German Law/Obligation to provide information about income
I'm a German citizen but I live in Atlanta. I have no residence in Germany. Do I have to tell the German Youth-Office about my income? And if so, according to what law?
If the German Youth Office has been asked to legally represent your child by your child's representative, then yes.
According to § 1605 I BGB: https://dejure.org/gesetze/BGB/1605.html
If you fail to comply, you might get sued either in Germany or in the US. The plaintiff can choose between these two jurisdictions. A German court may also order you to pay child support based on an estimated income if you were given a chance to provide information but didn't do so.
German court orders in child support matters are usually enforced in the US (and vice versa), actually often beyond what the German court ordered.
Because German child support calculations do take other obligations and debt and mortgage payments into account, it is also in your own interest to provide as much information as possible. You have to consider that if there is a child, there is a financial obligation. The judicial cooperation between Germany and the US works quite well. And if you ignore letters now, you'll have to pay later and only the lawyers will be happy. On the other hand, German child support is based on your disposable income, so if you don't make much, you may not even have to pay much or possibly nothing at all.
If you want, I can give you an assessment of your exact obligations and if there are ways to reduce them, but I would need to get a complete picture of your financial situation. My e-mail is firstname.lastname@example.org, but I do charge 400 EUR for a consultation with child support calculation.