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Register a new userAlexandrov geometry: preliminary version no. 1
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This is a preliminary version of our book. It goes up to the definition of dimension, which is about 30% of the material we plan to include. If you use it as a reference, do not forget to include the version number since the numbering will be changed.
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Let a compact Lie group act isometrically on a non-collapsing sequence of compact Alexandrov spaces with fixed dimension and uniform lower curvature and upper diameter bounds. If the sequence of actions is equicontinuous and converges in the equivariant Gromov--Hausdorff topology, then the limit space is equivariantly homeomorphic to spaces in the tail of the sequence. As a consequence, the class of Riemannian orbifolds of dimension $n$ defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is finite up to orbifold homeomorphism.
In this paper we discuss the sufficient and necessary conditions for multiple Alexandrov spaces being glued to an Alexandrov space. We propose a Gluing Conjecture, which says that the finite gluing of Alexandrov spaces is an Alexandrov space, if and only if the gluing is by path isometry along the boundaries and the tangent cones are glued to Alexandrov spaces. This generalizes Petrunins Gluing Theorem. Under the assumptions of the Gluing Conjecture, we classify the $2$-point gluing over $(n-1,epsilon)$-regular points as local separable gluing and the gluing near un-glued $(n-1,epsilon)$-regular points as local involutional gluing. We also prove that the Gluing Conjecture is true if the complement of $(n-1,epsilon)$-regular points is discrete in the glued boundary. In particular, this implies the general Gluing Conjecture as well as a new Gluing Theorem in dimension 2.
In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yaus gradient estimate for harmonic functions is also obtained on Alexandrov spaces.
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Chengs maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the volume upper bound is achieved. Our theorem also can be applied to Riemannian manifolds with non-smooth boundary, which generalizes Heintze and Karchers classical volume comparison theorem. Our main tool is the gradient flow of semi-concave functions.
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