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Integral estimates for the trace of symmetric operators

Let $Φ:TM\to TM$ be a positive-semidefinite symmetric operator of class $C^1$ defined on a complete non-compact manifold $M$ isometrically immersed in a Hadamard space $\bar{M}$. In this paper, we given conditions on the operator $Φ$ and on the second fundamental form to guarantee that either $Φ\equiv 0$ or the integral $\int_M \mathrm{tr}\,ΦdM$ is infinite. We will given some applications. The first one says that if $M$ admits an integrable distribution whose integrals are minimal submanifolds in $\bar{M}$ then the volume of $M$ must be infinite. Another application states that if the sectional curvature of $\bar{M}$ satisfies $\bar{K}\leq -c^2$, for some $c\geq 0$, and $λ:M^m\to [0,\infty)$ is a nonnegative $C^1$ function such that gradient vector of $λ$ and the mean curvature vector $H$ of the immersion satisfy $|H+p\nabla λ|\leq (m-1)c λ$, for some $p\geq 1$, then either $λ\equiv 0$ or the integral $\int_M λ^s dM$ is infinite, for all $1\leq s\leq p$.