I would like to open a LLC in the Grand cayman islands.I live in N.Y.I will not be doing business in the caymans.

You will need to consult with a registered agent in Grand Cayman (such as Citco) to assist you in forming the entity. You should note that tax-avoidance opportunities that you may imagine from such an arrangement are limited. Moreover, it will be a crime to fail to disclose to the IRS your control over a foreign bank account. If you intend to use the vehicle to buy income-producing property in the U.S., you would probably be better served forming the entity in New York.