We prove a new rigorous upper bound on the vertical heat transport for Benard-Marangoni convection of a two- or three-dimensional fluid layer with infinite Prandtl number. Precisely, for Marangoni number $Ma gg 1$ the Nusselt number $Nu$ is bounded asymptotically by $Nu lesssim Ma^{2/7}(ln Ma)^{-1/7}$. Key to our proof are a background temperature field with a hyperbolic profile near the fluids surface, and new estimates for the coupling between temperature and vertical velocity.
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