论文标题
Lipschitz标量曲率的刚度
Lipschitz rigidity for scalar curvature
论文作者
论文摘要
让$ m $成为均匀的平滑连接的旋转型$ n $，让$ g $是$ m $的Riemannian定期$ w^{1，p} $，$ p> n $，在$ m $上，其分布标量表曲率以Lee-Lefloch的意义在下面限制在$ n（n-1）$的情况下，并让$ f \ f \ f \ colon（M） \ Mathbb {s}^n $是$ 1 $ -LIPSCHITZ连续（不一定是平滑的）非零度的地图，$ n $ n $ -sphere。然后$ f $是度量标准。这概括了Llarull（1998）的结果，并在Gromov（2019）的“四个讲座”中回答。 我们的证明是基于低规律性riemannian指标的狄拉克运算符的光谱特性，并用Lipschitz束扭曲。我们认为，从Reshetnyak的意义上讲，非零的谐波旋转型场$ f $的存在是Quasiregular的，并且通过这种方式将Quasiregular地图的强大理论与Atiyah-Singer索引定理联系起来。
Let $M$ be a closed smooth connected spin manifold of even dimension $n$, let $g$ be a Riemannian metric of regularity $W^{1,p}$, $p > n$, on $M$ whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by $n(n-1)$, and let $f \colon (M,g) \to \mathbb{S}^n$ be a $1$-Lipschitz continuous (not necessarily smooth) map of non-zero degree to the unit $n$-sphere. Then $f$ is a metric isometry. This generalizes a result of Llarull (1998) and answers in the affirmative a question of Gromov (2019) in his "Four lectures". Our proof is based on spectral properties of Dirac operators for low regularity Riemannian metrics and twisted with Lipschitz bundles. We argue that the existence of a non-zero harmonic spinor field forces $f$ to be quasiregular in the sense of Reshetnyak, and in this way connect the powerful theory for quasiregular maps to the Atiyah-Singer index theorem.
