Tubes and comparison theorems in Riemannian and Kählerian geometry
by
Vicente Miquel
Universidad de Valencia, Spain

Plenary Lecture

In this talk we intend to give an account of some results got in the last 10 years which have been influenced by the the papers [Gr1] and [Gr2]. As a sample of the kind of results that we shall review, we give the following:

Let (M;<, >) be a compact riemannian manifold with boundary \partialM. Let us denote by \mu₁(M) the first Dirichlet eigenvalue of the Laplacian on this manifold.

Theorem 1.- Let us suppose that the Ricci curvature of M, the mean curvature of \partialM, and some partial Ricci curvatures of M and partial mean curvatures of \partialM are bounded from below by the corresponding ones of a tube S^q_r of radius r around a totally geodesic sphere S^q in the round sphere Sⁿ. Then

\mu1(M) >= \mu1(Sqr),

and the equality holds if and only if M is an Eschenburg's tube of radius r.

Theorem 2.- Now, if M is also Kähler, and the bounds from below are the corresponding curvatures of a tube IC P^q_r of radius r around a totally geodesic complex projective space in a complex projective space IC Pⁿ, then

\mu1(M) >= \mu1(IC Pqr),

and the equality holds if and only if M is Kähler model tube of radius r.

Theorem 3.- Now, if M is still Kähler, but the bounds from below are the corresponding curvatures of a tube IRPⁿ_r of radius r around a lagrangian and totally geodesic real projective space in a complex projective space IC Pⁿ, then

\mu1(M) >= \mu1(IR Pnr),

and the equality holds if and only if M is a Kⁿ_r model.

An intersting fact of these theorems with similar hypotheses in different contexts is that equality gives a very different degree of rigidity. In fact, Eschenburg's tubes may be constructed on any Riemannian vector bundle on any riemannian manifold. However, a study of the structure of Kähler models reveals that they can be constructed only oncomplex vector bundles on complex submanifolds of a complex projective space and, for real codimension two or for simply connected submanifolds, the posible complex vector bundles to be used are determined by the submanifold through the cohomology class of its Kähler form. The structure of the Kⁿ_r models is even more rigid: the centre of such a tube must be a riemannian manifold of constant sectional curvature, and the tube is uniquely determined by its centre.

References

[Gr1] A. Gray "Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula" Topology vol. 21 (1982) 201-228.

[Gr2] A. Gray "Volumes of tubes about complex submanifolds of complex projective space" Trans. Amer.Math. Soc. vol. 291 (1985) 437-449.

Date received: May 31, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-02.