We prove that the sigma functions of Weierstrass (g = 1) and Klein (g = 2) are the unique solutions (up to multiplication by a complex constant) of the corresponding systems of 2g linear differential heat equations in a nonholonomic frame (for a function of 3g variables) that are holomorphic in a neighborhood of at least one point where all modular variables vanish. We also show that all local holomorphic solutions of these systems can be extended analytically to entire functions of angular variables. For g =1, we give a complete description of the envelopes of holomorphy of such solutions.

中文翻译：

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二维Sigma函数的唯一性定理

我们证明了Weierstrass（g = 1）和Klein（g = 2）的sigma函数是非完整框架中2 g线性微分热方程的相应系统的唯一解（直至乘以复数常数）3 g变量的函数），在至少一个点上所有模块变量均消失的附近是全纯的。我们还表明，这些系统的所有局部全纯解都可以解析地扩展到角度变量的整个函数。对于g = 1，我们给出了此类解的全同性包络的完整描述。