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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.

##### MSC:
 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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