Open Problems
Open Problems
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DOI：
10.1090/surv/222/10
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发表时间：
2022-01
期刊：
影响因子：
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通讯作者：
Kefeng Liu
中科院分区：
文献类型：
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作者：
Kefeng Liu
计算平坦环面的格林函数。环面的热核可以通过观察定义环面的对偶晶格来计算（参见[1]的§9.5）。格林函数可以通过对热核进行时间积分，减去常数项后得到。但是用一个好的形式来表达它是很复杂的。对于二维环面，格林函数可以用自同构形式表示（参见Serge Lang的书[2]的第二章）。D 'Hoker和Phong[3]研究了高属曲线上的保形不变格林函数，它允许用函数表示（更准确地说，用“素数形式”表示）。它很有吸引力，因为它只依赖于复杂结构而不是度规，并且在弦理论中经常使用。在IMU b[4]的调查文章中，我建议研究特征函数、格林函数和热核的零点和临界点。最好的测试问题是在平坦环面上，在那里问题已经很深很困难了。C.-L。Wang和c - s。林b[5]在二维情况下取得了根本性的进展。c。林能够把它与椭圆的算术联系起来
Compute the Green’s functions for a flat torus. The heat kernel for the torus can be computed by looking into the dual lattice that defines the torus (see §9.5 of [1]). Green’s function can be obtained by integrating the heat kernel in time, after subtracting the constant term. But this is complicated to be expressed in a nice form. For two dimensional tori, Green’s function can be expressed in terms of automorphic forms (see Chapter 2 of Serge Lang’s book [2]). D’Hoker and Phong [3] studied the conformally invariant Green’s function on curves of higher genus, which admits an expression in terms of theta functions (more precisely, in terms of the “prime form”). It is attractive since it depends only on the complex structure and not the metric, and it is used frequently in string theory. In my survey article for IMU [4], I suggested to study zeros and critical points of eigenfunctions, Green’s function and heat kernel. The best testing problem is on flat tori where the problem is already deep and difficult. C.-L. Wang and C.-S. Lin [5] made fundamental progress in the two dimensional case. C.-S. Lin was able to relate it to arithmetic of elliptic