Real Algebraic and Analytic Geometry Preprint Server

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Submission: 2003, February 11.

Abstract:
Let $\f_\lambda$ be a family of codimension $p$ foliations defined on a family $M_\lambda$ of manifolds and let $X_\lambda$ be a family of compact subsets of $M_\lambda$. Suppose that $\f_\lambda$, $M_\lambda$ and $X_\lambda$ are definable in an o-minimal structure and that all leaves of $\f_\lambda$ are closed. Given a definable family $\Omega_\lambda$ of differential $p$-forms satisfaying $i_Z\Omega_\lambda=0$ for any vector field $Z$ tangent to $\f_\lambda$, we prove that there exists a constant $A >0$ such that the integral of $|\Omega_\lambda|$ on any transversal of $\f_\lambda$ intersecting each leaf in at most one point is bounded by $A$. We apply this result to prove that $p$-volumes of transverse sections of $\f_\lambda$ are uniformly bounded.

Mathematics Subject Classification (2000): 53C12, 49Q15, 03C64, 32B20.

Keywords and Phrases: real foliations, o-minimal structures, integration of differential forms.

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