The references provided by Carlo Beenakker solve the problem only in special cases. The article "Global existence and convergence of Yamabe flow" by R. Ye assumes -- for example -- conformal flatness. The problem was solved completely within the last years. Important progress in dimension 3,4 and 5 was obtained by
- Schwetlick and Struwe, Convergence of the Yamabe flow for "large'' energies. J. Reine Angew. Math. 562 (2003), 59–100.
and then later, the general statement was proven by Simon Brendle in
- Brendle, Convergence of the Yamabe flow in dimension 6 and higher. Invent. Math. 170 (2007), no. 3, 541–576.
However, Brendle's proof only solves all cases, if one assumes the general version of the positive mass theorem. This general version was the subject of several preprints recently. Besides several articles by J. Lohkamp (see arXiv), there is also a preprint by Schoen and Yau https://arxiv.org/abs/1704.05490. To my knowledge no one of these preprints has been published so far.