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

Abstract

On a complete Riemannian manifold M with Ricci curvature satisfying $$\text{Ric}(\nabla r,\nabla r)\ge -A{r}^2(\log r{)}^2(\log (\log r){)}^2...({\log }^{k}r{)}^2$$ 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 $f:M\to \mathbb{R}$ satisfying $\Delta f\ge F(f)$ for a function $F:\mathbb{R}\to \mathbb{R}$.