Liouville-Type Theorems and Applications to Geometry on Complete Riemannian Manifolds

Abstract

On a complete Riemannian manifold M with Ricci curvature satisfying

$$\mathrm{Ric}(\nabla r,\nabla r) \geq -Ar^2(\log r)^2(\log(\log r))^2\cdots (\log^{k}r)^2$$

for r≫1, where A>0 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.