We present here a simplified version of recent results obtained with B. Helffer and M. Klein. They are concerned with the exponentally small eigenvalues of the Witten Laplacian on $0$-forms. We show how the Witten complex structure is better taken into account by working with singular values. This provides a convenient framework to derive accurate approximations of the first eigenvalues of ${\Delta }_{f,h}^{\left(0\right)}$ and solves efficiently the question of weakly resonant wells.

The conformal infinity of a quaternionic-Kähler metric on a $4n$-manifold with boundary is a codimension $3$ distribution on the boundary called quaternionic contact. In dimensions $4n-1$ greater than $7$, a quaternionic contact structure is always the conformal infinity of a quaternionic-Kähler metric. On the contrary, in dimension $7$, we prove a criterion for quaternionic contact structures to be the conformal infinity of a quaternionic-Kähler metric. This allows us to find the quaternionic-contact structures...