2018/02/11 by He, Weiyong
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1802.03841
We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of Chen-Cheng \citeCC1, CC2, CC3 on existence of Kähler metrics with constant scalar curvature (csck) on compact Kähler manifolds. Following their strategy, we prove that given a Sasaki structure (with Reeb vector field and complex structure on its cone fixed ), there exists a Sasaki structure with transverse constant scalar curvature (cscs) if and only if the K-energy is reduced proper modulo the identity component of the automorphism group which preserves both the Reeb vector field and transverse complex structure. Technically, the proof mainly consists of two parts. The first part is a priori estimates for scalar curvature type equations which are parallel to Chen-Cheng's results in \citeCC2, CC3 in Sasaki setting. The second part is geometric pluripotential theory on a compact Sasaki manifold, building up on profound results in geometric pluripotential theory on Kähler manifolds. There are notable, and indeed subtle differences in Sasaki setting (compared with Kähler setting) for both parts (PDE and pluripotential theory). The PDE part is an adaption of deep work of Chen-Cheng \citeCC1, CC2, CC3 to Sasaki setting with necessary modifications. While the geometric pluripotential theory on a compact Sasaki manifold has new difficulties, compared with geometric pluripotential theory in Kähler setting which is very intricate. We shall present the details of geometric pluripotential on Sasaki manifolds in a separate paper \citeHL (joint work with Jun Li).