Geodesic
Convexity of the 𝐾-energy on the space of Kähler metrics and uniqueness of extremal metrics
Berman, Robert1 ; Berndtsson, Bo1
1 Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden
Journal of the American Mathematical Society, Tome 30 (2017) no. 4, pp. 1165-1196

Voir la notice de l'article provenant de la source American Mathematical Society

We establish the convexity of Mabuchi’s $K$-energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.
DOI : 10.1090/jams/880

Berman, Robert 1 ; Berndtsson, Bo 1

1 Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden 
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Berman, Robert; Berndtsson, Bo. Convexity of the 𝐾-energy on the space of Kähler metrics and uniqueness of extremal metrics. Journal of the American Mathematical Society, Tome 30 (2017) no. 4, pp. 1165-1196. doi: 10.1090/jams/880

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