No, we are not. As a matter of fact there are several (more then 10) girls, claiming to be Princess Eugenie (both in Facebook and myspace).
Most of the mini bios are plain simple and childish - something along the lines "My parents are divorced but they are supet, my granny is the best in the world, I'm very close with my cousins Wills and Harry".
If Princess Eugenie, Beatrice or any other Royal wished to have a facebook account (quite a normal wish and means to communicate with friends), I would think they had enough sense to have write something other then 'Hi, I'm Eugenie. I'm a Princess and member of the Royal Family. My grandma is the Queen of England'.
