Deforming metrics of foliations

Vladimir Rovenski; Robert Wolak

Open Mathematics (2013)

  • Volume: 11, Issue: 6, page 1039-1055
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Vladimir Rovenski, and Robert Wolak. "Deforming metrics of foliations." Open Mathematics 11.6 (2013): 1039-1055. <http://eudml.org/doc/269448>.

@article{VladimirRovenski2013,
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.},
author = {Vladimir Rovenski, Robert Wolak},
journal = {Open Mathematics},
keywords = {Riemannian manifold; Distribution; Foliation; Flow of metrics; Second fundamental tensor; Mean curvature; Harmonic; Totally geodesic; Heat equation; Double-twisted product; distribution; foliation; flow of metrics; second fundamental tensor; mean curvature; harmonic; totally geodesic; heat equation; double-twisted product},
language = {eng},
number = {6},
pages = {1039-1055},
title = {Deforming metrics of foliations},
url = {http://eudml.org/doc/269448},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Vladimir Rovenski
AU - Robert Wolak
TI - Deforming metrics of foliations
JO - Open Mathematics
PY - 2013
VL - 11
IS - 6
SP - 1039
EP - 1055
AB - 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.
LA - eng
KW - Riemannian manifold; Distribution; Foliation; Flow of metrics; Second fundamental tensor; Mean curvature; Harmonic; Totally geodesic; Heat equation; Double-twisted product; distribution; foliation; flow of metrics; second fundamental tensor; mean curvature; harmonic; totally geodesic; heat equation; double-twisted product
UR - http://eudml.org/doc/269448
ER -

References

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