  We investigate the role of pressure, via its Hessian tensor $\mathbf{H}$, on
amplification of vorticity and strain-rate and contrast it with other inviscid
nonlinear mechanisms. Results are obtained from direct numerical simulations of
isotropic turbulence with Taylor-scale Reynolds number in the range $140-1300$.
Decomposing $\mathbf{H}$ into local isotropic ($\mathbf{H}^{\rm I}$) and
nonlocal deviatoric ($\mathbf{H}^{\rm D}$) components reveals that
$\mathbf{H}^{\rm I}$ depletes vortex stretching (VS), whereas $\mathbf{H}^{\rm
D}$ enables it, with the former slightly stronger. The resulting inhibition is
significantly weaker than the nonlinear mechanism which always enables VS.
However, in regions of intense vorticity, identified using conditional
statistics, contribution from $\mathbf{H}$ dominates over nonlinearity, leading
to overall depletion of VS. We also observe near-perfect alignment between
vorticity and the eigenvector of $\mathbf{H}$ corresponding to the smallest
eigenvalue, which conforms with well-known vortex-tubes. We discuss the
connection between this depletion, essentially due to (local) $\mathbf{H}^{\rm
I}$, and recently identified self-attenuation mechanism [Buaria et al. {\em
Nat. Commun.} 11:5852 (2020)], whereby intense vorticity is locally attenuated
through inviscid effects. In contrast, the influence of $\mathbf{H}$ on
strain-amplification is weak. It opposes strain self-amplification, together
with VS, but its effect is much weaker than VS. Correspondingly, the
eigenvectors of strain and $\mathbf{H}$ do not exhibit any strong alignments.
For all results, the dependence on Reynolds number is very weak. In addition to
the fundamental insights, our work provides useful data and validation
benchmarks for future modeling endeavors, for instance in Lagrangian modeling
of velocity gradient dynamics, where conditional $\mathbf{H}$ is explicitly
modeled.
