I know it's not an issue in practice but I don't like seeing this kind of quadratic loop. It's barely longer to do with a single substring call:

if (name != null) {
  int notSlash;
  while (notSlash <= name.length() && name.charAt(notSlash) == '/')
    notSlash++;
  name = name.substring(notSlash);
}

(or substitute your favorite for loop variant)