October 20, 2025Open Access

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

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This analysis reveals the sigma invariant's role in describing K3 surfaces and Euclidean 3D manifolds, indicating their Kazdan-Warner type II status.

The study shows that the torus, K3 surface, and Euclidean 3D manifolds exhibit Kazdan-Warner type II properties.
It establishes that these manifolds admit Ricci flat metrics minimizing functionals of mean curvature and second fundamental form.
An elliptic 3D manifold has a specific sigma invariant value, dependent on its underlying group structure.
Diffeomorphism between elliptic manifolds is determined by the invariant properties of their corresponding harmonic spaces.

Santiago R. Simanca (2025) studied this question.

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This analysis reveals the sigma invariant's role in describing K3 surfaces and Euclidean 3D manifolds, indicating their Kazdan-Warner type II status.