Can a ''natural'' three-generation neutrino mixing scheme satisfy everything?

We examine the potential for a ''natural'' three-neutrino mixing scheme to satisfy available data and astrophysical arguments. By ''natural'' we mean no sterile neutrinos, and a neutrino mass hierarchy similar to that of the charged leptons. We seek to satisfy (or solve) (1) accelerator and reactor neutrino oscillation constraints, including LSND, (2) the atmospheric muon neutrino deficit problem, (3) the solar neutrino problem, (4) r-process nucleosynthesis in neutrino-heated supernova ejecta, and (5) cold+hot dark matter models. We argue that putative supernova r-process nucleosynthesis bounds on two-neutrino flavor mixing can be applied directly to three-neutrino mixing in the case where one vacuum neutrino mass eigenvalue difference is dominated by the others. We show that in this ''one mass scale dominance'' limit, a natural three-neutrino oscillation solution meeting all the above constraints exists only if the atmospheric neutrino data and the LSND data can be explained with one neutrino mass difference. In this model, an explanation for the solar neutrino data can be effected by employing the other independent neutrino mass difference. Such a solution is only marginally allowed by the current data, and proposed long-baseline neutrino oscillation experiments can definitively rule it out. If it were ruled out, the simultaneous solution of the above constraints by neutrino oscillations would then require sterile neutrinos and/or a neutrino mass hierarchy of a different nature than that of the charged leptons.