On the existence of minimal tori in S(3) of arbitrary spectral genus.

摘要

Harmonic mappings are a generalisation of geodesics, and are defined as the solutions to a natural variational problem. Interest in them began in 1873 with Plateau's problem of finding surfaces of minimal area bounded by given closed space curves. The field has been studied by mathematicians and physicists ever since, and is now both broad and extremely active. In this dissertation I consider harmonic maps which can be studied using integrable systems, and thus by algebro-geometric means. In particular I focus upon a simple case of both geometric and physical interest, namely harmonic maps f from a 2-torus (with conformal structure τ) to the 3-sphere. In [10] Hitchin showed that (except in the case of a conformal map to a totally geodesic S2 ⊆ S 3) the data (f, τ) is in one-to-one correspondence with certain algebro-geometric data. This data consists of a hyperelliptic curve X (called the spectral curve) together with a projection map $p:X\to CP1$, a pair of holomorphic functions on $X-p-10,\infty$, and a line bundle on X, all satisfying certain conditions. He proved (case-by-case) that for g ≤ 3, there are curves of genus g that support the required data, and hence describe harmonic maps f : (T2, τ) → S 3. Of especial interest are conformal harmonic maps as their images are minimal surfaces. I show that for each g ≥ 0, there are countably many conformal harmonic maps f : ( T2, τ) → S3 whose spectral curves have genus g.