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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2509.26079 |
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- 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.