Harmonic Maps and the Topology of Manifolds with Positive Spectrum and Stable Minimal Hypersurfaces
نویسندگان
چکیده
Harmonic maps are natural generalizations of harmonic functions and are critical points of the energy functional defined on the space of maps between two Riemannian manifolds. The Liouville type properties for harmonic maps have been studied extensively in the past years (Cf. [Ch], [C], [EL1], [EL2], [ES], [H], [HJW], [J], [SY], [S], [Y1], etc.). In 1975, Yau [Y1] proved that any harmonic function bounded from one side on a complete Riemannian manifold with non-negative Ricci curvature must be a constant. Schoen and Yau [SY] have shown that a harmonic map of finite energy from a complete Riemannian manifold with non-negative Ricci curvature to a complete manifold with non-positive sectional curvature is constant. This Liouville theorem of Schoen-Yau was used [SY] to show the important result which states that any smooth map of finite energy from a complete Riemannian manifold with non-negative Ricci curvature to a compact manifold with non-positive sectional curvature is homotopic to constant on each compact set. In this paper, we use the same idea of Schoen-Yau to study complete non-compact manifolds with Ricci curvature bounded from below and stable minimal hypersurfaces
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