On Hyperelliptic Minimal Surfaces With Even Genus

Abstract

In this paper, we announce our results for compact oriented minimal surfaces in flat 4-tori which are given in our preprint [ 7 ]. We consider (i) the Morse index of a one-parameter family, (ii) the existence of non-holomorphic hyperelliptic minimal surfaces with even genus. Key tools are established by the first author [ 3 , 4 ], and we continue the work related to our previous papers [ 5 , 6 ] for the higher genus and codimensional case. Our key tools can be applied for the higher genus and codimensional case, but the situation is different from that of the previous paper. It is crucial to solve such difficulties, and we give a partial answer for it. We will give a clear explanation about the long story.