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An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a “Weltkonstante”-or-how Ramanujan split temperatures. (English) Zbl 1468.35080

Summary: In this work we investigate the heat kernel of the Laplace-Beltrami operator on a rectangular torus and the according temperature distribution. We compute the minimum and the maximum of the temperature on rectangular tori of fixed area by means of Gauss’ hypergeometric function \(_2F_1\) and the elliptic modulus. In order to be able to do this, we employ a beautiful result of S. Ramanujan [Quart. J. 45, 350–372 (1914; JFM 45.1249.01)], connecting hypergeometric functions, the elliptic modulus and theta functions. Also, we investigate the temperature distribution of the heat kernel on hexagonal tori and use Ramanujan’s corresponding theory of signature 3 to derive analogous results to the rectangular case. Lastly, we show connections to the problem of finding the exact value of E. Landau’s “Weltkonstante” [Math. Z. 30, 608–634 (1929; JFM 55.0770.03)], a universal constant arising in the theory of extremal holomorphic mappings; and for a related, restricted extremal problem we show that the conjectured solution is the second lemniscate constant.

MSC:

35K08 Heat kernel
33C05 Classical hypergeometric functions, \({}_2F_1\)
35-03 History of partial differential equations
51M16 Inequalities and extremum problems in real or complex geometry
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