Narrow Results

In this paper, we extend the classical result of Lyusternik and Fet on the existence of closed geodesics to singular spaces. We show that if $X$ is a compact geodesic metric space satisfying the CAT($\kappa$) condition for some fixed $\kappa >0$ and ${\pi }_n(X)\ne 0$ for some $n>0$ then $X$ has a periodic geodesic. This condition is satisfied for example by locally CAT($\kappa$) manifolds. Our result applies more generally to compact locally uniquely geodesic spaces.

In this paper, we give some generalized packing radius theorems of an $n$-dimensional Alexandrov space $M$ with curvature $\ge 1$. Let $\{q_1,\dots ,q_k\}$ be any $\frac{\pi }{2}$-separated subset in $M$ (i.e. the distance $\left|q_i{q}_j\right|\ge \frac{\pi }{2}$ for any $i\ne j$). Under the condition “$\left|q_i{q}_j\right|<\pi$” (after [K. Grove and F. Wilhelm, Hard and soft packing radius theorems, Ann. of Math.142 (1995) 213–237]), we give the upper bound of $k$ (which depends only on $n$), and classify the geometric structure of $M$ when $k$ attains the upper bound. As a corollary, we get an isometrical sphere theorem in Riemannian case.

We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e. small loops at $p\in X$ generate a subgroup of the fundamental group of the unit ball $B_1(p)$ that contains a nilpotent subgroup of index $\le w(n)$, where $w(n)$ is a constant depending only on the dimension $n$. The proof is based on the main ideas of V. Kapovitch, A. Petrunin and W. Tuschmann, and the following results:

(1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration. We also prove that such fibration is uniquely determined up to a homotopy equivalence.

(2) We give a detailed proof on the gradient push, improving the universal pushing time bound given by V. Kapovitch, A. Petrunin and W. Tuschmann, and justifying in a specific way that the gradient push between regular points can always keep away from extremal subsets.

It is shown that the Hodge theory for metric spaces based on the Alexander Spanier coboundary operator, in the presence of a measure previously developed in [4], holds for the class of compact Alexandrov spaces with curvature bounded from above. In particular, the real cohomology of the space is isomorphic to the corresponding space of harmonic co-chains.

We study sequences of integral current spaces $(X_j,d_j,T_j)$ such that the integral current structure $T_j$ has weight $1$ and no boundary and, all $(X_j,d_j)$ are closed Alexandrov spaces with curvature uniformly bounded from below and diameter uniformly bounded from above. We prove that for such sequences either their limits collapse or the Gromov–Hausdorff and Sormani–Wenger Intrinsic Flat limits agree. The latter is done showing that the lower $n$-dimensional density of the mass measure at any regular point of the Gromov–Hausdorff limit space is positive by passing to a filling volume estimate. In an appendix, we show that the filling volume of the standard $n$-dimensional integral current space coming from an $n$-dimensional sphere of radius $r>0$ in Euclidean space equals $r^n$ times the filling volume of the $n$-dimensional integral current space coming from the $n$-dimensional sphere of radius $1$.

Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently large compared to the weakness of singularities of $X$, then $f_j$ is a locally trivial fibration. Moreover, we show some properties on the intrinsic metric and the volume of the fibers of $f_j$.