Variational formulas of higher order mean curvatures

In this paper,we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional M2p of a submanifold M n in a general Riemannian manifold N n+m for p = 0,1,...,[n 2 ].As an example,we prove that closed complex submanifolds in complex projective spaces are critical points of the functional M2p,called relatively 2p-minimal submanifolds,for all p.At last,we discuss the relations between relatively 2p-minimal submanifolds and austere submanifolds in real space forms,as well as a special variational problem.     

On mean curvatures in submanifolds geometry

By using moving frame theory,first we introduce 2p-th mean curvatures and(2p 1)-th mean curvature vector fields for a submanifold.We then give an integral expression of them that characterizes them as mean values of symmetric functions of principle curvatures.Next we apply it to derive directly the celebrated Weyl-Gray tube formula in terms of integrals of the 2p-th mean curvatures and some Minkowski-type integral formulas.     

Partitions with prescribed mean curvatures

We consider a certain variational problem on Caccioppoli partitions with countably many components, which models immiscible fluids as well as variational image segmentation, and generalizes the well-known problem with prescribed mean curvature. We prove existence and regularity results, and finally show some explicit examples of minimizers. Received: 7 June 2001 / Revisied version: 8 October 2001     

Differentiation in abstract spaces

A new definition of differentiation for mappings between topological vector spaces is introduced. It does not require the mapping to be defined on a linear manifold, nor does it require it to be continuous. All of the main theorems of differential calculus hold and all other known definitions of differentiation are included. The new definition can be used for singular mappings and those defined on arbitrary sets. Applications are given.     

On the curvatures of Einstein spaces

For a pseudo-Riemannian manifold (M, g) of dimensionn3, we introduce a scalar curvature functionS(V) for non-degenerate subspacesV ofT _pM which is a generalization of the scalar curvature, and give some characterizations of Einstein spaces in terms of this scalar curvature function. We also give a characterization for spaces of constant curvature. As an application of our results, we show that the Ricci curvature or the sectional curvature of a Lorentz manifold is constant if the scalar curvature function for non-degenerate subspaces is bounded.Partially supported by the grants from TGRC.     

Variational properties of the Gauss–Bonnet curvatures

The Gauss–Bonnet curvature of order 2k is a generalization to higher dimensions of the Gauss–Bonnet integrand in dimension 2k, as the scalar curvature generalizes the two dimensional Gauss–Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metrics with unit volume are generalized Einstein metrics and once restricted to a pointwise conformal class of metrics are metrics with constant Gauss–Bonnet curvature.     

New abstract Hardy spaces

The aim of this paper is to propose an abstract construction of spaces which keep the main properties of the (already known) Hardy spaces H¹. We construct spaces through an atomic (or molecular) decomposition. We prove some results about continuity from these spaces into L¹ and some results about interpolation between these spaces and the Lebesgue spaces. We also obtain some results on weighted norm inequalities. Finally we present partial results in order to understand a characterization of the duals of Hardy spaces.     

Variational principles in non-metrizable spaces

We study generic variational principles in optimization when the underlying topological space X is not necessarily metrizable. It turns out that, to ensure the validity of such a principle, instead of having a complete metric which generates the topology in the space X (which is the case of most variational principles), it is enough that we dispose of a complete metric on X which is stronger than the topology in X and fragments the space X.     

Total curvatures of Kaehler manifolds in complex projective spaces

Teufel showed that total absolute curvature of a submanifold in a sphere or a hyperbolic space equals to the mean value of the number of critical points of level functions. This is an extension of the classical work of Chern and Lashof. In this paper we shall prove a similar result holds for the total absolute curvature of Kaehler manifold in a complex projective space. We shall also express the total curvature by the Euler numbers.The present research was supported by Grant in Aid for Scientific Research No. 5754005.     

Asymptotic kuhn-tucker conditions in abstract spaces

Asymptotic extensions of the Kuhn-Tucker conditions, in which both the adjoint equation and the complementary slackness condition are solved asymptotically, are given for vector-valued mathematical programming problems in locally convex spaces. Under appropriate hypotheses, the conditions are both necessary and sufficient for optimality. In particular, they characterize optimality for linear programs. An asymptotic dual program is also given.     

On the mean curvatures sharp estimates of hypersurfaces

Sharp estimates for the mean curvatures of hypersurfaces in Riemannian manifolds are known from the works of Jorge-Xavier ^[3], Markvorsen ^[6] and Vlachos ^[11]. We first give a simplified proof of these estimates. This proof shows that a similar original result holds for hypersurfaces in Einstein manifolds which are warped product of by Ricci-flat manifolds.     

Weak curvatures of irregular curves in high-dimensional Euclidean spaces

Annals of Global Analysis and Geometry - We deal with a robust notion of weak normals for a wide class of irregular curves defined in Euclidean spaces of high dimension. Concerning polygonal...     

Some abstract maximal ideal-like spaces

Presented by R. S. Pierce.     

Semigroups and abstract Gevrey spaces

Conditions are found under which a closed linear operator A in a Banach space X generates a continuous semigroup in a linear topological space Y which is dense in X. The space Y is an abstract Gevrey space associated with the operator A. This is an abstract setting for some results for hyperbolic systems with data in spaces of Gevrey functions.     

Horo-tightness and total (absolute) curvatures in hyperbolic spaces

We prove Gauß-Bonnet-type and Chern-Lashof-type formulas for immersions in hyperbolic space. Moreover we investigate the notion of tightness with respect to horospheres introduced by T.E. Cecil and P.J. Ryan. We introduce the notions of top-set and drop-set, and we prove fundamental properties of horo-tightness in hyperbolic spaces.     

Fixed point theorems in ordered abstract spaces

We extend some fixed point theorems in -spaces, obtaining extensions of the Banach fixed point theorem to partially ordered sets.