Positive Quaternionic Kaehler manifolds and symmetry rank: II

Let M be a positive quaternionic Kaehler manifold of dimension 4m. If the isometry group Isom(M) has rank at least m/2 +3, then M is isometric to HP^m or Gr_2(C^{m+2}). The lower bound for the rank is optimal if m is even.Comment: 10 page

Finite isometry groups of 4-manifolds with positive sectional curvature

Let M be an oriented compact positively curved 4-manifold. Let G be a finite subgroup of the isometry group of $M$. Among others, we prove that there is a universal constant C (cf. Corollary 4.3 for the approximate value of C), such that if the order of G is odd and at least C, then G is either abelian of rank at most 2, or non-abelian and isomorphic to a subgroup of PU(3) with a presentation \{A, B| A^m=B^n=1, BAB^{-1}=A^r, (n(r-1), m)=1, r\ne r^3=1(\text{mod}m) \}. Moreover, M is homeomorphic to CP^2 if G is non-abelian, and homeomorphic to S^4 or CP^2 if G is abelian of rank 2.Comment: 17 page

Positive quaternionic Kaehler manifolds and symmetry rank

A quaternionic K\"ahler manifold M is called {\it positive} if it has positive scalar curvature. The main purpose of this paper is to prove several connectedness theorems for quaternionic immersions in a quaternionic K\"ahler manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic submanifolds in a positive quaternionic K\"ahler manifold. As applications we prove that, among others, a 4m-dimensional positive quaternionic K\"ahler manifold with symmetry rank at least (m-2) must be either isometric to \Bbb HP^m or Gr_2(\Bbb C^{m+2}), if m\ge 10.Comment: 21 page

Positively curved manifolds with maximal discrete symmetry rank

Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a constant p(n)>0 such that (1) If M^{2n} admits an effective isometric \Bbb Z_p^k-action for a prime p\ge p(n), then k\le n and ``='' implies that M^{2n} is homeomorphic to a sphere or a complex projective space. (2) If M^{2n+1} admits an isometric S^1 x \Bbb Z_p^k-action for a prime p\ge p(n), then k\le n and ``='' implies that M is homeomorphic to a sphere. (3) For M in (1) or (2), if n\ge 7 and k\ge [\frac{3n}4]+2, then M is homeomorphic to a sphere or homotopic to a complex projective space.Comment: 18 page

Collapsed 5-manifolds with pinched positive sectional curvature

Let M be a closed 5-manifold of pinched curvature 0<\delta\le \text{sec}_M\le 1. We prove that M is homeomorphic to a spherical space form if M satisfies one of the following conditions: (i) \delta =1/4 and the fundamental group is a non-cyclic group of order at least C, a constant. (ii) The center of the fundamental group has index at least w(\delta), a constant depending on \delta. (iii) The ratio of the volume and the maximal injectivity radius is less than \epsilon(\delta). (iv) The volume is less than \epsilon(\delta) and the fundamental group \pi_1(M) has a center of index at least w, a universal constant, and \pi_1(M) is either isomorphic to a spherical 5-space group or has an odd order.Comment: 41 page

Convergence of Kaehler-Ricci flow with integral curvature bound

Let $g(t)$, $t\in [0, +\infty)$, be a solution of the normalized K\"ahler-Ricci flow on a compact K\"ahler $n$-manifold $M$ with $c_{1}(M)>0$ and initial metric $g (0)\in 2\pi c_{1}(M)$. If there is a constant $C$ independent of $t$ such that $$\int_{M}|Rm(g(t))|^{n}dv_{t}\leq C,$$ then, for any $t_{k}\to \infty$, a subsequence of $(M, g(t_{k}))$ converges to a compact orbifold $(X, h)$ with only finite many singular points $\{q_{j}\}$ in the Gromov-Hausdorff sense, where $h$ is a K\"ahler metric on $X\backslash \{q_{j}\}$ satisfying the K\"ahler-Ricci soliton equation, i.e. there is a smooth function $f$ such that $$Ric(h)-h=\nabla\bar{\nabla}f, {\rm and}\it \nabla \nabla f=\bar{\nabla} \bar{\nabla} f=0. $

Reflection groups in non-negative curvature

We provide an equivariant description/classification of all complete (compact or not) non-negatively curved manifolds M together with a co-compact action by a reflection group W, and moreover, classify such W. In particular, we show that the building blocks consist of the classical constant curvature models and generalized open books with non negatively curved bundle pages, and derive a corresponding splitting theorem for the universal cover.Comment: 20 page

Homeomorphism Classification of positively curved manifolds with almost maximal symmetry rank

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m>4) of positive sectional curvature on which a (m-1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By a result of Wilking, these results imply a homeomorphism classification for positively curved n-manifolds (n>7) of almost maximal symmetry rank [\frac{n-1}2].Comment: 20 page

Secondary Brown-Kervaire Quadratic forms and π\pi-manifolds

In this paper we define a secondary Brown-Kervaire quadratic forms. Among the applications we obtain a complete classification of (n-2)-connected 2n-dimensional framed manifolds up to homeomorphism and homotopy equivalence, . In particular, we prove that the homotopy type of such manifolds determine their homeomorphism type

An almost flat manifold with a cyclic or quaternionic holonomy group bounds

A long-standing conjecture of Farrell and Zdravkovska and independently S.~T.~Yau states that every almost flat manifold is the boundary of a compact manifold. This paper gives a simple proof of this conjecture when the holonomy group is cyclic or quaternionic. The proof is based on the interaction between flat bundles and involutions.Comment: 8 pages, to appear in the Journal of Differential Geometry. New version of Lemma 2.5: A manifold bounds if there is an involution on TM whose fixed bundle is full ran