2025/09/30 by Santiago R. Simanca, Simanca, Santiago R.
Mathematics · #53C20 #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2509.26079
openalex publication_date 2025/09/30 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28
On the space of isometric embeddings fg of metrics g on a manifold Mn into the standard (\mbS\tn=\tn(n),\tg), we consider the total exterior scalar curvature Θfg(M), and squared L2 norm of the mean curvature vector Φfg(M) and second fundamental form Πfg(M) functionals of fg, respectively. Then \mcWfg(M) =(1-δn,1)(n/(n-1)) Θ_fg(M) + Φ_fg)(M) and \mcDfg(M)=(1-δn,1) (1/(n-1))Θfg(M)+Π_fg)(M) are functionals intrinsically defined in the space of metrics in the conformal class of g, and \mcSg(M):=∫ sg dμg=\mcWfg(M)- \mcDfg(M). We extend the notions of σ invariant and Kazdan-Warner type to manifolds of dimension n≥ 1. M is a manifold of type II if, and only if, it admits a Ricci flat metric g with minimal isometric embedding fg that minimizes \mcW_fg'(M) and \mcD_fg'(M) among metrics g' in conformal classes [g'] with scalar flat representatives. We show that the torus Tn, 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 \mcWfg(M)= \mcDfg(M), with Euclidean 3d manifolds of isomorphic π1 being diffeomorphic iff the values of \mcWfg(M) for their canonical gs are the same. An elliptic 3d manifold (M,ΓM) of underlying group π1(M)≅ ΓM ⊂ \mbS\mbO(4) has σ(M) =6(2π2)(2)/(3)/|π1(M)|(2)/(3), and if (M,ΓM) and (M',ΓM') are two of them of isomorphic π1, M is diffeomorphic to M' iff the spaces of ΓM and ΓM' invariant homogeneous spherical harmonics of degree |π1| are the same.