Liouville-type theorems and applications to geometry on complete Riemannian manifolds

摘要

On a complete Riemannian manifold M with Ricci curvature satisfying Ric(r, r)≥-Ar~2(logr)~2(log(logr))~2? (log~kr)~2 for r ?1, where A0 is a constant, and r is the distance from an arbitrarily fixed point in M, we prove some Liouville-type theorems for a C~2 function f:M→R satisfying Δf≥F(f) for a function F:R→R. As an application, we obtain a C~0 estimate of a spinor satisfying the Seiberg-Witten equations on such a manifold of dimension 4. We also give applications to the conformal transformation of the scalar curvature and isometric immersions of such a manifold.