The paper “Lieb–Thirring constant on the sphere and on the torus” co-authored by A. Ilyin, A. Laptev, and S. Zelik, has been published in the Journal of Functional Analysis. The paper proves, on the 2-sphere and on the 2-torus, the Lieb–Thirring inequalities with improved constants, for orthonormal families of scalar and vector functions. A link to the published version is here, and the screenshot below shows the main theorem.