论文标题
正面标量曲率和等效的callias型指数定理
Positive scalar curvature and an equivariant Callias-type index theorem for proper actions
论文作者
论文摘要
对于本地紧凑型组$ g $在具有$ g $ equivariant旋转结构的歧管$ m $上的适当行动，我们会妨碍存在具有均匀阳性曲率的完整$ g $ invariant riemannian指标。我们专注于$ m/g $不合时宜的情况。障碍物来自Callias型指数定理，并与$ M $中的Hypersurfaces附近的正标曲率相关。我们还推断出该索引定理的其他一些应用。如果$ g $是一个连接的谎言组，则在对动作的轻度假设下，构成正标曲率的障碍物消失。在这种情况下，我们概括了Lawson和Yau的建筑，以获得具有均匀的正标性曲率的完整的$ G $ - invariant Riemannian指标，该指标是在等效的有限几何学假设下的。
For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.
