VOLUME AND ENERGY STABILITY FOR IMMERSIONS (HARMONIC, MINIMAL)

Much work has been done on minimal submanifolds of a given manifold and also on harmonic maps between manifolds. It is known, for instance, that the critical sets of the volume functional and the energy functional coincide for a Riemannian immersion f between two Riemannian manifolds.;We find a minimal (harmonic) immersion f that is energy stable but not volume stable by looking at maps f: M (--->) N where N is a flat Riemannian manifold, M is a compact manifold without boundary, and dim N = dim M + 1, dim M (GREATERTHEQ) 2. We show that f is a minimal, volume stable immersion if f is totally geodesic. On the other hand, for f: M (--->) N with f harmonic and N flat, we get that f is automatically energy stable.;For oriented surfaces M of genus g immersed in the torus T('3), we show that for g = 0 there are no minimal immersions of M in T('3). For g = 1, we get that M must be totally geodesic and thus a sub-torus. For g (GREATERTHEQ) 2, we show that f cannot be totally geodesic. Thus, a minimal (harmonic) surface in T('3) of genus g (GREATERTHEQ) 2 must be area unstable, but energy stable.;One such minimally immersed surface of genus 9 can be constructed from Schwarz's tetrahedral surface. This surface is a minimal surface immersed in R('3) that is triply periodic, but not oriented. By taking an 8-fold covering on this surface, and then dividing out by the periodic action, we get a minimal surface in T('3) of genus 9 that is orientable. This surface will then be area unstable, but energy stable. Several other examples of surfaces with this stability behavior for energy and volume are also discussed.;An interesting question is whether the stability of f coincides for volume and energy. The second variation formulas for volume and energy show us that for variations of f in normal directions, energy stability implies volume stability. Thus, one suspects that there are energy stable harmonic immersions that are volume unstable minimal immersions.