Differential inequalities on complete Riemannian manifolds and applications (Q791898)

We prove two sharp inequalities involving the Laplacian of a complete Riemannian manifold. These inequalities depend on the asymptotic rate of growth of V(r), the volume of a geodesic ball of radius r, and have various applications. For example, it is shown that (a) if \((M^ n,ds^ 2)\) has scalar curvature S(x) that satisfies \(S(x)\geq -C(1+dist^ 2(x,x_ 0))\) then \((M^ n,ds^ 2)\) cannot be isometrically minimally immersed in a bounded set in \({\mathbb{R}}^{n+k}\), and (b) Brownian motion is non-explosive on a complete properly embedded minimal submanifold of \({\mathbb{R}}^ N\) while it is explosive on a bounded minimal submanifold of \({\mathbb{R}}^ N\).