Non-immersion problems in Riemannian geometry
In [3], besides results on minimal surfaces, we proved a sharp estimate on the radius of the smallest ball that can enclose a complete submanifold of scalar curvature bounded from below. Our proof uses ideas ordinary differential equations. Similar results were also obtained by Karp using partial differential equations.
It has been conjectured that the hyperbolic n-space admits no isometric immersion in the (2n-1)-dimensional Euclidean space. The case n=2 is a classical theorem of Hilbert. In [7] we proved, using the local theory developed by Cartan and the growth of the fundamental group, that a hyperbolic manifold with non-abelian fundamental group (most of them have this property) cannot be immersed in the (2n-1)-dimensional Euclidean space.
In [9] we extended the Efimov theorem to complete hypersurfaces of Euclidean space of dimension n+1, with negative Ricci curvature, as follows: a) If n=3 then the infimum of the length of the second fundamental form of M is zero (this is the three dimensional version of a conjecture of Milnor for surfaces). b) If n>3 and the sectional curvature does not assume every real number, then the Ricci curvature is not bounded away from zero. The argument uses the structure theorem for Euclidean hypersurfaces of non-negative curvature (due to H. Wu and others). The idea is to pass from negative to positive curvature by taking parallel hypersurfaces obtained by going beyond the focal set.
Related Publications
[3] F. Xavier (with L. P.Jorge), An inequality between the exterior diameter and the mean curvature of bounded immersions, Mathematische Zeitschrift, 178, 77-82, (1981).
[7] F. Xavier, A non-immersion theorem for hyperbolic manifolds, Comm. Math.Helv. 60, 280-283, (1985).
[9] F. Xavier (with B. Smyth), Efimov's theorem in dimension greater than two, Invent. Math. 90, 443-450, (l987).