A fundamental inequality for 3-dimensional CAT(ε) hypersurfaces

By J.F. Nash’s Theorem, any Riemannian manifold can be embedded into a Euclidean ambient space with dimension sufficiently large. In 1968, S.-S. Chern pointed out that a key technical element in applying Nash’s Theorem effectively is finding useful relationships between intrinsic and extrinsic elements that are characterizing immersions. After 1993, when a groundbreaking work written by B.-Y. Chen on this theme was published, many explorations pursued this important avenue of inquiry. Bearing in mind this historical context, in our present paper we point out a new relationship involving intrinsic and extrinsic curvature invariants, under natural geometric conditions. We conclude our work with an obstruction to minimality, in the spirit of S.-S. Chern’s reflection from 1968.