Lower volume growth and total σkσk-scalar curvature estimates

Let M   be a complete noncompact immersed submanifold in a Hadamard manifold M¯ and Φ a positive-semidefinite symmetric endomorphism on M  . Under our assumptions, we obtain that either Φ≡0Φ≡0 or the growth of the integral ∫Bμtrace(Φ)dVol is at least logarithmic. As the main application, we given conditions to guarantee that the total σkσk-scalar curvature is infinite. Further applications to convex functions and certain ruled manifolds are given.