Let M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D ⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D ⊥-closed. Assuming that D ⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H.
@article{bwmeta1.element.doi-10_2478_s11533-013-0231-y,
author = {Vladimir Rovenski and Robert Wolak},
title = {Deforming metrics of foliations},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {1039-1055},
zbl = {1275.53062},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0231-y}
}
Vladimir Rovenski; Robert Wolak. Deforming metrics of foliations. Open Mathematics, Tome 11 (2013) pp. 1039-1055. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0231-y/
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