超Kähler空间H上的全纯函数与正则四元数函数
Holomorphic functions and regular quaternionic functions on the hyperkähler space H
摘要：
设H是四元数空间，具有标准超复数结构。设R（D）是D上的正则函数的模。对于S^2中的每一个酉向量p，R（D）包含全纯函数空间w.r. t.证明了在任意有界区域D上，对任意p都不是J_p-全纯的正则函数的存在性.我们的出发点是Chen和Li关于超Kaehler流形之间映射的一个结果，其中对一类限制较少的四元数映射也得到了类似的结果.基于全纯映射的能量极小性质，给出了判别J_p-全纯函数是否为正则函数的一个判据.
英文摘要：
Let H be the space of quaternions, with its standard hypercomplex structure. Let R(D) be the module of regular functions on D. For every unitary vector p in S^2, R(D) contains the space of holomorphic functions w.r.t. the complex structure J_p induced by p. We prove the existence, on any bounded domain D, of regular functions that are not J_p-holomorphic for any p. Our starting point is a result of Chen and Li concerning maps between hyperkaehler manifolds, where a similar result is obtained for a less restricted class of quaternionic maps. We give a criterion, based on the energy-minimizing property of holomorphic maps, that distinguishes J_p-holomorphic functions among regular functions.
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