Stability of Valuations and Koll\'ar Components

We prove that among all Koll\'ar components obtained by plt blow ups of a klt singularity $o \in (X, D)$, there is at most one that is (log-)K-semistable. We achieve this by showing that if such a Koll\'ar component exists, it uniquely minimizes the normalized volume function introduced in [Li15a] among all divisorial valuations. Conversely, we show any divisorial minimizer of the normalized volume function yields a K-semistable Koll\'ar component. We also prove that for any klt singularity, the infimum of the normalized function is always approximated by the normalized volumes of Koll\'ar components.