Minimalité des sous-variétés totalement géodésiques en géométrie de Finsler

Berck, Gautier
(2004)

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  • Berck, GautierUCLouvain
    author
Supervisors
Lambrechts, Pascal
 ;
Álvarez, Juan Carlos
Abstract
During the last years, modern methods of convex, differential and integral geometry have enlightened the study of areas and volumes on normed and Finsler spaces originally initiated by Busemann in the 40's. One big difference with the Euclidean case is the existence of several natural notions of areas and volumes. Among them, two stand out thanks to very nice properties. On one hand, normed and Finsler spaces are metric spaces, hence the Hausdorff measure is a natural choice. On the other hand, the Holmes-Thompson volume, introduced for purely geometrical purposes, appears to be the good notion to generalize classical formulas of integral geometry. Armed with areas and volumes, the problem of the characterization of minimal submanifolds naturally arises. Since we also hold a norm, one is then led to compare the uni and multidimensional variational problems. The main result of this work lies naturally on this context. Indeed, we proove that totally geodesic submanifolds of Finsler spaces are minimal for the Holmes-Thompson volume. To establish this, a large part of this manuscript is focused on the construction of geometrical tools to deal with the multidimensional variational problem. More specifically, we generalize to general densities the well known Legendre's map and Hilbert's form. By the way, we get other interesting results such as an explicit construction of a mean curvature covector whose nullity characterizes the minimality of submanifolds. Also, we obtain a nice integral representation of the Holmes-Thompson density allowing to compute it without knowing the dual norm. Finally, we give sufficient conditions, generalizing the usual Hamel's equations, for the minimality of affine subspaces.
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Berck, G. (2004). Minimalité des sous-variétés totalement géodésiques en géométrie de Finsler. https://hdl.handle.net/2078.5/97528