Affine symplectic geometry. I: Applications to geometric inequalities
Following the ideas of [\textit{C. B. Croke}, Ann. Sci. Éc. Norm. Super., IV. Ser. 13, 419-435 (1980; Zbl 0465.53032)] and [\textit{M. Berger}, Differ. Geom. Relativ., Vol. Honour A. Lichnerowicz 60th Birthday, 33-42 (1976; Zbl 0342.53038)], the author applies the canonical symplectic structure in the space of geodesics of a Riemannian manifold to obtain various geometric inequalities. He gives a sharp estimate from below of the area of bubble, spanned on a given curve in \(\mathbb{R}^ 3\), and investigates an `` inverse Croke inequality'' relating the area of the complex domain boundary in a Riemannian manifold to the injectivity radius and the volume of the domain. An estimate of the first eigenvalue of the Dirichlet problem (in the case of non-positive curvature) and a sharp estimate of the first eigenvalue of an operator corresponding to the harmonic oscillator on a Hadamard manifold are given.
scientific article; zbMATH DE number 219415
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