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Spectral, stochastic and curvature estimates for submanifolds of highly negative curved spaces

We prove spectral, stochastic and mean curvature estimates for complete $m$-submanifolds $φ\colon M \to N$ of $n$-manifolds with a pole $N$ in terms of the comparison isoperimetric ratio $I_{m}$ and the extrinsic radius $r_φ\leq \infty$. Our proof holds for the bounded case $r_φ< \infty$, recovering the known results, as well as for the unbounded case $r_φ=\infty$. In both cases, the fundamental ingredient in these estimates is the integrability over $(0, r_φ)$ of the inverse $I_{m}^{-1}$ of the comparison isoperimetric radius. When $r_φ=\infty$, this condition is guaranteed if $N$ is highly negatively curved.