The sigma invariant of the $n$ torus, the K3 surface, and Euclidean and elliptic 3d manifolds

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${\Theta }_{f_g}(M)$, and squared $L^2$ norm of the mean curvature vector ${\Phi }_{f_g}(M)$ and second fundamental form ${\Pi }_{f_g}(M)$ 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 metric 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 $\sigma$ invariant and Kazdan-Warner type to manifolds of dimension $n\ge 1$. $M$ 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 $g^{\prime }$ in conformal classes $[g^{\prime }]$ 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 $\sigma$ invariant and said minimal value $\mc{W}_{f_g}(M)= \mc{D}_{f_g}(M)$, with Euclidean 3d manifolds of isomorphic ${\pi }_1$ being diffeomorphic iff the values of $\mc{W}_{f_g}(M)$ for their canonical $g$s are the same. An elliptic 3d manifold $(M,{\Gamma }_M)$ of underlying group $π_1(M)\cong Γ_M \subset \mb{S}\mb{O}(4)$ has $\sigma (M)=6(2{\pi }^2{)}^{\frac{2}{3}}/|{\pi }_1(M){|}^{\frac{2}{3}}$, and if $(M,{\Gamma }_M)$ and $(M^{\prime },{\Gamma }_{M^{\prime }})$ are two of them of isomorphic ${\pi }_1$, $M$ is diffeomorphic to $M^{\prime }$ iff the spaces of ${\Gamma }_M$ and ${\Gamma }_{M^{\prime }}$ invariant homogeneous spherical harmonics of degree $|{\pi }_1|$ are the same.

Last part of Theorem 1 statement changed to what we were actually proving, with this portion of the proof clarified in detail. Some portions of the argument in the proof of Theorem 3 were expanded as they were extremely brief in the original. Edited the file overall to improve readability, removing some annoying misprints