We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau
manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kahler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.
The $pi_2$-diffeomorphism finiteness result (cite{FR1,2}, cite{PT}) asserts that the diffeomorphic types of compact $n$-manifolds $M$ with vanishing first and second homotopy groups can be bounded above in terms of $n$, and upper bounds on the absolu
te value of sectional curvature and diameter of $M$. In this paper, we will generalize this $pi_2$-diffeomorphism finiteness by removing the condition that $pi_1(M)=0$ and asserting the diffeomorphism finiteness on the Riemannian universal cover of $M$.
We will present a new proof for the Gromovs theorem on almost flat manifolds ([Gr], [Ru]).
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows techniques, we partially extend the
nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci boundedcovering geometry.
In the study manifolds of Ricci curvature bounded below, a stumbling obstruction is the lack of links between large-scale geometry and small-scale geometry at a fixed reference point. There have been few links (volume, dimension) when the unit ball a
t the point is not collapsed, that is, $mathrm{vol}(B_1(p))ge v>0$. In this paper, we conjecture a new link in terms of isometries: if the maximal displacement of an isometry $f$ on $B_1(p)$ is at least $delta>0$, then the maximal displacement of $f$ on the rescaled unit ball $r^{-1}B_r(p)$ is at least $Phi(delta,n,v)>0$ for all $rin(0,1)$. We call this scaling $Phi$-nonvanishing property at $p$. We study the equivariant Gromov-Hausdorff convergence of a sequence of Riemannian universal covers with abelian $pi_1(M_i,p_i)$-actions $(widetilde{M}_i,tilde{p}_i,pi_1(M_i,p_i))overset{GH}longrightarrow(widetilde{X},tilde{p},G)$, where $pi_1(M_i,p_i)$-action is scaling $Phi$-nonvanishing at $tilde{p_i}$. We establish a dimension monotonicity on the limit group associated to any rescaling sequence. As one of the applications, we prove that for an open manifold $M$ of non-negative Ricci curvature, if the universal cover $widetilde{M}$ has Euclidean volume growth and $pi_1(M,p)$-action on $R^{-1}widetilde{M}$ is scaling $Phi$-nonvanishing at $tilde{p}$ for all $R$ large, then $pi_1(M)$ is finitely generated.
The Milnor Problem (modified) in the theory of group growth asks whether any finite presented group of vanishing algebraic entropy has at most polynomial growth. We show that a positive answer to the Milnor Problem (modified) is equivalent to the Nil
potency Conjecture in Riemannian geometry: given $n, d>0$, there exists a constant $epsilon(n,d)>0$ such that if a compact Riemannian $n$-manifold $M$ satisfies that Ricci curvature $op{Ric}_Mge -(n-1)$, diameter $dge op{diam}(M)$ and volume entropy $h(M)<epsilon(n,d)$, then the fundamental group $pi_1(M)$ is virtually nilpotent. We will verify the Nilpotency Conjecture in some cases, and we will verify the vanishing gap phenomena for more cases i.e., if $h(M)<epsilon(n,d)$, then $h(M)=0$.
In this paper, we prove the Soul Conjecture in Alexandrov geometry in dimension $4$, i.e. if $X$ is a complete non-compact $4$-dimensional Alexandrov space of non-negative curvature and positive curvature around one point, then a soul of $X$ is a point.
Given two $n_i$-dimensional Alexandrov spaces $X_i$ of curvature $ge 1$, the join of $X_1$ and $X_2$ is an $(n_1+n_2+1)$-dimensional Alexandrov space $X$ of curvature $ge 1$, which contains $X_i$ as convex subsets such that their points are $frac pi2
$ apart. If a group acts isometrically on a join that preserves $X_i$, then the orbit space is called quotient of join. We show that an $n$-dimensional Alexandrov space $X$ with curvature $ge 1$ is isometric to a finite quotient of join, if $X$ contains two compact convex subsets $X_i$ without boundary such that $X_1$ and $X_2$ are at least $frac pi2$ apart and $dim(X_1)+dim(X_2)=n-1$.
This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=pm 1$ or $0$ is diffeomorphic to a $H$-space f
orm if for every ball of definite size on $M$, the lifting ball on the Riemannian universal covering space of the ball achieves an almost maximal volume, provided the diameter of $M$ is bounded for $H e 1$. In [CRX], we verified the conjecture for the case that $M$ or its Riemannian universal covering space $tilde M$ is not collapsed for $H=1$ or $H e 1$ respectively. In the present paper, we will verify this conjecture for the case that Ricci curvature is also bounded above, while the above non-collapsing condition is not required.
Let $M$ be a compact $n$-manifold of $operatorname{Ric}_Mge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (
i) There is $rho>0$ such that for any $xin M$, the open $rho$-ball at $x^*$ in the (local) Riemannian universal covering space, $(U^*_rho,x^*)to (B_rho(x),x)$, has the maximal volume i.e., the volume of a $rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H=-1$, the volume entropy of $M$ is maximal i.e. $n-1$ ([LW1]). The main results of this paper are quantitative space form rigidity i.e., statements that $M$ is diffeomorphic and close in the Gromov-Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H=1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].
