Let $M^n$ be a complete Riemannian manifold and assume that there is a constant $a \geq 0$ such that $\operatorname{sec}_M \geq -a^2$. Do you know whether this implies an upper bound on the volume growth of area minimizing submanifolds i.e. whether there exists a function $f: [0,\infty) \rightarrow [0,\infty)$ such that for any area minizing submanifold $\Sigma^k \subseteq M$ and any $p\in M$ it holds that $\mathcal{H}^k(B(p,r)\cap \Sigma) \leq f(r)$ for every $r\geq 0$. Here $\mathcal{H}^k$ is the $k$-dimensional Hausdorff measure.

Let $M^n$ be a complete simply connected Riemannian manifold with $\operatorname{sec}_M \leq 0$ (i.e. a Hadamard manifold) and assume that there is a constant $a \geq 0$ such that $\operatorname{sec}_M \geq -a^2$. Do you know whether this implies an upper bound on the volume growth of area minimizing submanifolds i.e. whether there exists a function $f: [0,\infty) \rightarrow [0,\infty)$ such that for any area minizing submanifold $\Sigma^k \subseteq M$ and any $p\in M$ it holds that $\mathcal{H}^k(B(p,r)\cap \Sigma) \leq f(r)$ for every $r\geq 0$. Here $\mathcal{H}^k$ is the $k$-dimensional Hausdorff measure.

I would be very grateful for a reference or a counterexample in the case when $M = \mathbb{R}^n$.