An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a “Weltkonstante”-or-how Ramanujan split temperatures

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 2F1 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.

A variational principle for Gaussian lattice sums

“…This is in accordance with the conjecture in [27], where also the value of L Z 2 has been established numerically (see also [4]). Besides the fact that lattices seem to play an important role for Landau's problem, there is another relation to our results: for the hexagonal lattice Λ 2 , it follows [35] that…”

“…It can be seen as the 2-dimensional standard torus with a flat metric induced by the lattice Λ. Using our result, we can affirm a conjecture raised in [35], closely related to the problem posed by Baernstein in 1997 [3]. Our main result implies that among all tori T Λ the minimal temperature of the heat distribution is maximized by the hexagonal torus T Λ 2 .…”

“…The absolute constant L is known as Landau's constant and its solution is conjectured to have been found by Rademacher [66] in 1943 and comes from the universal covering map of the once-punctured hexagonal torus. Due to the findings in [35], we believe that our result could be of some relevance for finding the value of L or at least has some connection to this problem. The problem of determining the exact value of the absolute constant L, described above as Landau's constant, can be seen as a holomorphic packing problem, as we want to pack a disc of a given size into f (D).…”

“…This fact also holds for the product of the respective constants if we replace Λ 2 by Z 2 in the respective formulas [29,35]. Since min z E Z 2 (z; 1) ≤ min z E Λ 2 (z; 1) it follows that L Z 2 ≥ L Λ 2 .…”

“…Baernstein suggested to study heat kernels on flat tori T Λ = R 2 /Λ of fixed covering radius in order to solve Landau's problem [2,3,5]. In [35] a conjecture was raised that if the surface area is fixed instead, the minimal temperature should be maximal only for the hexagonal torus. It is well-known that the eigenvalues of the Laplace-Beltrami operator ∆ Λ on T Λ (with appropriately chosen sign) are given by κ λ = 4π 2 |λ ⊥ | 2 , where λ ⊥ ∈ Λ ⊥ is an element of the dual lattice.…”