Rotate GameObject on specific axis

Hello together

I'm a Unity beginner, so my question is may easy to answer to you. But I'm struggling now for a while and don't know what to do else.

I want to implement a "Holographic Puzzle" for the Windows HoloLens. For that I want to rotate my puzzle Parts. Simplified my puzzle parts are just cubes. The user target these puzzle parts with "Gaze", a ray coming out of the users head in the direction the user is looking. If the ray hits an object (puzzle part), I draw an indicator to show that the user can rotate this object. This Indicator is on transformed to the plane which is hit. See Image below. For that I transform the indicator:

     // Assign gameObject's transform up vector equals GazeManager's instance Normal.
     gameObject.transform.up = GazeManager.Instance.Normal;

Now my problem: I want to rotate the GameObject if the user make a Gesture. This rotation should be around this GazeManager's instance Normal AND the rotation should be arround the center of the GameObject. See Image below.

I tried a lot and searched a lot, but nothing worked. I paste my actual code below. If someone can help me it would be very friendly

Thanks a lot, Lukas

Code:

 public class GestureAction : MonoBehaviour
 {
 ....
 private Vector3 startUpdateNormalPosition = Vector3.zero;
     private Vector3 startUpdateGazeOriginPosition = Vector3.zero;
 ....
 void PerformNavigationStart(Vector3 position)
     {
         Debug.Log("PerformNavigationStart");
         startUpdateNormalPosition = GazeManager.Instance.Normal;
         startUpdateGazeOriginPosition = GazeManager.Instance.gazeOrigin;
     }
  void PerformNavigationUpdate(Vector3 position)
     {
         // Debug.Log("PerformManipulationUpdate");
         // rotationFactor = GestureManager.Instance.NavigationPosition.z * RotationSensitivity;
         Vector3 navigationDirection = AppContext.Instance.AppState.NavigationStartPosition - position;
         float rotationMaximum = Mathf.Max(navigationDirection.x, navigationDirection.y, navigationDirection.z);
         Debug.Log("rotationMaximum: " + rotationMaximum);
       
         // Apply rotation to object
         //transform.Rotate(new Vector3(rotationFactor_X, -1 * rotationFactor_Y, rotationFactor_Z));
         // transform.Rotate(new Vector3(0, -1 * rotationFactor, 0));
         // transform.Rotate(rotationVector, rotationSpeed);
        // Vector3 worldRotation = this.transform.TransformDirection(AppContext.Instance.AppState.NavigationStartRotationAxis);
         Vector3 worldRotation = this.transform.TransformDirection(AppContext.Instance.AppState.NavigationStartRotationAxis);
         //  transform.Rotate(worldRotation, rotationSpeed);
         transform.RotateAround(startUpdateGazeOriginPosition, startUpdateNormalPosition, rotationMaximum*20*Time.time);
        
     }
 }