Using Manipulate with ParametricPlot3D: buttons does not appear

Question

I'm trying to use Manipulate with ParametricPlot3D, but the buttons does not appear. Someone has an idea why? Here's the code:

Remove["Global`*"]
ClearAll["Global``*"]

Manipulate[
 ξ0 = 1 + 1*I ;
 F1[x_, y_] :=    
  1/2*Re[(N[Log[x + I*y], 3] - (x + I*y)^4/4) - (N[Log[ξ0], 
        3] - (ξ0)^4/4)];
 F2[x_, y_]   :=   -1/2*
   Im[(N[Log[x + I*y], 3] + (x + I*y)^4/4) - (N[Log[ξ0], 
        3] + (ξ0)^4/4)] ;
 F3[x_, y_]   :=    2*Re[(x + I*y)^2 - (ξ0)^2] ;

 ParametricPlot3D[{F1[x, y], F2[x, y], F3[x, y]}, {x, a, b}, {y, c, 
   d}], {a, -2, -4, Appearance → "Open"}, {b, 2, 4, 
  Appearance → "Open"}, {c, -2, -4, 
  Appearance → "Open"}, {d, 2, 4, 
  Appearance → "Open"}]

Answer 1

Here the problem is with the character you are using instead of Rule . I'm afraid ControlType -> Manipulator doesn't make a difference.

ToCharacterCode["\[RightArrow]"]
(* {8594} *)

ToCharacterCode["\[Rule]"]
(* {62754} *)

Your code works fine if you use -> instead of \[RightArrow].

Manipulate[
 ξ0 = 1 + 1*I;
 F1[x_, y_] := 
  1/2*Re[(N[Log[x + I*y], 3] - (x + I*y)^4/4) - (N[Log[ξ0], 
        3] - (ξ0)^4/4)];
 F2[x_, y_] := -1/2*
   Im[(N[Log[x + I*y], 3] + (x + I*y)^4/4) - (N[Log[ξ0], 
        3] + (ξ0)^4/4)];
 F3[x_, y_] := 2*Re[(x + I*y)^2 - (ξ0)^2];
 ParametricPlot3D[{F1[x, y], F2[x, y], F3[x, y]}, {x, a, b}, {y, c, d}]
 , {a, -2, -4, Appearance -> "Open"}
 , {b, 2, 4, Appearance -> "Open"}
 , {c, -2, -4, Appearance -> "Open"}
 , {d, 2, 4, Appearance -> "Open"}
 ]

By the way, your code would be better if you didn't redefine your functions each time Manipulate refreshes.

With[
 {ξ0 = 1 + 1*I},
 With[{
   F1 = Function[
     {x, y},
     1/2*Re[(N[Log[x + I*y], 3] - (x + I*y)^4/4) - (N[Log[ξ0], 
           3] - (ξ0)^4/4)]
     ],
   F2 = Function[
     {x, y},
     -1/2*
      Im[(N[Log[x + I*y], 3] + (x + I*y)^4/4) - (N[Log[ξ0], 
           3] + (ξ0)^4/4)]
     ],
   F3 = Function[
     {x, y},
     2*Re[(x + I*y)^2 - (ξ0)^2]
     ]
   },
  Manipulate[
   ParametricPlot3D[
    {
     F1[x, y],
     F2[x, y],
     F3[x, y]
     }
    , {x, a, b}
    , {y, c, d}
    ]
   , {a, -2, -4, Appearance -> "Open"}
   , {b, 2, 4, Appearance -> "Open"}
   , {c, -2, -4, Appearance -> "Open"}
   , {d, 2, 4, Appearance -> "Open"}
   ]
  ]
 ]