@article{Li2018, 
author = {Chi Li and Chenyang Xu},
title = {Stability of Valuations: Higher Rational Rank},
year = {2018},
journal = {Peking Mathematical Journal},
volume = {1},
number = {1},
pages = {1-79},
keywords = {Quasi-monomial valuation, Normalized volume, K-stability, Metric tangent cone},
url = {https://www.sciopen.com/article/10.1007/s42543-018-0001-7},
doi = {10.1007/s42543-018-0001-7},
abstract = {Given a klt singularity  x∈(X,D), we show that a quasi-monomial valuation v with a finitely generated associated graded ring is a minimizer of the normalized volume function  vol^(X,D),x, if and only if v induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a minimizer is unique among all quasi-monomial valuations up to rescaling. As a consequence, we prove that for a klt singularity  x∈X on the Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, the intermediate K-semistable cone associated with its metric tangent cone is uniquely determined by the algebraic structure of  x∈X, hence confirming a conjecture by Donaldson–Sun.}
}