This paper proves that the existence of Kähler-Einstein metrics implies the stability of the underlying Kähler manifold, thereby disproving a long-standing conjecture that a compact Kähler manifold admits Kähler-Einstein metrics if it has positive first Chern class and no nontrivial holomorphic vector fields. The paper also establishes an analytic criterion for the existence of Kähler-Einstein metrics, which is satisfied on stable Kähler manifolds if a certain $ C^0 $-estimate is true. In 1977, Yau solved the Calabi conjecture, which states that any $(1,1)$-form representing $ c_1(M) $ is the Ricci form of some Kähler metric on M. Aubin and Yau independently proved the existence of Kähler-Einstein metrics on Kähler manifolds with negative first Chern class. Calabi also proved the uniqueness of Kähler-Einstein metrics for Kähler manifolds with nonpositive first Chern class. In 1986, Bando and Mabuchi proved the uniqueness of Kähler-Einstein metrics on compact Kähler manifolds with positive first Chern class. In 1957, Matsushima proved that a Kähler-Einstein metric with positive scalar curvature exists on M only if the Lie algebra $ \eta(M) $ of holomorphic fields is reductive. This implies that the positivity of the first Chern class is not sufficient for the existence of Kähler-Einstein metrics. In 1983, Futaki introduced the Futaki invariant $ f_M $, which is a character of the Lie algebra $ \eta(M) $. He proved that $ f_M $ is zero if M has a Kähler-Einstein metric. In 1989, the author solved the problem for complex surfaces, proving that any complex surface M with $ c_1(M) > 0 $ has a Kähler-Einstein metric if and only if $ \eta(M) $ is reductive. This result provides new insight into the geometric aspects of the Calabi problem. However, the conjecture cannot be generalized to Kähler orbifolds, as there exists a two-dimensional Kähler orbifold S with $ c_1(S) > 0 $, $ \eta(S) = \{0\} $, and no Kähler-Einstein orbifold metrics. This orbifold has an isolated singularity, which may be responsible for the nonexistence of Kähler-Einstein orbifold metrics.This paper proves that the existence of Kähler-Einstein metrics implies the stability of the underlying Kähler manifold, thereby disproving a long-standing conjecture that a compact Kähler manifold admits Kähler-Einstein metrics if it has positive first Chern class and no nontrivial holomorphic vector fields. The paper also establishes an analytic criterion for the existence of Kähler-Einstein metrics, which is satisfied on stable Kähler manifolds if a certain $ C^0 $-estimate is true. In 1977, Yau solved the Calabi conjecture, which states that any $(1,1)$-form representing $ c_1(M) $ is the Ricci form of some Kähler metric on M. Aubin and Yau independently proved the existence of Kähler-Einstein metrics on Kähler manifolds with negative first Chern class. Calabi also proved the uniqueness of Kähler-Einstein metrics for Kähler manifolds with nonpositive first Chern class. In 1986, Bando and Mabuchi proved the uniqueness of Kähler-Einstein metrics on compact Kähler manifolds with positive first Chern class. In 1957, Matsushima proved that a Kähler-Einstein metric with positive scalar curvature exists on M only if the Lie algebra $ \eta(M) $ of holomorphic fields is reductive. This implies that the positivity of the first Chern class is not sufficient for the existence of Kähler-Einstein metrics. In 1983, Futaki introduced the Futaki invariant $ f_M $, which is a character of the Lie algebra $ \eta(M) $. He proved that $ f_M $ is zero if M has a Kähler-Einstein metric. In 1989, the author solved the problem for complex surfaces, proving that any complex surface M with $ c_1(M) > 0 $ has a Kähler-Einstein metric if and only if $ \eta(M) $ is reductive. This result provides new insight into the geometric aspects of the Calabi problem. However, the conjecture cannot be generalized to Kähler orbifolds, as there exists a two-dimensional Kähler orbifold S with $ c_1(S) > 0 $, $ \eta(S) = \{0\} $, and no Kähler-Einstein orbifold metrics. This orbifold has an isolated singularity, which may be responsible for the nonexistence of Kähler-Einstein orbifold metrics.