Area contraction of k-dimensional surfaces

TL;DRAbstract

and denote byt(x) the solution of the above differential equations with initial condition �0(x) = x. Further, we assume that there is a (positive definite) metric g. With this metric, every regular k-dimensional surface can be assigned a k-dimensional area, which we will denote byk. In the following section we will relate the evolution of this area to properties of the dynamical system and the metric g. This results in an upper bound for the dimension of regular bounded invariant sets with a finite area. In the next sec- tion we assume that the dynamical system has p known first integrals and we show how the evolution of k-dimensional surfaces is related to the evolution of k p-dimensional sur- faces in the level set of the first integrals. We conclude by giving some applications of the stated results.

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