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A geometric theory of harmonic and semi-conformal maps

100%

Kock A.

Open Mathematics

2004

tom 2

nr 5

708-724

We describe for any Riemannian manifold M a certain scheme M L, lying in between the first and second neighbourhood of the diagonal of M. Semi-conformal maps between Riemannian manifolds are then analyzed as those maps that preserve M L; harmonic maps are analyzed as those that preserve the (Levi-Civita-) mirror image formation inside M L.

Deforming metrics of foliations

84%

Rovenski V., Wolak R.

Open Mathematics

2013

tom 11

nr 6

1039-1055

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.

Ograniczanie wyników

2 Open Mathematics

1 Kock A.

1 Rovenski V.

1 Wolak R.

1 2013

1 2004

Wyniki wyszukiwania

A geometric theory of harmonic and semi-conformal maps

Deforming metrics of foliations