A Cheng-Yau type estimate for positive biharmonic functions

Abstract

We establish a Cheng-Yau type estimate for positive biharmonic functions on complete Riemannian manifolds with Ricci curvature satisfies $Ri{c}_g\ge -(n-1)Kg$. If $u$ is a positive biharmonic function in $B_{2R}(p)$, then $$-\frac{{\Delta }_{g}u}{u}+\frac{1}{8n}\frac{|\nabla u{|}_g^2}{u^2}\le C_n\left(R^{-2}+K\right)\text{on }{B}_R(p).$$ Further, if the Ricci curvature is nonnegative, every global positive biharmonic function satisfies ${\Delta }_{g}u\equiv c$ and the sharp estimate $|\nabla u{|}_g^2\le 2cu$ for some nonnegative constant $c$. We also show that every positive $k$-polyharmonic function has nonnegative constant $(k-1)$-st Laplacian and growth of order at most $2k-2$.

13 Pages