The sigma invariant of the $n$ torus, the K3 surface, and Euclidean and elliptic 3d manifolds
Santiago R. Simanca
TL;DR
This work extends the sigma invariant to all dimensions $n\ge1$ via two intrinsic functionals ${\mathcal W}_{f_g}(M)$ and ${\mathcal D}_{f_g}(M)$ arising from Nash isometric embeddings, linking them to the scalar curvature through $s_g = n(n-1)+\|H_{f_g}\|^2-\|\alpha_{f_g}\|^2$ and to the Yamabe problem via $\lambda(M,g)=\mu_g^{-2/N}{\mathcal S}_g(M)$. It proves a unifying classification: manifolds of Kazdan–Warner type II admit a Ricci-flat metric with a minimal embedding that minimizes ${\mathcal W}$ and ${\mathcal D}$ within scalar-flat conformal classes, leading to vanishing $\sigma(M)$ and canonical realisers. The paper computes explicit realizers for the $n$-torus, the K3 surface, Euclidean 3‑manifolds, and elliptic 3‑manifolds, with a closed-form formula for elliptic spaces: ${\sigma(M)=6(2\pi^2)^{2/3}/|π_1(M)|^{2/3}}$. It further shows that for elliptic manifolds, the diffeomorphism type is determined by spaces of invariant homogeneous spherical harmonics ${\mathcal S}^{inv}_{Γ_M}$ when $π_1$ is non-Abelian, and it furnishes concrete examples (e.g., Lens spaces and the square Kummer surface) illustrating the interplay between topology, conformal geometry, and minimal embeddings.
Abstract
On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\mb{S}^{\tn=\tn(n)},\tg)$, we consider the total exterior scalar curvature $Θ_{f_g}(M)$, and squared $L^2$ norm of the mean curvature vector $Φ_{f_g}(M)$ and second fundamental form $Π_{f_g}(M)$ functionals of $f_g$, respectively. Then $\mc{W}_{f_g}(M) =(1-δ_{n,1})(n/(n-1)) Θ_{f_{g}}(M) + Φ_{f_{g)}}(M)$ and $\mc{D}_{f_g}(M)=(1-δ_{n,1}) (1/(n-1))Θ_{f_g}(M)+Π_{f_{g)}}(M)$ are functionals intrinsically defined in the space of metrics in the conformal class of $g$, and $\mc{S}_g(M):=\int s_g dμ_g=\mc{W}_{f_g}(M)- \mc{D}_{f_g}(M)$. We extend the notions of $σ$ invariant and Kazdan-Warner type to manifolds of dimension $n\geq 1$. $M$ is a manifold of type II if, and only if, it admits a Ricci flat metric $g$ with minimal isometric embedding $f_g$ that minimizes $\mc{W}_{f_{g'}}(M)$ and $\mc{D}_{f_{g'}}(M)$ among metrics $g'$ in conformal classes $[g']$ with scalar flat representatives. We show that the torus $T^n$, the K3 surface, and any Euclidean 3d manifold are manifolds of Kazdan-Warner type II, exhibiting in each case the canonical Ricci flat $g$ that realizes the vanishing $σ$ invariant and said minimal value $\mc{W}_{f_g}(M)= \mc{D}_{f_g}(M)$, with Euclidean 3d manifolds of isomorphic $π_1$ being diffeomorphic iff the values of $\mc{W}_{f_g}(M)$ for their canonical $g$s are the same. An elliptic 3d manifold $(M,Γ_M)$ of underlying group $π_1(M)\cong Γ_M \subset \mb{S}\mb{O}(4)$ has $σ(M) =6(2π^2)^{\frac{2}{3}}/|π_1(M)|^{\frac{2}{3}}$, and if $(M,Γ_M)$ and $(M',Γ_{M'})$ are two of them of isomorphic $π_1$, $M$ is diffeomorphic to $M'$ iff the spaces of $Γ_M$ and $Γ_{M'}$ invariant homogeneous spherical harmonics of degree $|π_1|$ are the same.