  We consider a diffusion on a potential landscape which is given by a smooth
Hamiltonian $H:\mathbb {R}^n\to \mathbb {R}$ in the regime of low temperature
$\varepsilon$. We proof the Eyring-Kramers formula for the optimal constant in
the Poincar\'{e} (PI) and logarithmic Sobolev inequality (LSI) for the
associated generator $L=\varepsilon \Delta -\nabla H\cdot\nabla$ of the
diffusion. The proof is based on a refinement of the two-scale approach
introduced by Grunewald et al. [Ann. Inst. Henri Poincar\'{e} Probab. Stat. 45
(2009) 302-351] and of the mean-difference estimate introduced by Chafa\"{i}
and Malrieu [Ann. Inst. Henri Poincar\'{e} Probab. Stat. 46 (2010) 72-96]. The
Eyring-Kramers formula follows as a simple corollary from two main ingredients:
The first one shows that the PI and LSI constant of the diffusion restricted to
metastable regions corresponding to the local minima scales well in
$\varepsilon$. This mimics the fast convergence of the diffusion to metastable
states. The second ingredient is the estimation of a mean-difference by a
weighted transport distance. It contains the main contribution to the PI and
LSI constant, resulting from exponentially long waiting times of jumps between
metastable states of the diffusion.
