Abstract
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.
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Rovenski, V., Wolak, R. Deforming metrics of foliations. centr.eur.j.math. 11, 1039–1055 (2013). https://doi.org/10.2478/s11533-013-0231-y
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DOI: https://doi.org/10.2478/s11533-013-0231-y