12 research outputs found
Biwave Maps into Manifolds
We generalize wave maps to biwave maps. We prove that the composition of a biwave map and a totally geodesic map is a biwave map. We give examples of biwave nonwave maps. We show that if f is a biwave map into a Riemannian manifold under certain circumstance, then f is a wave map. We verify that if f is a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then f is a wave map. We finally obtain a theorem involving an unstable biwave map
Orbifolds, geometric structures and foliations. Applications to harmonic maps
In recent years a lot of attention has been paid to topological spaces which
are a bit more general than smooth manifolds - orbifolds. Orbifolds are
intuitively speaking manifolds with some singularities. The formal definition
is also modelled on that of manifolds, an orbifold is a topological space which
locally is homeomorphic to the orbit space of a finite group acting on .
Orbifolds were defined by Satake, as V-manifolds, then studied by W. Thurston,
who introduced the term "orbifold". Due to their importance in physics, and in
particular in the string theory, orbifolds have been drawing more and more
attention. In this paper we propose to show that the classical theory of
geometrical structures, easily translates itself to the context of orbifolds
and is closely related to the theory of foliated geometrical structures, cf.
\cite{Wo0}. Finally, we propose a foliated approach to the study of harmonic
maps between Riemannian orbifolds based on our previous research into
transversely harmonic maps
Algebraic quantum theory
The main objective consists in endowing the elementary particles with an
algebraic space-time structure in the perspective of unifying quantum field
theory and general relativity: this is realized in the frame of the Langlands
global program based on the infinite dimensional representations of algebraic
groups over adele rings. In this context, algebraic quanta, strings and fields
of particles are introduced.Comment: 137 p
On the normal stability of triharmonic hypersurfaces in space forms
This article is concerned with the stability of triharmonic maps and in
particular triharmonic hypersurfaces. After deriving a number of general
statements on the stability of triharmonic maps we focus on the stability of
triharmonic hypersurfaces in space forms, where we pay special attention to
their normal stability.
We show that triharmonic hypersurfaces of constant mean curvature in
Euclidean space are weakly stable with respect to normal variations while
triharmonic hypersurfaces of constant mean curvature in hyperbolic space are
stable with respect to normal variations.
For the case of a spherical target we show that the normal index of the small
proper triharmonic hypersphere is equal to one and make some comments on the
normal stability of the proper triharmonic Clifford torus
Polyharmonic hypersurfaces into pseudo-Riemannian space forms
In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form N-t(m+1)(c) of dimension m + 1 and index t (m >= 2 and 1 <= t <= m). We shall study hypersurfaces M-t'(m) which are polyharmonic of order r (briefly, r-harmonic), where r >= 3 and either t' = t or t' = t - 1. Let A denote the shape operator of M-t'(m). Under the assumptions that M-t'(m) is CMC and TrA(2) is a constant, we shall obtain the general condition which determines that M-t'(m) is r-harmonic. As a first application, we shall deduce the existence of several new families of proper r-harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper r-harmonic hypersurfaces (r >= 3). Finally, we shall obtain the complete classification of proper r-harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form