\| |\sigma_{\varepsilon}|^0_{\lambda_{\varepsilon}} - |\Sigma|^0_{\lambda_{\varepsilon}} \|_{C^{k,\alpha}_0(M)} \leq C \quad\text{ and }\quad \| |\sigma_{\varepsilon}|^0_{\lambda_{\varepsilon}} - |\Sigma|^0_{\lambda_{\varepsilon}} \|_{C^{k,\alpha}_0(M)} \leq C