Moderate smoothness of most Alexandrov surfaces

Abstract

We show that, in the sense of Baire categories, a typical Alexandrov surface with curvature bounded below by κ has no conical points. We use this result to prove that, on such a surface (unless it is flat), at a typical point, the lower and the upper Gaussian curvatures are equal to κ and ∞ respectively.Math. Subj. Classification (2010): 53C45

Farthest points on most Alexandrov surfaces

“…Denote by R (κ) the set of all closed Riemannian surfaces with Gauss curvature at least κ, and by P (κ) the set of κ-polyhedra. A formal proof for the next result can be found, for instance, in [8].…”

“…It is known that, endowed with the topology induced by the Gromov-Hausdorff distance, the set A(κ) is a Baire space [8]. In any Baire space, one says that most elements enjoy, or that a typical element enjoys, a given property if the set of those elements which do not satisfy it is of first category.…”

“…The investigation of typical properties of Alexandrov surfaces from Baire category viewpoint is very recent, see [1], [8], [15]; it generalizes a similar, well-established research direction for convex surfaces, see e.g. the survey [6].…”

Hyperbolization of cusps with convex boundary

“…By a theorem of Alexandrov and Zalgaller, (S, m) can be decomposed into non-overlapping geodesic triangles. Replacing each triangle by a comparison triangle in the space of constant curvature k, we obtain a polyhedral CBB(k) metric on S see [Ric12,IRV15] for details. In particular, we obtain the following.…”

Simple closed geodesics on most Alexandrov surfaces

“…It is also known that, endowed with topology induced by the Gromov-Hausdorff distance, A(κ) is a Baire space [15]. In any Baire space, one says that most elements or a typical element enjoys a property P if the set of those elements which do not satisfy P it is of first category.…”