A Kähler-Einstein metric is a special type of geometric structure on a Kähler manifold, which is a complex manifold equipped with a symplectic form. This metric satisfies Einstein's equations in the context of differential geometry, meaning it has constant scalar curvature. These metrics are important in both mathematics and theoretical physics, particularly in the study of string theory and algebraic geometry. They help in understanding the geometry of complex spaces and have applications in various fields, including mathematical physics and complex differential geometry.