Minimal measures and minimizing closed normal one-currents.

*(English)*Zbl 0973.58004The stable norm of a real homology class \(h\in H_{q}(M,\mathbb{R})\) on a compact Riemannian manifold \((M,g)\) is the infimum of the sums \(\sum |r_{i}|\text{vol}_{q}^g(\sigma_{i})\), where \(\sum r_{i}\sigma_{i}\) is a real Lipschitz cycle representing \(h\).

Basic questions concerning stable norms are: (i) to characterize classes where the infimum is attained and, (ii) to understand the nature of the minimizing objects.

It is known that the infimum is attained in the class of closed normal \(q\)-currents for every \(q\in\mathbb{N}\). For \(q=1\), both questions have been answered in the theory of probability measures on \(TM\) invariant under the geodesic flow \(\varphi\) (and satisfying an additional integrability condition) develoved by J. Mather [Comment. Math. Helv. 64, No. 3, 375-394 (1989; Zbl 0689.58025) and Math. Z. 207, No. 2, 169-207 (1991; Zbl 0696.58027)].

In this paper the case \(q=1\) is considered. A theory of measures in the set of complete Lipschitz curves in \(M\) invariant under shifts of the parameter is developed so that closed normal \(1\)-currents and \(\varphi\)-invariant measures on \(TM\) can be interpreted within this theory. Additionally, an alternative approach considering closed measures on \(TM\) is developed to relate the two classes of objects.

As remarked by the author, the statements hold for Finsler metrics not necessarily symmetric.

Basic questions concerning stable norms are: (i) to characterize classes where the infimum is attained and, (ii) to understand the nature of the minimizing objects.

It is known that the infimum is attained in the class of closed normal \(q\)-currents for every \(q\in\mathbb{N}\). For \(q=1\), both questions have been answered in the theory of probability measures on \(TM\) invariant under the geodesic flow \(\varphi\) (and satisfying an additional integrability condition) develoved by J. Mather [Comment. Math. Helv. 64, No. 3, 375-394 (1989; Zbl 0689.58025) and Math. Z. 207, No. 2, 169-207 (1991; Zbl 0696.58027)].

In this paper the case \(q=1\) is considered. A theory of measures in the set of complete Lipschitz curves in \(M\) invariant under shifts of the parameter is developed so that closed normal \(1\)-currents and \(\varphi\)-invariant measures on \(TM\) can be interpreted within this theory. Additionally, an alternative approach considering closed measures on \(TM\) is developed to relate the two classes of objects.

As remarked by the author, the statements hold for Finsler metrics not necessarily symmetric.

Reviewer: Manuel RitorĂ© (Granada)

##### MSC:

58E10 | Variational problems in applications to the theory of geodesics (problems in one independent variable) |

49Q20 | Variational problems in a geometric measure-theoretic setting |

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |

53C99 | Global differential geometry |