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# Communications in Analysis and Geometry

## Volume 12 (2004)

### Number 3

### An Optimal Loewner-type Systolic Inequality and Harmonic One-forms of Constant Norm

Pages: 703 – 732

DOI: http://dx.doi.org/10.4310/CAG.2004.v12.n3.a8

#### Authors

#### Abstract

We present a new optimal systolic inequality for a closed Riemannian manifold *X*, which generalizes a number of earlier inequalities, including that of *C*. Loewner. We characterize the boundary case of equality in terms of the geometry of the Abel-Jacobi map, *A*_{X}, of *X*. For an extremal metric, the map *A*_{X} turns out to be a Riemannian submersion with minimal fibers, onto a flat torus. We characterize the base of *A*_{X} in terms of an extremal problem for Euclidean lattices, studied by A.-M. Bergé and J. Martinet. Given a closed manifold *X* that admits a submersion F to its Jacobi torus *T*^{b1(X)}, we construct all metrics on *X* that realize equality in our inequality. While one can choose arbitrary metrics of fixed volume on the fibers of *F*, the horizontal space is chosen using a multi-parameter version of J. Moser's method of constructing volume-preserving flows.