The paper “On the Lieb-Thirring constant on the torus“, co-authored by Sergey Zelik, Alexei Illyin (Keldysh, Moscow), and Ari Laptev (Imperial College London), has been published in the Russian journal Математические заметки. The case of the two-torus is treated in the paper. A link to the Russian version is here, and a link to the English translation (in Mathematical Notes) is here. A screenshot of the introduction is shown below.