Recently, I have been studying the monotonicity of the Cheeger constant under Ricci flow on surfaces. In fact, I want to use the monotonicity to prove the convergence of Ricci flow on $S^2$, which was first proven by Bennett Chow. Then, Hamilton proved it once more.
When I dealt with it, I found a way to prove the attainability of the Cheeger constant. Although I haven't done the exact calculations yet, I think the proof for compact Riemannian surfaces is at least correct.
So I searched on Google for information on the attainability of the Cheeger constant, but I couldn't find anything. Therefore, I want to know if this problem has been studied.
I am a beginner in Geometrical Analysis and I do not have an adviser to guide me. This platform is my only option to seek assistance. I appreciate any help provided.