We derive an upper bound on the free energy of a Bose gas at density $ \varrho $ and temperature T. In combination with the lower bound derived previously by Seiringer (Commun. Math. Phys. 279(3): 595–636, 2008), our result proves that in the low density limit, i.e., when $ a^3 \varrho \ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, the leading term of $\Delta f$, the free energy difference per volume between interacting and ideal Bose gases, is equal to $ 4\pi a(2\varrho^{2}-[\varrho-\varrho_{c}]^{2}_{+}) $. Here, $\varrho_c (T)$ denotes the critical density for Bose–Einstein condensation (for the ideal Bose gas), and $ [⋅]_{+}= max \{⋅ , 0\}$ denotes the positive part.

A hyperbolic 3-manifold M which fibers over the circle admits a flow called the suspension flow. We show that such a flow can be isotoped to be uniformly quasigeodesic in the hyperbolic metric on M; i.e., the flow lines lifted to hyperbolic space are K-bilipschitz embeddings of R (K > 1 fixed).

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In this paper, I study the group of analytic automorphisms of a bounded product domain in the space$C(S,C)$of continuous functions on a compact space$S$. I prove that its automorphism group is a Lie group and I am able to prove which are the bounded symmetric ones.