I am applying for a PhD at a certain university that requires the student to provide and evidence of support from a potential supervisor before submitting the application. After mailing some professors, one of them agreed to support my application. It was a very brief email saying "I am happy to support you". I immediately prepared a research proposal and sent it to her. She replied by saying "Noted". I prepared my application and submitted it two months later.
Is it appropriate to email her as a reminder of myself? If so, what would be a good email? This university is very important to me and I want to do everything I can to increase my chances of admission.