyperbolic space form. We deduce that the Mostow-Siu surfaces (and their three-dimensional analogues constructed by the first author) do not admit K\"ahler metrics with pinching close to 1/4.

Biharmonic Riemannian submersions from 3-manifolds

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa [CI] and Jiang [Ji] states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper [CMO2], Cadeo-Monttaldo-Oniciuc shown that the theorem remains true if the target Euclidean space is replaced by a 3-dimensional hyperbolic space form. In this paper, we prove the dual results for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional space form of non-positive curvature into a surface is biharmonic if and only if it is harmonic.

On Stability of the Hyperbolic Space Form under the Normalized Ricci Flow

Abstract This paper studies the normalized Ricci flow from a slight per turbation of the hy-perbolic metric on H n . It’s proved that if the perturbation is small and decays sufficiently fastat the infinity, then the flow will converge exponentially fas t to the hyperbolic metric when thedimension n > 5. 1 Introduction The Ricci flow of Hamilton evolves the metric of a Riemannian m anifold in the direction ofan Einstein metric. There is a natural question that if one starts from a small perturbation ofan Einstein metric, or without the a priori knowledge of the existence of an Einstein metric,from a sufficiently (Ricci) pinched metric, can we show the flo w converges to (the) Einsteinmetric? The problem is addressed by R. Ye in [7]. Ye proved several theorems in this directionin the case of closed Riemannian manifolds. Central to his proof is a concept of stability. Onone hand, if the solution to the Ricci flow remains stable, the L 2 -norm of traceless Ricci tensordecays exponentially. On the other hand, if the solution remains pinched, it is stable. Since thetraceless Ricci tensor is almost the right hand side of the Ricci flow equation, Ye was able tocombine the above two observations to conclude if the initial metric is sufficiently pinched thenthe solution will remain so for any later time and converge to some Einstein metric.In this paper, we try to study a similar problem for complete noncompact manifolds. Moreprecisely, we are concerned with the normalized Ricci flow∂g

Complete hypersurfaces in the hyperbolic space form with parallel shape operator

Let M n be a hypersurface in H n+1 . We give a classification for hypersurfaces with parallel shape operator.

On holomorphic immersions into kähler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f:X →M from a complex manifoldX into a Kahler manifold of constant holomorphic sectional curvatureM, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. ForX compact we show that the tangent sequence splits holomorphically if and only iff is a totally geodesic immersion. ForX not necessarily compact we relate an intrinsic cohomological invariantp(X) onX, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariantsp(X) and?(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially whenX is a complex surface andM is of complex dimension 4, under the assumption thatX admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.

Orbifold compactification and solutions of M-theory from Milne spaces

In this paper, we consider solutions and spectral functions of M-theory from Milne spaces with extra free dimensions. Conformal deformations to the metric associated with the real hyperbolic space forms are derived. For the three-dimensional case, the orbifold identifications $SL(2,{\mathbb Z}+i{\mathbb Z})/\{\pm Id\}$, where $Id$ is the identity matrix, is analyzed in detail. The spectrum of a eleven-dimensional field theory can be obtained with the help of the theory of harmonic functions in the fundamental domain of this group and it is associated with the cusp forms and the Eisenstein series. The supersymmetry surviving for supergravity solutions involving real hyperbolic space factors is briefly discussed.

Integral Congruence Two Hyperbolic 5-Manifolds

In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form H 5/Γ where Γ is a torsion-free subgroup of minimal index of the congruence two subgroup Γ 5 2 of the group Γ 5 of positive units of the Lorentzian quadratic form x 2/1 +... +x 5/2 -x 6/2. We also show that Γ 5 2 is a reflection group with respect to a 5-dimensional right-angled convex polytope in H 5. As an application, we construct a hyperbolic 5-manifold of smallest known volume 7ζ (3)/4.