  We study the thermal conductance of ballistic point contacts. These contacts
are realized as few nanometer long pillars in so called air-gap
heterostructures (AGHs). The pillar length being much smaller that the mean
free path of the phonons up to room temperature. Due to the small dimension and
the low density of the pillars the thermal conductance of the AGHs is several
orders of magnitude reduced in comparison to bulk structures. The measurement
results are in quantitative agreement with a simple model that based on the
Boltzmann transport equation.
