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Let $(X,\omega)$ be a compact K\"ahler manifold and $\mathcal H$ the space ofK\"ahler metrics cohomologous to $\omega$. If a cscK metric exists in $\mathcalH$, we show that all finite energy minimizers of the extended K-energy aresmooth cscK metrics, partially confirming a conjecture of Y.A. Rubinstein andthe second author. As an immediate application, we obtain that existence of acscK metric in $\mathcal H$ implies J-properness of the K-energy, thusconfirming one direction of a conjecture of Tian. Exploiting this propernessresult we prove that an ample line bundle $(X,L)$ admitting a cscK metric in$c_1(L)$ is $K$-polystable.

Regularity of weak minimizers of the K-energy and applications to properness and K-stability