Let $T$ be a $k$-dimensional varifold in a Riemannian manifold $M$. Assume that $f$ is a smooth function on $M$ which is weakly (sub-)harmonic on $T$; that means that $$ \int \langle \nabla_\omega f, \nabla_\omega \varphi \rangle dT(x,\omega) = 0 \,\,(\leq 0) $$ for every compactly supported smooth function $\varphi\geq 0$ on $M$. A prime example is when $M = \mathbb{R}^n$, $T$ is stationary and $f$ is a coordinate function; then $f$ is a weakly harmonic function on $T$. Are there any maximum principles that apply for weakly (sub-)harmonic functions on varifolds? I would be grateful for a reference where I can learn about this.
Added: What I am particularly interested in knowing is the following: Let $T$ be a stationary integer rectifiable varifold in a Riemannian manifold $M$ and let $f$ be a smooth function on $M$ which is weakly sub-harmonic on $T$. Assume that there exists a point $x \in \operatorname{spt}(T)$ such that $f(x) = \sup_{y \in \operatorname{spt}(T)} f(y)$. Can one conclude that $f$ is a constant on the connected component of $\operatorname{spt}(T)$ which contains $x$?
