6. If acute angles $A, B, C$ satisfy $\sin ^{2} A+\sin ^{2} B+\sin ^{2} C=2$, then the minimum value of $\frac{1}{\sin ^{2} A \cos ^{4} B}+\frac{1}{\sin ^{2} B \cos ^{4} C}+\frac{1}{\sin ^{2} C \cos ^{4} A}$ is $\qquad$
$$
\begin{array}{l}
\frac{1}{\sin ^{2} A \cos ^{4} B}+\frac{1}{\sin ^{2} B \cos ^{4} C}+\frac{1}{\sin ^{2} C \cos ^{4} A} \\
=\frac{\frac{1}{2}}{\frac{1}{2} \sin ^{2} A\left(1-\sin ^{2} B\right)^{2}}+\frac{\frac{1}{2}}{\frac{1}{2} \sin ^{2} B\left(1-\sin ^{2} C\right)^{2}}+\frac{\frac{27}{2}}{\frac{1}{2} \sin ^{2} C\left(1-\sin ^{2} A\right)^{2}} \\
\geqslant \frac{27}{\left(2-2 \sin ^{2} B+\frac{1}{2} \sin ^{2} A\right)^{3}}+\frac{\frac{27}{2}}{\left(2-2 \sin ^{2} C+\frac{1}{2} \sin ^{2} B\right)^{3}}+\frac{1}{\left(2-2 \sin ^{2} A+\frac{1}{2} \sin ^{2} C\right)^{3}} \\
\geqslant \frac{\left[\left(\frac{27}{2}\right)^{\frac{1}{4}} \times 3\right]^{4}}{3^{3}}=\frac{81}{2}, \text { equality holds when } \sin ^{2} A=\sin ^{2} B=\sin ^{2} C=\frac{2}{3} .
\end{array}
$$

Attachment: Proof of the Power Mean Inequality.
(Hölder's Inequality) Let $p>1, q>1, \frac{1}{p}+\frac{1}{q}=1, x_{k}>0, y_{k}>0(1 \leqslant k \leqslant n)$,
then $\sum_{k=1}^{n} x_{k} y_{k} \leqslant\left(\sum_{k=1}^{n} x_{k}^{p}\right)^{\frac{1}{p}}\left(\sum_{k=1}^{n} y_{k}^{q}\right)^{\frac{1}{q}}$.
Taking $p=m+1, q=\frac{m+1}{m}, x_{k}=\frac{a_{k}}{b_{k}^{\frac{m}{m+1}}}, y_{k}=b_{k}^{\frac{m}{m+1}}$, substituting into Hölder's inequality we get
$$
\begin{array}{l}
\sum_{k=1}^{n} a_{k} \leqslant\left(\sum_{k=1}^{n} \frac{a_{k}^{m+1}}{b_{k}^{m}}\right)^{\frac{1}{m+1}}\left(\sum_{k=1}^{n} b_{k}\right)^{\frac{m}{m+1}} \\
\Rightarrow\left(\sum_{k=1}^{n} a_{k}\right)^{m+1} \leqslant\left(\sum_{k=1}^{n} \frac{a_{k}^{m+1}}{b_{k}^{m}}\right)\left(\sum_{k=1}^{n} b_{k}\right)^{m} \Rightarrow \sum_{k=1}^{n} \frac{a_{k}^{m+1}}{b_{k}^{m}} \geqslant \frac{\left(\sum_{k=1}^{n} a_{k}\right)^{m+1}}{\left(\sum_{k=1}^{n} b_{k}\right)^{m}}
\end{array}
$$