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arXiv:1810.04661 [math.DG]AbstractReferencesReviewsResources

Uniform convexity in $L^p$ Mabuchi geometry, the space of rays, and geodesic stability

Tamás Darvas, Chinh H. Lu

Published 2018-10-10Version 1

Suppose $(X,\omega)$ is a compact K\"ahler manifold. We consider the $L^p$ type Mabuchi metric spaces $(\mathcal H_{\omega},d_p), \ p \geq 1$, and show that their completions $(\mathcal E^p_{\omega},d_p)$ are uniformly convex for $p >1$, immediately implying that these spaces are uniquely geodesic. Using these findings we show that $\mathcal R^p_{\omega}$, the space of $L^p$ geodesic rays emanating from a fixed K\"ahler potential, admits a chordal metric $d_p^c$, making $(\mathcal R^p_{\omega},d_p^c)$ a complete geodesic metric space for any $p \geq 1$. We also show that the radial K-energy is convex along the chordal geodesic segments of $(\mathcal R^p_{\omega},d_p^c)$. Using the relative Ko{\l}odziej type estimate for complex Monge-Amp\`ere equations, we point out that any ray in $\mathcal R^p_{\omega}$ can be approximated by rays of bounded potentials, with converging radial K-energy. Most importantly, we use these results to verify (the uniform version of) Donaldson's geodesic stability conjecture for rays of bounded potentials.

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