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The space of Kähler metrics. (English) Zbl 1041.58003
In 1987, T. Mabuchi introduced a “Riemannian distance” in the infinite dimensional space of Kähler metrics of a fixed Kähler class and defined the Levi-Civita connection in it [Osaka J. Math. 24, 227–252 (1987; Zbl 0645.53038)]. S. K. Donaldson conjectured that each fixed Kähler class with Mabuchi distance is, in fact, a geodesically convex metric space. The author partially solves Donaldson’s conjecture by proving that any two Kähler metrics of a fixed Kähler class can be joined by a \(C^{1,1}\)-geodesic (Theorem 3); further it is proved that the length of such geodesic equals the Mabuchi distance between its “end” Kähler metrics – the least upper bound of lengths of all possible curves joining the same pair of metrics (Corollary 3). In Proposition 2 the author proves that the length of each geodesic joining different metrics does not vanish. Hence, the Mabuchi distance is, in fact, a metric – a positive definite symmetric distance satisfying the triangle inequality. By applying the foregoing results, the author proves uniqueness results for constant scalar curvature metrics.

58D17 Manifolds of metrics (especially Riemannian)
53C22 Geodesics in global differential geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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