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arXiv2026

Odd-dimensional lens-space Yamabe counterexamples

Let $n\ge5$ be odd and let $L_{3q}^n=S^n/C_{3q}$ be the standard lens space, where $q>3$ is prime. We prove that for all sufficiently large $q$, \[ σ(L_{3q}^n) > (3q)^{-2/n}Y(S^n)=Y(L_{3q}^n,[g_{\rm rd}]). \] Here $[g_{\rm rd}]$ denotes the quotient-round conformal class. Combining spin bordism with the surgery theorem of Ammann--Dahl--Humbert, we establish \[ σ(L_{3q}^n)\ge \min \{q^{-2/n}Y(S^n),Λ_n\}, \] where $Λ_n>0$ depends only on $n$. For sufficiently large $q$, this exceeds the quotient-round value, showing that the quotient-round conformal class need not maximize the smooth Yamabe invariant in any odd dimension $n\ge5$.

math.DG↗

arXiv2026

Sharp Yamabe Bounds for Spherical Three-Space Forms via Four-Dimensional Poincaré--Einstein Compactness

Let $Γ<SO(4)$ be a nontrivial finite group acting freely on $S^3$, and let $M=S^3/Γ$. We prove that every smooth conformal class $C$ on $M$ satisfies \[ Y(M,C)\le |Γ|^{-2/3}Y(S^3), \] with equality for the quotient-round conformal class. Consequently, \[ σ(M)=|Γ|^{-2/3}Y(S^3). \] Our approach uses four-dimensional Poincaré--Einstein fillings of the ball. We first establish compactness for fillings whose conformal infinities range over a compact family of positive-scalar-curvature metrics. Since compactness alone does not guarantee continuation through degenerate Einstein metrics, we use equivariant Fredholm parity to obtain a filling with the prescribed boundary conformal class and group action. A common interior fixed point then makes the Li--Qing--Shi boundary test function invariant, and its descent to $M$ gives the sharp covering factor.

math.DG↗