How do I clean a stainless steel gas cooktop without harming the surface?
Try white vinegar with 5% or 10% acidity first. Weiman makes some great cleaners for stainless steel you can try too. 

When I've burned food into my stainless steel pots the only thing that would get it out was to manually rub them with varying roughness's of stainless steel wool. This of course is going to alter the surface..