$I$-properness of Mabuchi's $K$-energy

Abstract

Over the space of Kähler metrics associated to a fixed Kähler class, we first prove the lower bound of the energy functional ${\tilde{E}}^{\beta }$, then we provide the criterions of the geodesics rays to detect the lower bound of ${\tilde{\mathfrak{J}}}^{\beta }$-functional. They are used to obtain the properness of Mabuchi's $K$-energy. The criterions are examined by showing the convergence of the negative gradient flow of ${\tilde{\mathfrak{J}}}^{\beta }$-functional.

References in corpus (1)

• Kähler non-collapsing, eigenvalues and the Calabi flow

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