A Sphere Theorem for 2-dimensional Cat(1)-spaces

The problems of sphere theorems in Riemannian geometry have yielded the beautiful results and the fruitful techniques for the study of global geometry (cf. [22]). The main purpose of this paper is to study sphere theorems for CAT(1)-spaces: When are CAT(1)-spaces homeomorphic to the sphere? The notion of CAT(κ)-spaces is introduced by Gromov ([11]) based on Alexandrov’s original notion, i.e., spaces with curvature bounded above by κ ∈ R. The research for CAT(1)-spaces is important since the space of directions at a given point in a CAT(κ)-space, which has the most local geometric information, is a CAT(1)-space. Furthermore, the ideal boundary of a given CAT(0)-space (the so-called, Hadamard space), which has the most global one, is a CAT(1)-space. In addition, all spherical buildings are CAT(1)-spaces (cf. [13], [23]). Throughout this paper, we always assume that CAT(κ)-spaces have the local compactness and the geodesical completeness. Nevertheless, the local metric structure may be complicated. For example, it is known by Kleiner that a CAT(κ)-space X may admit no triangulation even if X is 2-dimensional (cf. [12], [14]). We require the careful treatment of the local structure. IfX is a compact, geodesically complete CAT(1)-space, then the diameter of X is not smaller than π. There exist many examples of compact, geodesically complete CAT(1)-spaces possessing the minimal diameter π which are not homeomorphic to each other: Ballmann and Brin [5] have classified the isometry classes of the 2-dimensional spherical polyhedra in some sense which are such CAT(1)-spaces of the minimal diameter π. In this paper, we shall study volume sphere theorems for compact, geodesically complete CAT(1)-spaces.