Horo-tightness and total (absolute) curvatures in hyperbolic spaces

We prove Gauß-Bonnet-type and Chern-Lashof-type formulas for immersions in hyperbolic space. Moreover we investigate the notion of tightness with respect to horospheres introduced by T.E. Cecil and P.J. Ryan. We introduce the notions of top-set and drop-set, and we prove fundamental properties of horo-tightness in hyperbolic spaces.     

Finite metric spaces of strictly negative type

We prove that, if a finite metric space is of strictly negative type, then its transfinite diameter is uniquely realized by the infinite extender (load vector). Finite metric spaces that have this property include all spaces on two, three, or four points, all trees, and all finite subspaces of Euclidean spaces. We prove that, if the distance matrix is both hypermetric and regular, then it is of strictly negative type. We show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two pairs of antipodal points. In connection with an open problem raised by Kelly, we conjecture that all finite subspaces of hyperbolic spaces are hypermetric and regular, and hence of strictly negative type.     

Parabolic approximations of diffusive–dispersive equations

We consider a lower-order approximation for a third-order diffusive–dispersive conservation law with nonlinear flux. It consists of a system of two second-order parabolic equations; a coupling parameter is also added. If the flux has an inflection point it is well-known, on the one hand, that the diffusive–dispersive law admits traveling-wave solutions whose end states are also connected by undercompressive shock waves of the underlying hyperbolic conservation law. On the other hand, if the diffusive–dispersive regularization vanishes, the solutions of the corresponding initial-value problem converge to a weak solution of the hyperbolic conservation law. We show that both of these properties also hold for the lower-order approximation. Furthermore, when the coupling parameter tends to infinity, we prove that solutions of initial value problems for the approximation converge to a weak solution of the diffusive–dispersive law. The proofs rely on new a priori energy estimates for higher-order derivatives and the technique of compensated compactness.     

Cusp closing in rank one symmetric spaces

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily–Borel resp.Siu–Yau compactification of finite volume quotients of the complex hyperbolic space. Oblatum 2-IX-1994 & 7-VIII-1995     

Relative hyperbolicity and Artin groups

This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a two-dimensional Artin group the Deligne complex is Gromov hyperbolic precisely when the corresponding Davis complex is Gromov hyperbolic, that is, precisely when the underlying Coxeter group is a hyperbolic group. For Artin groups of FC type we give a sufficient condition for hyperbolicity of the Deligne complex which applies to a large class of these groups for which the underlying Coxeter group is hyperbolic. The key tool in the proof is an extension of the Milnor-Svarc Lemma which states that if a group G admits a discontinuous, co-compact action by isometries on a Gromov hyperbolic metric space, then G is weakly hyperbolic relative to the isotropy subgroups of the action.      

Cusp closing in rank one symmetric spaces

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily-Borel resp. Siu-Yau compactification of finite volume quotients of the complex hyperbolic space.Oblatum 2-IX-1994 & 7-VIII-1995     

Mean curvature rigidity of horospheres,hyperspheres, and hyperplanes

Archiv der Mathematik - We prove that horospheres, hyperspheres, and hyperplanes in a hyperbolic space $${\mathbb {H}}^n,\,n\ge 3$$ , admit no perturbations with compact support which increase...     

Partially hyperbolic geodesic flows

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such an example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.     

Quasi-isometries between visual hyperbolic spaces

We prove that a power quasi-symmetric (or PQ-symmetric) homeomorphism between two complete metric spaces can be extended to a quasi-isometry between their hyperbolic approximations. This result can be used to prove that two visual Gromov hyperbolic spaces are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundaries with bounded visual metrics. Also, in the case of trees, we prove that two geodesically complete trees are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundaries with visual metrics based at infinity. We also give a characterization for a map to be PQ-symmetric based on the relative distortion of subsets.     

Robust Weak Ergodicity And Stable Ergodicity

In this paper, we define robust weak ergodicity and study the relation between robust weak ergodicity and stable ergodicity for conservative partially hyperbolic systems. We prove that a C^r(r > 1) conservative partially hyperbolic diffeomorphism is stably ergodic if it is robustly weakly ergodic and has positive (or negative) central exponents on a positive measure set. Furthermore, if the condition of robust weak ergodicity is replaced by weak ergodicity, then the diffeomophism is an almost stably ergodic system. Additionally, we show in dimension three, a C^r(r > 1) conservative partially hyperbolic diffeomorphism can be approximated by stably ergodic systems if it is robustly weakly ergodic and robustly has non-zero central exponents.     

Short separating geodesics for multiply connected domains

We consider the following questions: given a hyperbolic plane domain and a separation of its complement into two disjoint closed sets each of which contains at least two points, what is the shortest closed hyperbolic geodesic which separates these sets and is it a simple closed curve? We show that a shortest geodesic always exists although in general it may not be simple. However, one can also always find a shortest simple curve and we call such a geodesic a meridian of the domain. We prove that, although they are not in general uniquely defined, if one of the sets of the separation of the complement is connected, then they are unique and are also the shortest possible geodesics which separate the complement in this fashion.     

Hyperbolically convex constricted domains

A subdomain G in the unit disk D is called hyperbolically convex if the non-euclidean segment between any two points in G also lies in G. We introduce the concept of constricted domain relative to the hyperbolic geometry of D and prove that a hyperbolic convex domain is constricted if and only if it is not a quasidisk. Also examples are given to illustrate these ideas.     

Perturbation of semi-linear evolution equations under weak assumptions at initial time

We prove perturbation results for abstract semi-linear evolution equations in a Banach space. The main feature is that only very weak assumptions are needed at initial time. This allows to prove weak continuity properties and to deal with rather general domain perturbation problems for semi-linear parabolic and hyperbolic boundary value problems with various boundary conditions. The theory also implies the well known theory on parameter dependent equations.     

On the optimality of some classes of invex problems

In this paper we define two notions: Kuhn–Tucker saddle point invex problem with inequality constraints and Mond–Weir weak duality invex one. We prove that a problem is Kuhn–Tucker saddle point invex if and only if every point, which satisfies Kuhn–Tucker optimality conditions forms together with the respective Lagrange multiplier a saddle point of the Lagrange function. We prove that a problem is Mond–Weir weak duality invex if and only if weak duality holds between the problem and its Mond–Weir dual one. Additionally, we obtain necessary and sufficient conditions, which ensure that strong duality holds between the problem with inequality constraints and its Wolfe dual. Connections with previously defined invexity notions are discussed.     

Einstein hypersurfaces of the Cayley projective plane

We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a certain radius; this completes the classification of Einstein hypersurfaces in rank-one symmetric spaces.     

2-dimensional combinatorial Calabi flow in hyperbolic background geometry

For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small enough. Assuming the curvature has a uniform upper bound less than 2π, we prove that combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists.     

Integrals,Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems

We generalize the notions of Laplace transformations and Laplace invariants for systems of hyperbolic equations and study conditions for their existence. We prove that a hyperbolic system admits the Laplace transformation if and only if there exists a matrix of rank k mapping any vector whose components are functions of one of the independent variables into a solution of this system, where k is the defect of the corresponding Laplace invariant. We show that a chain of Laplace invariants exists only if the hyperbolic system has a entire collection of integrals and the dual system has a entire collection of solutions depending on arbitrary functions. An example is given showing that these conditions are not sufficient for the existence of a Laplace transformation.     

Local existence of stratified solutions to systems of balance laws

We define a class of weak solutions to hyperbolic systems of balance laws, in one space dimension; they are called here stratified solutions. For such solutions we prove some results about the propagation, the life span and the initial-value problem. To the memory of Lamberto Cattabriga     

Complete submanifolds in manifolds of partially non-negative curvature

We prove that a complete non-compact submanifold in a complete manifold of partially non-negative sectional curvature has only one end if the Sobolev inequality holds on it and if its total curvature is not very big by showing a Liouville theorem for harmonic maps and by using a existence theorem of constant harmonic functions with finite energy. We also generalize a result by Cao–Shen–Zhu saying that a complete orientable stable minimal hypersurface in a Euclidean space has only one end to submanifolds in manifolds of partially non-negative sectional curvature. Some related results about the structure of the same kind of submanifolds are also obtained.     

Rigidity of minimal submanifolds in hyperbolic space

We prove that if an n-dimensional complete minimal submanifold M in hyperbolic space has sufficiently small total scalar curvature then M has only one end. We also prove that for such M there exist no nontrivial L ² harmonic 1-forms on M.