A rigidity theorem for the pair ${\cal q}{\Bbb C} P^n$ (complex hyperquadric, complex projective space)

Given a compact Kahler manifold M of real dimension 2n, let P be either a compact complex hypersurface of M or a compact totally real submanifold of dimension n. Let \(\cal q\) (resp. \({\Bbb R} P^n\)) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space \({\Bbb C} P^n\) of constant holomorphic sectional curvature 4\( \lambda \). We prove that if the Ricci and some (n-1)-Ricci curvatures of M (and, when P is complex, the mean absolute curvature of P) are bounded from below by some special constants and volume (P) / volume (M) \(\leq \) volume (\(\cal q\))/ volume \(({\Bbb C} P^n)\) (resp. \(\leq \) volume \(({\Bbb R} P^n)\) / volume …

Total curvatures of convex hypersurfaces in hyperbolic space

We give sharp upper estimates for the difference circumradius minus inradius and for the angle between the radial vector (respect to the center of an inball) and the normal to the boundary of a compact $h$-convex domain in the hyperpolic space. We apply these estimates to get the limit at the infinity for the quotients Volume/Area and (Total $k$-mean curvature)/Area of a family of $h$-convex domains which expand over the whole space. The theorem for the first quotient gives an extension to arbitrary dimension of a result of Santalo and Yanez for the hyperbolic plane.

On nonimmersibility of compact hypersurfaces into a ball of a simply connected space form

We give a nonimmersibility theorem of a compact manifold with nonnegative scalar curvature bounded from above into a geodesic ball of a simply connected space form.

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

Reilly's type inequality for the Laplacian associated to a density related with shrinkers for MCF

Let $(\bar{M},,e^\psi)$ be a Riemannian manifold with a density, and let $M$ be a closed $n$-dimensional submanifold of $\bar{M}$ with the induced metric and density. We give an upper bound on the first eigenvalue $\lambda_1$ of the closed eigenvalue problem for $\Delta_\psi$ (the Laplacian on $M$ associated to the density) in terms of the average of the norm of the vector ${\vec{H}}_{{\psi}} + {\bar \nabla}$ with respect to the volume form induced by the density, where ${\vec{H}}_{{\psi}}$ is the mean curvature of $M$ associated to the density $e^\psi$. When $\bar{M}=\Bbb R^{n+k}$ or $\bar{M}=S^{n+k-1}$, the equality between $\lambda_1$ and its bound implies that $e^\psi$ is a Gaussian den…

Comparing the relative volume with a revolution manifold as a model

Given a pair (P, M), whereM is ann-dimensional connected compact Riemannian manifold andP is a connected compact hypersurface ofM, the relative volume of (P, M) is the quotient volume(P)/volume(M). In this paper we give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature ofM and the mean curvature ofP, with respect to that of a model pair\(\left( {\mathcal{P},\mathcal{M}} \right)\) where ℳ is a revolution manifold and\(\mathcal{P}\) a “parallel” of ℳ.

A comparison theorem for the mean exit time from a domain in a K�hler manifold

Let M be a Kahler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. Let ω be a domain in M with some bounds on the mean and JN-mean curvatures of its boundary ∂ω. The main result of this paper is a comparison theorem between the Mean Exit Time function defined on ω and the Mean Exit Time from a geodesic ball of the complex projective space ℂℙ n (λ) which involves a characterization of the geodesic balls among the domain ω. In order to achieve this, we prove a comparison theorem for the mean curvatures of hypersurfaces parallel to the boundary of ω, using the Index Lemma for Submanifolds.

Hermitian natural differential operators

Kähler Tubes of Constant Radial Holomorphic Sectional Curvature

We determine (up to holomorphic isometries) the family of Kahler 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 Kahler) of the Kostant-Souriau prequantization.

Feuilletages Riemanniens singuliers

Abstract We prove that a singular foliation on a compact manifold admitting an adapted Riemannian metric for which all leaves are minimal must be regular. To cite this article: V. Miquel, R.A. Wolak, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

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 formM λ n of sectional curvature λ. LetP 0 be a totally geodesic hypersurface ofM λ n throughc(0) and orthogonal toc(t). LetC 0 be a hypersurface ofP 0. LetC be the hypersurface ofM λ n obtained by a motion ofC 0 alongc(t). We shall denote it byC PorC Fif 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 ofC 0, then volume(C) ≥ volume(C),P),and the equality holds whenC 0 is contained in a geodesic sphere or…

Pappus type theorems for motions along a submanifold

Abstract We study the volumes volume( D ) of a domain D and volume( C ) of a hypersurface  C obtained by a motion along a submanifold P of a space form  M n λ . We show: (a) volume( D ) depends only on the second fundamental form of  P , whereas volume( C ) depends on all the i th fundamental forms of  P , (b) when the domain that we move D 0 has its q -centre of mass on  P , volume( D ) does not depend on the mean curvature of  P , (c) when D 0 is q -symmetric, volume( D ) depends only on the intrinsic curvature tensor of  P ; and (d) if the image of  P by the ln of the motion (in a sense which is well-defined) is not contained in a hyperplane of the Lie algebra of SO ( n − q − d ), and C …

Evolution by mean curvature flow of Lagrangian spherical surfaces in complex Euclidean plane

We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane $\mathbb{C}^2$. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in $\mathbb{C}^2$ such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the Clifford torus.

Bounds for the first Dirichlet eigenvalue of domains in Kaehler manifolds

Non-preserved curvature conditions under constrained mean curvature flows

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.

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 …

Comparison theorems for the volume of a geodesic ball with a product of space forms as a model

We prove two comparison theorems for the volume of a geodesic ball in a Riemannian manifold taking as a model a geodesic ball in a product of two space forms.

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 .

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 Kahler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M),M being a compact Riemannian or Kahler manifold andP being a compact real hypersurface ofM.

Volumes of certain small geodesic balls and almost-Hermitian geometry

Let D be the characteristic connection of an almost-Hermitian manifold, V D m (r) the volume of a small geodesic ball for the connection D and C C D 1 the first non-trivial term of the Taylor expansion of V D m (r). NK-manifolds are characterized in terms of C C D 1 and a family of Hermitian manifolds for which ∫ M C C D 1 dvol is a spectral invariant is given and one proves that C C D 1 and 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

AbstractWe 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.