On the Yamabe invariant of certain compact manifolds with boundary

TL;DR

This paper extends the understanding of $ ext{Yamabe}$ invariants for certain compact manifolds with boundary, providing explicit calculations and revealing new geometric properties related to connected sums and boundary modifications.

Contribution

It generalizes Kobayashi's connected-sum inequality to $ ext{Yamabe}$ invariants and computes these invariants for complex manifold combinations, highlighting boundary effects.

Findings

Calculated $ ext{Yamabe}$ invariants for specific manifold sums.

Showed $ ext{RP}^n$ minus finitely many balls shares invariants with the hemisphere.

Contrasted boundary effects with Bray-Neves results on $ ext{RP}^3$.

Abstract

We generalize Kobayashi's connected-sum inequality to the $\lambda$-Yamabe invariants. As an application, we calculate the $\lambda$-Yamabe invariants of $\#m_1\mathbb{RP}^n\# m_2(\mathbb{RP}^{n-1}\times S^1)\#lH^n\#kS_+^n$, for any $\lambda\in [0,1]$, $n\geq 3$, provided $k+l\geq 1$. As a corollary, we prove that $\mathbb{RP}^n$ minus finitely many disjoint $n$-balls have the same $\lambda$-Yamabe invariants as the hemi-sphere, which forms an interesting contrast with the famous Bray-Neves results on the Yamabe invariants of $\mathbb{RP}^3$.