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Geodesics in the space of Kähler cone metrics, I
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 137, Number 5, October 2015
- pp. 1149-1208
- 10.1353/ajm.2015.0036
- Article
- Additional Information
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In this paper, we study the Dirichlet problem of the geodesic equation in
the space of K\"ahler cone metrics ${\cal H}_{\cal B}$; that is equivalent
to a homogeneous complex Monge-Amp\`ere equation whose boundary values
consist of K\"ahler metrics with cone singularities. Our approach concerns
the generalization of the function space defined by Donaldson to the case
of K\"ahler manifolds with boundary; moreover we introduce a subspace
${\cal H}_C$ of ${\cal H}_{\cal B}$ which we define by prescribing
appropriate geometric conditions. Our main result is the existence,
uniqueness and regularity of $C^{1,1}_{\cal B}$ geodesics whose boundary
values lie in ${\cal H}_C$. Moreover, we prove that such geodesic is the
limit of a sequence of $C^{2, \alpha}_{\cal B}$ approximate geodesics
under the $C^{1,1}_{\cal B}$-norm. As a geometric application, we prove
the metric space structure of ${\cal H}_C$.