THE EXISTENCE OF HYPERELLIPTIC MINIMAL SURFACES WITH EVEN GENUS AND THEIR GEOMETRIC INVARIANTS

抄録

In this paper, we will show the existence of non-holomorphic hyperelliptic minimal surfaces of genus four in flat 4-tori. Moreover, we compute two kinds of geometric invariants of such minimal surfaces, namely, the Morse index and the signature. This work treats the higher-genus and -codimension case of the previous papers, and the situation is quite different. We solve the difficulty by the Micallef–Wolfson technique. As its application, we obtain the existence of non-holomorphic hyperelliptically stable minimal surfaces of genus four in flat 4-tori.