Abstract
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 Ramanujan, 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 Landau’s “Weltkonstante”, 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.
Highlights
This article is inspired by a problem posed and investigated in the article of Baernstein et al [7] and, again, in Eremenko’s preprint [24]
The problem discussed in [7,24] is about finding the exact value of a constant, closely related to a universal constant arising from a problem in geometric function theory, posed by Landau [38]
We solely focus on tori identifiable with rectangular lattices or a hexagonal lattice
Summary
This article is inspired by a problem posed and investigated in the article of Baernstein et al [7] and, again, in Eremenko’s preprint [24]. In [24], Eremenko studies Landau’s problem for rectangular lattices with lattice parameters 2ω and 2ω and the restriction that the covering radius is fixed to 4ω2 + 4ω 2 = 1. This leads to the main idea in this work, which is to study the heat kernel as a function of the elliptic modulus k and the complementary elliptic modulus k with k2 + k 2 = 1. Baernstein suggested to study the heat distribution on the torus to get a better understanding of Landau’s problem [4–6] This is the main motivation for this article. We will show a similar connection of L+ to the minimal temperature on the hexagonal torus
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