6. Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length $\sqrt{3}$, there is a cylinder inside it, and this cylinder is exactly aligned with the line $A C_{1}$. Then the maximum volume of the cylinder is $\qquad$
Answer $\frac{\pi}{2}$.
Analysis From the problem, we only need to consider the case where the base of the cylinder is tangent to the surface of the cube.
By the symmetry of the figure, the top base of the cylinder must be tangent to the three faces passing through point $A$, and the points of tangency are on $A B_{1}, A C, A D_{1}$, respectively.
Let the point of tangency on segment $A B_{1}$ be $E$, the center of the top base of the cylinder be $O_{1}$, and the radius $O_{1} E=r$. From $\triangle A O_{1} E \sim \triangle A_{1} B_{1} C_{1}$, we get $A O_{1}=\sqrt{2} r$, so the height of the cylinder is
$$
3-2 A O_{1}=3-2 \sqrt{2} r, \quad V=\pi r^{2}(3-2 \sqrt{2} r)
$$

Using the derivative method or the AM-GM inequality, we get $V_{\max }=\frac{\pi}{2}$.