Volumes of flows
Author:
David L. Johnson
Journal:
Proc. Amer. Math. Soc. 104 (1988), 923931
MSC:
Primary 53C20; Secondary 58F17
MathSciNet review:
964875
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Abstract: If is an oriented nonsingular flow on a Riemannian manifold , the volume of is defined as the dimensional measure of the unit vector field tangent to , as a section of with the induced metric. It is shown that, for any metric of the twodimensional torus, and for any homotopy class of flows on the torus, there is a unique smooth flow of minimal volume within the homotopy class. It has been shown that the Hopf foliation on the round threesphere absolutely minimizes the volume of flows on . In higher dimensions this is not the case; the Hopf fibrations are not even local minima of the volume functional for flows on the round fivesphere. It is not known whether a volumeminimizing flow on exists.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198809648754
PII:
S 00029939(1988)09648754
Article copyright:
© Copyright 1988 American Mathematical Society
