Index theory with bounded geometry, the uniformly finite A-hat class, and infinite connected sums

Infinite connected sums, K-area and positive scalar curvature

Whyte used the index theory of Dirac operators and Block-Weiberger uniformly finite homology to show that certain infinite connected sums do not carry a metric with nonnegative scalar curvature in their bounded geometry class. His proof uses a coarse version of the $\hat{A}$-class to obstruct such metrics. In this note we prove a version of Whyte's result where a variant of the notion of infinite $K$-area, originally due to Gromov, is used to obstruct metrics with positive scalar curvature.

Deforming 3-manifolds of bounded geometry and uniformly positive scalar curvature

We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the compact case. The proof uses Ricci flow with surgery as well as arguments involving performing infinite connected sums with control on the geometry.

相对eta不变量与非紧致流形上的均匀正标量曲率

Relative eta invariant and uniformly positive scalar curvature on non-compact manifolds

本文研究非紧致流形上Dirac-Schrödinger算子的相对eta不变量，推导非紧致流形上的谱流公式，并推广Atiyah-Patodi-Singer索引定理至非紧边界情况。

一致Temple图的存在性及其在零距离上的应用

Existence of uniform Temple charts and applications to null distance

本文证明Temple柱面未来零坐标图可以一致构造，并估计其光学函数的梯度，进而应用于研究时空转换为定度度量空间后的可求长性和Lorentz等距定理。

$S^1$-equivariant bordism, invariant metrics of positive scalar curvature, and rigidity of elliptic genera

We construct geometric generators of the effective $S^1$-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which $S^1$-manifolds admit invariant metrics of positive scalar curvature. It turns out that, up to taking connected sums with several copies of the same manifold, the only obstruction to the existence of such a metric is an $\hat{A}$-genus of orbit spaces. This $\hat{A}$-genus generalizes a previous definition of Lott for orbit spaces of semi-free $S^1$-actions. As a further application of our results, we give a new proof of…