Let $(M^{n},g)$ be a Riemannian manifold such that $B_{R}(p)$ has a compact closure in $B_{2R}(p)$. Let $K\subset M^{n}$ be a smooth submanifold with $\dim(K)=k_{0}\leq n-3$. If $u$ is harmonic in $B_{R}(p)\setminus K$ and there is some $\epsilon\in(0,1)$ such that

(6.41)

$$|u(x)|\leq\frac{C}{d_{g}(x,K)^{(n-2-k_{0})-\epsilon}},$$

then $u$ is harmonic in $B_{R}(p)$.