Liouville-Type Theorems and Applications to Geometry on Complete Riemannian Manifolds

On a complete Riemannian manifold M with Ricci curvature satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Ric}(\nabla r,\nabla r) \geq -Ar^2(\log r)^2(\log(\log r))^2\cdots (\log^{k}r)^2$$\end{document} for r≫1, where A>0 is a constant, and r is the distance from an arbitrarily fixed point in M, we prove some Liouville-type theorems for a C2 function f:M→ℝ satisfying Δf≥F(f) for a function F:ℝ→ℝ.