Restrictions on scalar curvature and first Dirac eigenvalue of closed manifolds

在黎曼度规的适当限制下，黎曼流形的标量曲率K不可能处处变得任意大。Lichnerowicz和Gromov-Lawson的结果表明，对于大族流形，min _()≤0。Gromov， Llarull等人的结果表明，其他流形的最小值有正上界，在某些特殊情况下是最优的。同样，在封闭黎曼自旋流形上，如果Â-genus不为零，则最小绝对值的狄拉克本征值λ1为0。在其他情况下，vfa - witten和Gromov给出的|λ1|存在普遍上界，同样是在黎曼度规的适当限制下。Herzlich等给出了某些流形|λ1|的最优上界。注意，上面的两个问题都与Friedrich不等式直接相关，该不等式将min从上面的边界限定为|λ1|。此外，这两个问题所采用的一些方法是相似的。在这个项目中，我们想要在更大的黎曼流形上找到标量曲率和第一狄拉克特征值的上界。

Under suitable restrictions on the Riemannian metric, the scalar curvature K of a Riemannian manifold cannot become arbitrarily large everywhere. The results of Lichnerowicz and Gromov-Lawson show that min к ≤ 0 for large families of manifolds. Results of Gromov, Llarull and others show that other manifolds admit positive upper bounds for min к, which are optimal in certain special cases. Similarly, on a closed Riemannian spin manifold, the Dirac eigenvalue λ1 of smallest absolute value is 0 if the Â-genus is nonzero. In other cases, there exist universal upper bounds for |λ1| by Vafa-Witten and Gromov, again under suitable restrictions on the Riemannian metric. Herzlich and others gave optimal upper bounds for |λ1| for certain manifolds.Note that both problems above are directly related by the Friedrich inequality, which bounds min к from above by |λ1|. Also, some of the methods employed for both problems are similar. In this project, we want to find good upper bounds for both the scalar curvature and the first Dirac eigenvalue on larger classes of Riemannian manifolds.