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We discuss the effects of resonant $\nu_e \to \nu_s$ and $\bar{\nu}_e \to\bar{\nu}_s$ ($\nu_s$ is a sterile neutrino) conversions in the dense medium ofa supernova. In particular, we assume the sterile neutrino $\nu_s$ to be in thehot dark matter few eV mass range. The implications of such a scenario for thesupernova shock re-heating, the detected $\bar\nu_e$ signal from SN1987A andfor the r-process nucleosynthesis hypothesis are analysed in some detail. Theresulting constraints on mixing and mass difference for the $\nu_e-\nu_s$system are derived. There is also an allowed region in the neutrino parameterspace for which the r-process nucleosynthesis can be enhanced.Comment: Latex file, 27 pages including 10 ps figures, uses psfig.sty. Few  references added, some change in the Acknowledgements and some minor  corrections in the tex

Supernova bounds on resonant active-sterile neutrino conversions