Vicente Miquel

Comparison Theorems on Convex Hypersurfaces in Hadamard Manifolds

In a Hadamard manifold with sectional curvaturebounded from below by −k22, we give sharp upper estimates for the difference circumradius minus inradiusof a compact k2-convex domain, and we getalso estimates for the quotient (Total d-mean curvature)/Area of a convex domain.

Compact Hopf hypersurfaces of constant mean curvature in complex space forms

We prove that every connected compact Hopf hypersurface of a complex space form, contained in a geodesic ball of radius strictly smaller than the injectivity radius of, having constant mean curvature and with if if λ < 0 is a geodesic sphere of.

On Pappus-Type Theorems on the Volume in Space Forms

Let c be a curve in a n-dimensional space form Mλn, let Pt bea totally geodesic hypersurface of Mλn orthogonal to c at c(t),and let D0 be a domain in P0. If D is thedomain in Mλn obtained by a ‘motion’ of D0 alongc and Dt is the domain in Pt obtained by the motion ofD0 from 0 to t, we show that the n-volume ofD depends only on the length of the curve c, its first curvature, themodified (n − 1)-volume of D0 and the moment of Dt with respect to the totally geodesic hypersurface of Ptorthogonal to the normal vector f2(t) of c. As a consequence, if c(0) is the center of mass of D0, then the n-volume ofD is the product of the modified (n − 1)-volume of D0 and the length of c. We get an analogous theorem for ahypersurface of Mλn obtained by ‘parallel motion’ of ahypersurface of P0 along c.

Bounds for the first Dirichlet eigenvalue attained at an infinite family of Riemannian manifolds

LetM be a compact Riemannian manifold with smooth boundary ∂M. We get bounds for the first eigenvalue of the Dirichlet eigenvalue problem onM in terms of bounds of the sectional curvature ofM and the normal curvatures of ∂M. We discuss the equality, which is attained precisely on certain model spaces defined by J. H. Eschenburg. We also get analog results for Kähler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M),M being a compact Riemannian or Kähler manifold andP being a compact real hypersurface ofM.

Immersions of compact riemannian manifolds into a ball of a complex space form

There are some classical theorems on non-immersibility of compact riemannian manifolds with sectional curvature bounded from above given by Tompkins, O’Neill, Chern, Kuiper and Moore (see [3], pages 221-226). More recently, attention has been paid to the case of immersions into a geodesic ball of a simply connected space form, and some conditions of non-immersibility in such a ball have been proved. In particular, estimates for the mean curvature of a complete immersion into a geodesic ball have been obtained by Jorge and Xavier [11] and a corresponding rigidity theorem for compact hypersurfaces has been proved by Markvorsen [14]. In this paper we give the Kahler analogs of the theorems of Jorge and Xavier (only for the compact case) and Markvorsen, and get some other new results for the Kahler case that have no Riemannian analog. In order to state our results we shall introduce some notation and terminology. Given a real number λ, let us consider the functions

Pappus type theorems for hypersurfaces in a space form

In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following situation. Letc(t) be a curve in a space formMλn of sectional curvature λ. LetP0 be a totally geodesic hypersurface ofMλn throughc(0) and orthogonal toc(t). LetC0 be a hypersurface ofP0. LetC be the hypersurface ofMλn obtained by a motion ofC0 alongc(t). We shall denote it byCPorCFif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC0, then volume(C) ≥ volume(C),P),and the equality holds whenC0 is contained in a geodesic sphere or the motion corresponds to a curve contained in a hyperplane of the Lie algebraO(n−1) (whenn=3, the only motion with these properties is the parallel motion).

On the index form of a geodesic in a pseudoriemannian almost-product manifold

A Riemannian almost-product manifold is a triple (M, g, P), where g is a Riemannian metric and P, the almost-product structure, is a (1, 1)-tensor field on M such that P2=identity and g (P S, P T)--g (S, T) for all vector fields S, T on M. Giving such a P is equivalent to giving two orthogonal distributions V and H on M, which are identified, at each point, with the eigenspaces of P, corresponding to its eigenvalues 1 and--1, and called vertical and horizontal, respectively. Then, we have a pseudoriemannian metric q~ defined by q~ (S, T)= = g ( P S , T), for all vector fields S, ~ on M. Conversely, if (M, q0 is a pseudoriemannian manifold, then there exist (.see Proposition 2.1) a Riemannian metric g and an almost-product structure P on M such that (M, g, P) is a Riemarmian almost-product manifold, and ~(S ,T)=g(PS, T). Then, it seems interesting to study the relations between the geometries of (M, g, P) and (M, ~p). If we assume that the vertical and horizontal distributions are integrable, then the relation between the symmetric connections adaped to g and q~, which we compute in the general case, is particularly simple, and if the integral submanifolds of the vertical distribution are totally geodesic, then, the geodesics of M whose initial

Volumes of certain small geodesic balls and almost-Hermitian geometry

Let D be the characteristic connection of an almost-Hermitian manifold, VDm(r) the volume of a small geodesic ball for the connection D and CCD1the first non-trivial term of the Taylor expansion of VDm(r). NK-manifolds are characterized in terms of CCD1and a family of Hermitian manifolds for which ∫MCCD1dvol is a spectral invariant is given and one proves that CCD1and the spectrum of the complex Laplacian, together, determine the class in which a compact Hermitian manifold lines.

A comparison theorem for the first Dirichlet eigenvalue of a domain in a Kaehler submanifold

We give a sharp lower bound for the first eigenvalue of the Dirichlet eigenvalue problem on a domain of a complex submanifold of a Kaehler manifold with curvature bounded from above. The bound on the first eigenvalue is given as a function of the extrinsic outer radius and the bounds on the curvature, and it is attained only on geodesic spheres of a space of constant holomorphic sectional curvature embedded in the Kaehler manifold as a totally geodesic submanifold.

Kähler tubes of constant radial holomorphic sectional curvature

We determine (up to holomorphic isometries) the family of Kähler tubes, around totally geodesic complex submanifolds, of constant radial holomorphic sectional curvature when the centreP of the tube is either simply connected or a complex hypersurface withH1 (P, R)=0. In the last case, these tubes have the topology of tubular neighbourhoods of the zero section of the complex lines bundles over symplectic manifolds (when they are Kähler) of the Kostant-Souriau prequantization.