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

TL;DR

This paper establishes Liouville-type theorems for functions on complete Riemannian manifolds with specific Ricci curvature bounds, extending classical results to more general geometric settings.

Contribution

It introduces new Liouville-type theorems under Ricci curvature conditions involving logarithmic growth, broadening the scope of geometric analysis on manifolds.

Findings

Liouville-type theorems for functions satisfying $ riangle f geq F(f)$

Curvature bounds involving iterated logarithms

Applications to geometric problems on Riemannian manifolds

Abstract

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 $\Delta f\geq F(f)$ for a function $F:\Bbb R\rightarrow \Bbb R$.