Entire holomorphic curves on a Fermat surface of low degree

The purpose of the paper is to study some problems raised by Hayman and Gundersen about the existence of non-trivial entire and meromorphic solutions for the Fermat type functional equation $f^n+g^n+h^n=1$. Hayman showed that no non-trivial meromorphic solutions and entire solutions exist when $n \ge 9$ and $n \ge 7$ respectively. By considering the entire holomorphic curves on the Fermat surface defined by $X^n+Y^n+Z^n=W^n$ on the complex projective space $\mathbb{P}^3$ and applying the method of jet differentials, we show that no non-trivial meromorphic solutions and entire solutions exist when $n \ge 8$ and $n \ge 6$ respectively. In particular, this completes the investigation of non-trivial entire solutions for all $n$ and respectively, meromorphic solutions for all cases except for $n=7$. Finally, for the generalized Fermat type functional equation $f^n+g^m+h^l=1$, we will also prove the non-existence of non-trivial meromorphic solutions when $1/n+1/m+1/l \le 3/8$, giving the strongest result obtained so far.