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Let $\Sigma$ be a closed orientable hyperbolic surface. We introduce thenotion of a \textit{geodesic current with corners} on $\Sigma$, which behaveslike a geodesic current away from certain singularities (the "corners"). Wetopologize the space of all currents with corners and study its properties.  We prove that the space of currents with corners shares many properties withthe space of geodesic currents, although crucially, there is no canonicalaction of the mapping class group nor is there a continuous intersection form.To circumvent these difficulties, we focus on those currents with cornersarising from harmonic maps of graphs into $\Sigma$. This leads to the space of\textit{marked harmonic currents with corners}, which admits a natural Borelaction by the mapping class group, and an analog of Bonahon's\cite{Bonahon}compactness criterion for sub-level sets of the intersection form against afilling current.  As an application, we consider an analog of a curve counting problem on$\Sigma$ for triangulations. Fixing an embedding $\phi$ of a weighted graph$\Gamma$ into $\Sigma$ whose image $\phi(\Gamma)$ is a triangulation of$\Sigma$, let $N_{\phi}(L)$ denote the number of mapping classes $f$ so that aweighted-length minimizing representative in the homotopy class determined by$f \circ \phi$ has length at most $L$. In analogy with theorems ofMirzakhani\cite{Mirzakhani}, Erlandsson-Souto\cite{ErlandssonSouto}, andRafi-Souto\cite{RafiSouto}, we prove that $N_{\phi}(L)$ grows polynomially ofdegree $6g-6$ and the limit \[ \lim_{L \rightarrow \infty}\frac{N_{\phi}(L)}{L^{6g-6}}\] exists and has an explicit interpretationdepending on the geometry of $\Sigma$, the vector of weights, and thecombinatorics of $\phi$ and $\Gamma$.

Currents with corners and counting weighted triangulations