Relations between threshold constants for Yamabe type bordism invariants

Low-dimensional surgery and the Yamabe invariant

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k\le n-3. The smooth Yamabe invariants σ(M) and σ(N) satisfy σ(N)\ge min (σ(M),Λ) for Λ>0. We derive explicit lower bounds for Λin dimensions where previous methods failed, namely for (n,k)\in {(4,1),(5,1),(5,2),(6,3),(9,1),(10,1)}. With methods from surgery theory and bordism theory several gap phenomena for smooth Yamabe invariants can be deduced.

Yamabe稳定性优化元的存在性

Existence of Yamabe stability optimizers

该研究证明了满足特定阈值条件的高维正Yamabe不变量闭Riemann流形上Yamabe不等式稳定性优化元的存在性，明确了不同情形下的紧性阈值与严格不等式成立的条件。

Surgery and the spinorial tau-invariant

We associate to a compact spin manifold M a real-valued invariant τ(M) by taking the supremum over all conformal classes over the infimum inside each conformal class of the first positive Dirac eigenvalue, normalized to volume 1. This invariant is a spinorial analogue of Schoen's $σ$-constant, also known as the smooth Yamabe number. We prove that if N is obtained from M by surgery of codimension at least 2, then $τ(N) \geq \min\{τ(M),Λ_n\}$ with $Λ_n>0$. Various topological conclusions can be drawn, in particular that τis a spin-bordism invariant below $Λ_n$. Below $Λ_n$, the values of $τ$ can…