Resumen de Extremal metrics and stabilities on polarized manifolds

Toshiki Mabuchi

• TheHitchin�Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Lübke, Uhlenbeck andYau, states that an indecomposable holomorphic vector bundle over a compact Kähler manifold is stable in the sense of Takemoto�Mumford if and only if the vector bundle admits a Hermitian-Einstein metric. Its manifold analogue known as Yau�s conjecture, which originated from Calabi�s conjecture, asks whether �stability� and �existence of extremal metrics� for polarized manifolds are equivalent. In this note the recent progress of this subject, by Donaldson, Tian and our group, together with its relationship to algebraic geometry will be discussed.