You really need to cut it up into flat quads (or triangles) to guide the surface into a shape you want. A giant n-gon won't be able to read your mind when it comes to creating a contiguous non-flat surface.
The surface needs to twist as it goes from one edge across to the other. I would use a Twist modifier, with a Volume Select below it encompassing those two edges. Then I could add rings as needed to get the deformation looking smooth.