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Our main result is that if a sequence of unitary representations $\\rho_j$ strongly converges, then their renormalized energies converge to $\\frac{\\pi}{4}|\\chi(S)|$ and the shape of their harmonic representatives converges to a unique rescaled hyperbolic metric. 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For any unitary representation $\\rho: \\pi_1(S) \\to U(N)$, we introduce its renormalized energy and its harmonic representatives, which are equivariant harmonic maps from the universal cover of $S$ to the unit sphere in $\\mathbb{C}^N$. Our main result is that if a sequence of unitary representations $\\rho_j$ strongly converges, then their renormalized energies converge to $\\frac{\\pi}{4}|\\chi(S)|$ and the shape of their harmonic representatives converges to a unique rescaled hyperbolic metric. 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