Price inequalities and Betti number growth on manifolds without conjugate points (Q2107252)

In Differential Geometry, Price inequalities for harmonic forms on Riemannian manifolds without conjugate points and with a negative Ricci upper bound were introduced by \textit{P. Price} [Manuscr. Math., 43, 131--166 (1983; Zbl 0521.58024)]. Price inequalities have had an important role in several classical problems of the geometric measure theory and mathematical physics. Several authors study the Price inequalities [\textit{P. Price}, Manuscr. Math. 43, 131--166 (1983; Zbl 0521.58024); \textit{M. M. Olbrich}, Doc. Math. 7, 219--237 (2002; Zbl 1029.58019); \textit{W. Luck}, Geom. Funct. Anal. 4, No. 4, 455--481 (1994; Zbl 0853.57021);\textit{W. Luck}, Topology 33, No. 2, 203--214 (1994; Zbl 0847.55004); \textit{J. Lohkamp}, Ann. Math. (2) 140, No. 3, 655--683 (1994; Zbl 0824.53033); \textit{M. Gromov} and \textit{W. Thurston}, Invent. Math. 89, 1--12 (1987; Zbl 0646.53037)]. The principal objective in this paper is to derive the Price inequalities for harmonic forms on Riemannian manifolds without conjugate points and with a negative Ricci upper bound. The authors employ these inequalities to study the asymptotic behavior of the Betti numbers of coverings of Riemannian manifolds without conjugate points, and give a vanishing result for \(L^{2}\)-Betti numbers of closed manifolds without conjugate points.