谱Geroch猜想与非紧面积可扩和乐分支
Spectral Geroch conjecture and noncompact area enlargeable summands
AI总结：
本研究将Wang--Zhang关于闭面积可扩流形连通和无一致正标量曲率完备度量的结论推广至非紧情形，还证明了广义Geroch猜想的谱类比版本，核心方法为覆盖连通和构造结合两类余腰不等式。
AI中文摘要：
我们证明了，可能非紧的面积可扩流形$M_1$与同维数的任意旋流形$M_2$的连通和，不存在具有一致正标量曲率的完备黎曼度量。这将Wang--Zhang的一个定理（其中$M_1$被假定为闭的）推广到了非紧可扩和乐分支的情形；在这种一般性下，一致正性假设以本质方式进入论证。我们还根据$γ$-谱常数证明了广义Geroch猜想的谱类比：当$γ>(\text{dim }M_1-1)/(4\text{dim }M_1)$时，此类连通和不承载具有正$γ$-谱常数的完备度量。证明基于覆盖连通和构造，结合标量余腰不等式与谱余腰不等式。
英文摘要：
We prove that the connected sum of a possibly noncompact area enlargeable manifold $M_1$ with an arbitrary spin manifold $M_2$ of the same dimension admits no complete Riemannian metric of uniformly positive scalar curvature. This extends a theorem of Wang--Zhang, where $M_1$ is assumed closed, to noncompact enlargeable summands; in this generality the uniform positivity hypothesis enters the argument in an essential way. We also prove a spectral analogue of the generalized Geroch conjecture in terms of the $γ$-spectral constant: for $γ>(\dim M_1-1)/(4\dim M_1)$, such a connected sum carries no complete metric with positive $γ$-spectral constant. The proofs are based on a covering connected sum construction together with the scalar-cowaist and spectral-cowaist inequalities.
