Fermat functional equations revisited

The problem of the existence of transcendental meromorphic or entire solutions for the Fermat functional equation $f^n+g^n+h^n=1$ was first studied by Walter Hayman in 1984. It is known that meromorphic (entire) solutions exist for $n\le 6$ ($n \le 5$) and no meromorphic (entire) solution exists when $n\ge 9$ ($n\ge 7$). In this talk we will revisit this problem from a more geometric view point. This is a joint work with Sai-Kee Yeung.