论文标题
在保形的EIN不变性
On the conformal Ein invariants
论文作者
论文摘要
对于紧凑的Riemannian $ n $ -Manifold $（m，g）的正标性曲率，$ g $的资本$ \ ein $不变式被定义为Ricci Curvature的最大特征值的标量曲率曲率特征的$ m $以上。 This is a re-scale invariant and belongs to the interval $(0,n]$. For a positive conformal class $[g]$, we define the conformal invariant $\Ein([g]):=\sup\{\Ein(g): g\in [g]\}$. In this paper, we prove vanihing theorems for Betti numbers and for the higher homotopy groups of $ m $在$ \ ein（[g]）上的最佳下限，假设$ g $是当地的，我们建立了与Schoen-yau形式不变的$ d（m，[g]）$的不平等。在$ \ ein（[g]）> k $的情况下，在连接总和的运行下，$ 0 <k <n-1。$ \\对于一般的正形式类别，我们以$ 4 $ $ 4 $与第一和第二个Yamabe Invariants相关的不平等$ 4 $证明。 本文证明了类似的保形不变的类似结果，即小型$ \ ein $不变。
For a compact Riemannian $n$-manifold $(M,g)$ of positive scalar curvature, the capital $\Ein$ invariant of $g$ is defined to be the infinimum over $M$ of the quotient of the scalar curvature by the maximal eigenvalue of the Ricci curvature. This is a re-scale invariant and belongs to the interval $(0,n]$. For a positive conformal class $[g]$, we define the conformal invariant $\Ein([g]):=\sup\{\Ein(g): g\in [g]\}$. In this paper, we prove vanihing theorems for Betti numbers and for the higher homotopy groups of $M$ under optimal lower bounds on $\Ein([g])$ assuming that $g$ is locally conformally flat. We establish an inequality relating our invariant to Schoen-Yau conformal invariant $d(M,[g])$ from which we deduce a classification result for locally conformally flat manifolds with higher $\Ein([g])$. We show that the class of locally conformally flat manifolds with $\Ein([g])>k$ is stable under the operation of connected sums for $0<k<n-1.$\\ For a general positive conformal class, we prove in dimension $4$ an inequality relating $\Ein([g])$ to the first and second Yamabe invariants. Similar results are proved in this paper for an analogous conformal invariant, namely the small $\ein$ invariant.
