{
"id": "1810.04661",
"version": "v1",
"published": "2018-10-10T17:46:48.000Z",
"updated": "2018-10-10T17:46:48.000Z",
"title": "Uniform convexity in $L^p$ Mabuchi geometry, the space of rays, and geodesic stability",
"authors": [
"Tamás Darvas",
"Chinh H. Lu"
],
"categories": [
"math.DG",
"math.CV"
],
"abstract": "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.",
"revisions": [
{
"version": "v1",
"updated": "2018-10-10T17:46:48.000Z"
}
],
"analyses": {
"keywords": [
"mabuchi geometry",
"uniform convexity",
"type mabuchi metric spaces",
"radial k-energy",
"complete geodesic metric space"
],
"note": {
"typesetting": "TeX",
"pages": 0,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}