No abstract

In this paper we study the (equivariant) topological types of a class of 3-dimensional closed manifolds (i.e., 3-dimensional small covers), each of which admits a locally standard $(\mathbb{Z}_2)^3$-action such that its orbit space is a simple convex 3-polytope. We introduce six equivariant operations on 3-dimensional small covers. These six operations are interesting because of their combinatorial natures. Then we show that each 3-dimensional small cover can be obtained from $\mathbb{R}P^3$ and $S^1\times\mathbb{R}P^2$ with certain $(\mathbb{Z}_2)^3$-actions under these six operations. As an application, we classify all 3-dimensional small covers up to $({\Bbb Z}_2)^3$-equivariant unoriented cobordism.Comment: 34 pages with 40 figures, final version for publicatio

In this paper, we prove that if an asymptotically Euclidean manifold with nonnegative scalar curvature has long time existence of Ricci flow, the ADM mass is nonnegative. In addition, we give an independent proof of positive mass theorem in dimension three. Contentsfor any partial derivative of order k as r → ∞, where r is the Euclidean distance function. We call the positive number σ i the order of end E i . The ADM mass [5] from general relativity of an AE manifold (M, g) is defined aswhere dA j = ∂ j dV g E and g E is the canonical Euclidean metric on R n .The definition of mass involves a choice of asymptotic coordinates. But it follows from Bartnik's result [6] that if the order σ > (n − 2)/2 and the scalar curvature is integrable, then the mass is finite and independent of AE coordinates. In other words, m(g) depends only on the metric g.The general positive mass conjecture is the following, see [26, Theorem 10.1].Conjecture 1.1 (Positive Mass Conjecture). Let (M n , g) be an AE manifold of dimension n ≥ 3 with the order σ > (n − 2)/2, and nonnegative integrable scalar curvature. Then m(g) ≥ 0 with equality if and only if (M, g) = (R n , g E ).In dimension three, the positive mass conjecture was first proved by Schoen and Yau [41] in 1979 by constructing a stable minimal surface and considering its stability inequality. In addition, Schoen and Yau showed that their method could be extended to the case when the dimension was less than eight [39,42]. In 1981, Witten [46] proved the positive mass conjecture for spin manifolds of any dimension. In 2001, Huisken and Ilmanen [21] proved the stronger Riemannian Penrose inequality in dimension three by using the inverse mean curvature flow. In 2015, Hein and LeBrun gave a proof of the positive mass conjecture for Kähler AE manifolds, see [23]. To the author's knowledge, there is no proof of the positive mass conjecture in general dimension.A natural question arises, can we prove the positive mass conjecture by using other geometric flows? Since Ricci flow is one of the most powerful geometric flows by which Perelman have completely solved Thurston's geometrization conjecture, see [33,34,35], it is of interest to know how Ricci flow interacts with AE manifolds and the ADM mass.Recall that Ricci flow is a geometric flow such that a family of metrics g(t) on a smooth manifold M are evolved under the PDE ∂ t g(t) = −2Rc(g(t)).( 1.2) We will focus on the case when (M, g (0)) is an AE manifold.It has been proved by Dai and Ma in [18] that Ricci flow preserves the ALE condition, nonnegative integrable scalar curvature and the ADM mass. Hence, it is important to understand the change of mass at possible singular times and infinity if long time existence of Ricci flow is assumed.One of the main theorems in this paper shows that if we have long time existence of Ricci flow, an AE manifold will converge to the Euclidean space in some strong sense. The proof is partially motivated by considering possible steady solitons on ALE manifolds, see Appendix. The convergence at ti...

This paper is the sequel to our study of heat kernels on Ricci shrinkers in [28]. In this paper, we improve many estimates in [28] and extend the recent progress of Bamler [2]. In particular, we drop the compactness and curvature boundedness assumptions and show that the theory of F-convergence holds naturally on any Ricci flows induced by Ricci shrinkers.