An application of the Duistermaat–Heckman theorem and its extensions in Sasaki geometry

Abstract

Building on an idea laid out by , we use the Duistermaat-Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein-Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Studying the leading terms we prove they are all proper. Among consequences we get that the Einstein-Hilbert functional attains its minimal value and each Sasaki cone possess at least one Reeb vector field with vanishing transverse Futaki invariant.

Localizing the Donaldson–Futaki invariant

“…for a proof of this see eg. [10]. Therefore, using Theorem 2.2 and (13) for f = 1 2 ∆ g µ a , we get that…”

Contact Structures of Sasaki Type and Their Associated Moduli

“…Nevertheless, applying the well known Duistermaat-Heckman theorem to Sasaki geometry, the authors [BHL18] proved (2) V R , S r , H, H 1 (R) all tend to +∞ as R approaches the boundary of the Sasaki cone t + (away from 0); (3) there exists a ray r min ⊂ t + that minimizes H 1 ; (4) if dim T > 1 and H 1 (r min ) ≤ 0, all Sasakian structures in t + are indefinite; (5) if H 1 (R min ) = 0 and the corresponding Sasaki metric is extremal, it must have constant scalar curvature.…”

On positivity in Sasaki geometry

“…However, both the volume V ξ and the total transverse scalar curvature S ξ vary with the point on the ray. It is, thus, convenient to consider the Einstein-Hilbert functional H(ξ) [BHLTF17,BHL18]. Actually, it is more convenient to consider the 'signed' version…”

“…when the dimension of the Sasaki manifold is 2n + 1. It was shown in [BHL18] that H 1 (ξ) tends to +∞ as ξ approaches the boundary, thus, H 1 (ξ) has a global minimum ξ min . The critical points of H 1 (ξ) in t + are S ξ = 0 and the zeros of the Sasaki-Futaki invariant [BHLTF17].…”