The proof follows from the standard elliptic regularity. Indeed, the eigenfunction $\varphi_{k}$ satisfies the elliptic equation

(4.147)

$$-\Delta_{h_{0}}\varphi_{k}=\Lambda_{k}\cdot\varphi_{k}.$$

It follows from the standard elliptic regularity that there exists some constant $C>0$ depending only the metric $h_{0}$ such that

(4.148)

$$\|\varphi_{k}\|_{W^{2,2}(Y^{3})}\leq C\cdot\Lambda_{k}\cdot\|\varphi_{k}\|_{L^{2}(Y^{3})}=C\cdot\Lambda_{k}.$$

Applying the Sobolev embedding theorem,

(4.149)

$$\|\varphi_{k}\|_{C^{0,\frac{1}{2}}(Y^{3})}\leq C\|\varphi_{k}\|_{W^{2,2}(Y^{3})}\leq C\cdot\Lambda_{k},$$

where $C>0$ depends only on the metric $h_{0}$. The proof is complete. ∎