Area growth and rigidity of surfaces without conjugate points (Q355068)

E. Hopf proved in 1943 that a Riemannian \(2\)-torus without conjugate points is flat. The authors use his method to study complete Riemannian metrics without conjugate points on the plane and the cylinder. The result in the case of the plane is the following: For every point \(p\) the area \(A_p(r)\) of the metric ball around \(p\) with radius \(r\) satisfies NEWLINE\[NEWLINE\liminf_{r\to \infty}\frac{A_p(r)}{\pi r^2}\geq 1.NEWLINE\]NEWLINE Equality holds if and only if the metric is flat. For a cylinder it is shown that the metric is flat if both ends have subquadratic area growth. It is remarkable that both estimates are optimal. Previous results with stronger assumptions can be found in [\textit{V. Bangert} and \textit{P. Emmerich}, Commun. Anal. Geom. 19, No. 4, 773--806 (2011; Zbl 1260.53065)], \textit{K. Burns} and \textit{G. Knieper} J. Differ. Geom. 34, No. 3, 623--650 (1991; Zbl 0723.53024); \textit{H. Koehler}, Asian J. Math. 12, No. 1, 35--45 (2008; Zbl 1147.53034)].