Let $U \subset \mathbf{R}^{n+k}$ be a bounded open set, and $T \in \mathbf{I}_n(U)$ be an $n$-dimensional integral rectifiable current. Say that $T$ is stationary through (homological) deformations if for any $\epsilon > 0$ there is $\delta > 0$ so that if $S \in \mathbf{I}_{n+1}(U)$ is a current with $\operatorname{spt} \partial S \subset \subset U$ and $\mathbf{M}(\partial S) < \delta$ then \begin{equation} \lvert \mathbf{M}(T + \partial S) - \mathbf{M}(T) \rvert < \epsilon \, \mathbf{M}(\partial S). \end{equation}
Is this a sensible notion of stationarity? Are there related definitions of minimality in the literature?
This is stronger than the classical 'weak' definition of minimality for surfaces with singularities. Recall that a current $T$ may be called stationary in $U$ if, when deforming it along the flow $(\Phi_t)$ of any vector field $X \in C_c^1(U)$, one has \begin{equation} \frac{\mathrm{d} }{\mathrm{d} t} \mathbf{M}(\Phi_{t\#}T) |_{t = 0} = 0. \end{equation} (This is nothing else but saying that the underlying varifold $\lvert T \rvert$ is stationary.)
- There are currents that are stationary, but not 'stationary through deformations': two straight lines meeting at an acute angle give one example.
- However, smoothly embedded minimal surfaces should remain stationary even in this stronger sense.
