Deforming metrics of foliations

• Autorzy: Vladimir Rovenski , Robert Wolak
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Riemannian manifold; Distribution; Foliation; Flow of metrics; Second fundamental tensor; Mean curvature; Harmonic; Totally geodesic; Heat equation; Double-twisted product

Kategorie tematyczne

• 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
• 53C12: Foliations (differential geometric aspects)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 6
• Strony: 1039-1055

Daty

wydano

2013-06-01

online

2013-03-28

Twórcy

autor

Vladimir Rovenski

• Department of Mathematics, University of Haifa, Mount Carmel, Haifa, 31905, Israel

autor

Robert Wolak

• Faculty of Mathematics and Computer Science, Institute of Mathematics of Jagiellonian University, Łojasiewicza 6, 30-348, Kraków, Poland

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-013-0231-y