"Geodesic Equations on asymptotically locally Euclidean K\""ahler manifolds"

arXiv2307.01991MaRDI QIDQ6510922

Author name not available (Why is that?)

Abstract: We solve the geodesic equation in the space of K"ahler metrics under the setting of asymptotically locally Euclidean (ALE) K"ahler manifolds and we prove global

m a t h c a l C^{1, 1}

regularity of the solution. Then, we relate the solution of the geodesic equation to the uniqueness of scalar-flat ALE metrics. To this end, we study the asymptotic behavior of

v a r e p s i l o n

-geodesics at spatial infinity. Under the assumption that the Ricci curvature of a reference ALE K"ahler metric is non-positive, convexity of the Mabuchi

K

-energy along

v a r e p s i l o n

-geodesics. However, we will also prove that on the line bundle

m a t h c a l O (- k)

over

m a t h b b C m a t h b b P^{n - 1}

with

n g e q 2

and

k e q n

, no ALE K"ahler metric can have non-positive (or non-negative) Ricci curvature.

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