Liouville-type theorems and applications to geometry on complete Riemannian manifolds

On a complete Riemannian manifold M with Ricci curvature satisfying $\textrm{Ric}(\nabla r,\nabla r) \geq -Ar^2(\log r)^2(\log(\log r))^2...(\log^{k}r)^2$ for $r\gg 1$, 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 C^2 function $f:M\rightarrow \Bbb R$ satisfying $Δf\geq F(f)$ for a function $F:\Bbb R\rightarrow \Bbb R$.

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