In "Riemannian Geometry, Peter Petersen, GTM171, Third Edition" page 430, there is an Anderson Lemma: For each $n\geq 2$, there is an $\varepsilon(n)>0$ such that any complete Ricci flat manifold (M,g) that satisfies $$Vol(B(p,r))\geq(1-\varepsilon)\omega_n r^n$$ for some $p\in M$ is isometric to Euclidean space.
I have problem about the proof in this book.
The book proved it by contradiction: for each $i$, it constructed a complete Ricci flat manifold $(M_i,g_i)$, with $\lim\limits_{i\rightarrow \infty}\frac{Vol(B(p_i,r)}{\omega_n r^n}\geq(1-\frac{1}{i})$, and $(M_i,g_i)$ is not isometric to Euclidean space.
But the book said that for all $r>0$, the $C^{1,\alpha}$ harmonic norm of $(M_i,g_i)$ of scale $r$ is nonzero. After scaling the metric suitably, it assumed the $C^{1,\alpha}$ harmonic norm of $(M_i,\bar{g_i})$ less than 1, and the pointed norm has positive lower bound. I think it is impossible, but I do not know how to correct it.
It is true that if $(M_i,g_i)$ is not isometric to Euclidean space, we can find $r_i$, such that the $C^{1,\alpha}$ harmonic norm of $(M_i,g_i)$ of scale $r_i$ is nonzero. But we do not know that it is uniformly bounded.
Remark Crossposted in MO.