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In this note, we use harmonic coordinates to prove an existence result on exhaustion functions by adapting the method in Tam (in: Recent Advances in Geometric Analysis, Advanced Lectures in Mathematics, vol. 11, pp. 211–215, International Press, Somerville, 2010). Then we apply it to prove short-time existence of Ricci flow using similar method in He (Existence and applications of Ricci flows via pseudolocality, arXiv:1610.01735) and Lee and Tam (On existence and curvature estimates of Ricci flow, arXiv:1702.02667) .

A Note on Existence of Exhaustion Functions and Its Applications