Sounds like you may have a lucrative idea. However, if you think you'll be earning significant money from thus endeavor I would consult a tax attorney. Based on my limited knowledge of US Tax law, if you bring money into the US or its territories you are subject to certain tax requirements. Better safe than sorry and hire a tax lawyer.
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I agree with my colleague. If you are a U.S. citizen or U.S. permanent resident, run, don't walk, to a qualified CPA and/or tax attorney in the U.S. and discuss your tax issues. You could find yourself in major problems if you don't proactively address your tax obligations.
This is general advice regarding immigration law and is not meant to be taken as specific advice to a particular person or circumstance.
US residents are liable for tax on their worldwide income whereever they reside. You will never be able to set up a Cayman Islands company without a lawyer so you can tell your tax dreams then and there. Good luck on your worldwide revenue.
The above is general legal and business analysis. It is not "legal advice" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
Agree with the last answer, if you are a U.S. citizen, you owe taxes on worldwide income. You are exempt from Cayman taxes likely because you are not conducting any business in the islands. Furthermore, it is a bad time right now to try to skirt taxes if you are a U.S. citizen with FACTA rules and regulations soon to be implemented. If you have accounts with any bank that maintains accounts in the U.S., they will likely be giving your information to the IRS. If you owe taxes, the IRS will discover this and come after you. You should pay the taxes and avoid potential fines, penalties, and/or jail time. A small price to pay to the country that developed the internet and has allowed you to earn revenues for your business across the globe.