We discuss how the neutrino material-heating rate and electron fraction in the region above the neutrino-sphere in a nascent type II supernova become very sensitive to neutrino flavor mixings. Matter-enhancement effects would ensure that a vμ or vτ (hereafter, vτ) neutrino with a vacuum mass of roughly 10 to 100 eV would have a mass-level-crossing with a light ve in this region. Coincidentaly, this is the neutrino mass range which is sometimes conjectured to provide the closure density for the universe. Since the supernova ντ neutrinos have considerably higher average energies than do the νe, transformations between these species at the level crossings result in more energetic electron neutrinos. This would imply an enhanced νe capture rate which, in turn, would result in a higher heating rate and would tend to increase the electron fraction, reducing the neutron excess. For vacuum mixing angle between ντ and νe of θ>10-4 the increase in early heating results in roughly a factor of two increase in supernova explosion energy, which may help solve the explosion-mechanism problem. However, late neutrino flavor conversion will drive the ejected material proton-rich and a promising site for r-process nucleosynthesis may be lost unless it is also true that θ < 10-3. © 1993.