K-energy on polarized compactifications of Lie groups

Abstract

In this paper, we study Mabuchi's K-energy on a compactification M of a reductive Lie group G, which is a complexification of its maximal compact subgroup K. We give a criterion for the properness of K-energy on the space of K × K-invariant Kähler potentials. In particular, it turns to give an alternative proof of Delcroix's theorem for the existence of Kähler-Einstein metrics in case of Fano manifolds M . We also study the existence of minimizers of K-energy for general Kähler classes of M .2000 Mathematics Subject Classification. Primary: 53C25; Secondary: 53C55, 58J05, 19L10.

G-Sasaki Manifolds and K-Energy

“…More recently, the Kähler geometry on G-manifolds (called for simplicity, ifĜ is smooth and Kählerian) has been extensively studied (cf. [3,19,20,33,32,21]). For examples, Delcroix proved the existence of Kähler-Einstein metrics on a Fano G-manifold under a sufficient and necessary condition [19], and later, Li, Zhou and Zhu gave another proof of Delcroix's result and generalize it to Kähler-Ricci solitons [33].…”

“…[3,19,20,33,32,21]). For examples, Delcroix proved the existence of Kähler-Einstein metrics on a Fano G-manifold under a sufficient and necessary condition [19], and later, Li, Zhou and Zhu gave another proof of Delcroix's result and generalize it to Kähler-Ricci solitons [33]. Moreover, Delcroix's condition can be explained in terms of K-stability [33] (also see [20]), and thus their results can be both regarded as direct proofs to Yau-Tian-Danaldson conjecture in case of G-manifolds [46,39,41,13].…”

“…Our goal in this paper is to extend the argument in [33] on G-manifolds to G-Sasaki manifolds. In particular, we prove a version of Delcroix's theorem for the existence of transverse Sasaki-Einstein metrics in case of G-Sasaki manifolds.…”

“…Proposition 6.4). Then by a construction of piecewise linear function in [33], the analytic obstruction implies (0.1).…”

“…For the sufficient part of Theorem 0.1, we use the argument in [33] to prove the properness of K-energy on the space of K × K-invariant potentials on π n+1 c B 1 (M ) through the reduced K-energy µ(·). Since µ(·) is defined on a class of convex functions on ι * (P + ) and P does not satisfy the Delzant condition in general [17], we shall modify the proof of main theorem in [33,Theorem 1.2]. Here ι * is an isomorphism from P + to a polytope P in a subspace of codimension one in a * (cf.…”