Best Constants in Sobolev Inequalities on Riemannian Manifolds in the Presence of Symmetries

Let (M,g) be a smooth compact Riemannian manifold, and G a subgroup of the isometry group of (M,g). We compute the value of the best constant in Sobolev inequalities when the functions are G-invariant. Applications to non-linear PDEs of critical or upper critical Sobolev exponent are also presented.     

Systolic volume of hyperbolic manifolds and connected sums of manifolds

The systolic volume of a closed n-manifold M is defined as the optimal constant σ(M) satisfying the inequality vol(M, g) ≥ σ(M) sys(M, g) ⁿ between the volume and the systole of every metric g on M. First, we show that the systolic volume of connected sums of closed oriented essential manifolds is unbounded. Then, we prove that the systolic volume of every sequence of closed hyperbolic (three-dimensional) manifolds is also unbounded. These results generalize systolic inequalities on surfaces in two different directions.      

Differential Harnack inequalities for heat equations with potentials under geometric flows

In the paper we consider a closed Riemannian manifold M with a time-dependent Riemannian metric g _ij (t) evolving by ? _t g _ij  = ?2S _ij , where S _ij is a symmetric two-tensor on (M,g(t)). We prove some differential Harnack inequalities for positive solutions of heat equations with potentials on (M,g(t)). Some applications of these inequalities will be obtained.     

Extension of symmetries on Einstein manifolds with boundary

We investigate the validity of the isometry extension property for (Riemannian) Einstein metrics on compact manifolds M with boundary ∂M. Given a metric γ on ∂M, this is the issue of whether any Killing field X of (∂M, γ) extends to a Killing field of any Einstein metric (M, g) bounding (∂M, γ). Under a mild condition on the fundamental group, this is proved to be the case at least when X preserves the mean curvature of ∂M in (M, g).     

Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature

In this article, we show that, for a biharmonic hypersurface (M, g) of a Riemannian manifold (N, h) of non-positive Ricci curvature, if ò_M|H|² v_g < ￥{\int_M\vert H\vert^2 v_g<\infty}, where H is the mean curvature of (M, g) in (N, h), then (M, g) is minimal in (N, h). Thus, for a counter example (M, g) in the case of hypersurfaces to the generalized Chen’s conjecture (cf. Sect. 1), it holds that ò_M|H|² v_g=￥{\int_M\vert H\vert^2 v_g=\infty} .     

Stiefel–Whitney surfaces and the tri-genus of non-orientable 3-manifolds

Every non-orientable 3-manifold M can be expressed as a union of three orientable handlebodies V ₁,V ₂,V ₃ whose interiors are pairwise disjoint. If g _i denotes the genus of ∂V _i and g ₃≤g ₂≤g ₃, then the tri-genus of M is the minimum triple (g ₁,g ₂,g ₃), ordered lexicographically. If the Bockstein of the first Stiefel–Whitney class βw ₁(M)=0, then M has tri-genus (0,2g,g ₃), where g is the minimal genus of a 2-sided Stiefel Whitney surface of M. In this paper it is shown that, if βw ₁(M)&\ne;0, then M has tri-genus (1,2g−1,g ₃), where g is the minimal genus of a (1-sided) Stiefel–Whitney surface. As an application the tri-genus of certain graph manifolds is computed. Received: 28 April 1999     

A new Lichnerowicz–Obata estimate in the presence of a parallel p-form

 Let (M ⁿ ,g) be a compact Riemannian manifold with a smooth boundary. In this paper, we give a Lichnerowicz-Obata type lower bound for the first eigenvalue of the Laplacian of (M ⁿ ,g) when M has a parallel p-form (2 ≤p≤n/2). This result follows from a new Bochner-Reilly's formula. Moreover, we give a characterization of the equality case when (M ⁿ ,g) is simply connected. Received: 1 June 2001     

Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple part, then it preserves some holomorphic Riemannian metric on M with constant sectional curvature; (ii) If the Killing Lie algebra of g is solvable, then, up to a finite unramified cover, M is a quotient Γ\G, where Γ is a lattice in G and G is either the complex Heisenberg group, or the complex SOL group. S. Dumitrescu was partially supported by the ANR Grant BLAN 06-3-137237.     

Second-order necessary optimality conditions for optimization problems involving set-valued maps

Second-order necessary optimality conditions are established under a regularity assumption for a problem of minimizing a functiong over the solution set of an inclusion system 0 F(x), x M, whereF is a set-valued map between finite-dimensional spaces andM is a given subset. The proof of the main result of the paper is based on the theory of infinite systems of linear inequalities.     

Porosity of Mutually Nearest and Mutually Furthest Points Problems in Geodesic Space

The notion of superframe in general Hilbert spaces was introduced in the context of multiplexing, which has been widely used in mobile communication network, satellite communication network and computer area network. The notion of oblique dual frame is a generalization of conventional dual frame. It has provided us with a frame-like expansion. Using oblique dual frames one can extend frame expansions to include redundant expansions in which the analysis and synthesis frames lie in different spaces. Given positive integers L, M and N, an N?-periodic set 𝕊 in ?, let 𝒢(g, N, M) be a frame for l ²(𝕊, ? ^L ), and let 𝒢(h, N, M) be a frame for ?(h, N, M) (generated by 𝒢(h, N, M)). This article addresses super Gabor duals of g in ?(h, N, M). We obtain a necessary and sufficient condition on h admitting super oblique Gabor duals of g, and present a parametrization expression of all super oblique Gabor duals and all oblique canonical Gabor duals of g. We also characterize the uniqueness of super oblique Gabor dual and oblique canonical Gabor dual of g. Some examples are also provided.     

Notes on Gromov’s systolic estimate

These notes cover some of the main results of Gromov’s paper Filling Riemannian manifolds. The goal of these notes is to make the results and proofs accessible to more people. The main result is that if (M,g) is a Riemannian manifold of dimension n, then there is a non-contractible curve in (M,g) of length at most C _n Vol(M,g)^1/n .     

Scattering Rigidity for Analytic Riemannian Manifolds with a Possible Magnetic Field

Consider a compact manifold M with boundary ∂ M endowed with a Riemannian metric g and a magnetic field Ω. Given a point and direction of entry at the boundary, the scattering relation Σ determines the point and direction of exit of a particle of unit charge, mass, and energy. In this paper we show that a magnetic system (M,∂ M,g,Ω) that is known to be real-analytic and that satisfies some mild restrictions on conjugate points is uniquely determined up to a natural equivalence by Σ. In the case that the magnetic field Ω is taken to be zero, this gives a new rigidity result in Riemannian geometry that is more general than related results in the literature.     

Complete noncompact manifolds with harmonic curvature

Let (M ⁿ , g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (M ⁿ , g) is a space form if it has sufficiently small L ^n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M ⁿ , g) with positive scalar curvature.     

Positive Twistor Bundle of a Kähler Surface

The aim of this paper is to characterize Kähler surfaces in terms oftheir positive twistor bundle. We prove that an oriented four-dimensionalRiemannian manifold (M, g) admits a complex structure J compatible with the orientation and such that (M, g, J is a Kähler manifold ifand only if the positive twistor bundle (Z ⁺(M), g _c ) admits a verticalKilling vector field.     

On the geometry and trigonometry of homogeneous 3-manifolds with 4-dimensional isometry group

In this article we study the geometry of the family of simply connected homogeneous 3-manifolds (M, g _K,τ ) given as a principal bundle over a 2-manifold of constant curvature such that the curvature form is constant. We give explicit results for the conjugate radius, normal Jacobi fields and the cut locus on (M, g _K,τ ). Moreover, we determine the trigonometry on (M, g _K,τ ) by a complete set of trigonometric laws. The author would like to thank Uwe Abresch for his advice.     

Lorentz Manifolds with the Three Largest Degrees of Symmetry

We show that if a Lorentz manifold (M, g) has a sufficiently large group of isometries and if, in addition, this group has no null orbits, then (M, g) is homogeneous. A list of Lorentz manifolds with the three largest groups of isometries is given.     

Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds

We completely classify three-dimensional homogeneous Lorentzian manifolds, equipped with Einstein-like metrics. Similarly to the Riemannian case (E. Abbena et al., Simon Stevin Quart J Pure Appl Math 66:173–182, 1992), if (M, g) is a three-dimensional homogeneous Lorentzian manifold, the Ricci tensor of (M, g) being cyclic-parallel (respectively, a Codazzi tensor) is related to natural reductivity (respectively, symmetry) of (M, g). However, some exceptional examples arise. The author is supported by funds of MURST, GNSAGA and the University of Lecce.     

Scalar and φ-Sectional Curvature of a Certain Type of Metric f-Structures

We consider a Riemannian manifold (M,g) equipped with an f-structure of constant rank with parallelizable kernel. We assume certain integrability conditions on such a manifold. We prove some inequalities involving the scalar and *-scalar curvature of g. We prove that the corresponding equalities characterize an -manifold, which is a generalization of a Sasakian manifold. We also give a method of constructing such structures on toroidal bundles. Dedicated to the memory of Professor Aldo Cossu Research supported by the Italian MIUR 60% and GNSAGA.     

Invariance of the Nayatani metrics for Kleinian manifolds

The Nayatani metric g _N is a Riemannian metric on a Kleinian manifold M which is compatible with the standard flat conformal structure. It is known that, for M corresponding to a geometrically finite Kleinian group, g _N has large symmetry: the isometry group of (M, g _N ) coincides with the conformal transformation group of M. In this paper, we prove that this holds for a larger class of M. In particular, this class contains such M that correspond to Kleinian groups of divergence type.      

Constant mean curvature spheres in Riemannian manifolds

We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points.