Regularity of the geodesic equation in the space of Sasakian metrics
type: Item
instance of: scholarly article
title: Regularity of the geodesic equation in the space of Sasakian metrics
zbMATH Open document ID: 1245.58016
author: Pengfei Guan
author: Xi Zhang
published in: Advances in Mathematics
publication date: 2012-04-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/0906.5591
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 58D17
Mathematics Subject Classification ID: 32W20
zbMATH DE Number: 6029077
zbMATH Keywords: Sasakian manifold
zbMATH Keywords: geodesic equation
zbMATH Keywords: constant scalar curvature
zbMATH Keywords: degenerate complex Monge-Ampère equation
MaRDI profile type: Publication
OpenAlex ID: W2010965113
arXiv ID: 0906.5591
cites work: A gradient estimate in the Calabi-Yau theorem
cites work: Q5445710
cites work: Einstein metrics on spheres
cites work: The dirichlet problem for nonlinear second-order elliptic equations. II. Complex monge-ampère, and uniformaly elliptic, equations
cites work: The space of Kähler metrics. II.
cites work: The space of Kähler metrics.
cites work: Space of Kähler metrics. III: On the lower bound of the Calabi energy and geodesic distance
cites work: On the Calabi flow
cites work: Geometry of Kähler metrics and foliations by holomorphic discs
cites work: Q4393470
cites work: Q5601113
cites work: Uniqueness and examples of compact toric Sasaki-Einstein metrics
cites work: Q4946338
cites work: Q3473832
cites work: The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation
cites work: Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds
cites work: Locally Sasakian manifolds
cites work: The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function
cites work: The extremal function associated to intrinsic norms.
cites work: Boundary-value problems on \(S^ n\) for surfaces of constant Gauss curvature
cites work: Q4647955
cites work: Boundary value problems for surfaces of constant Gauss Curvature
cites work: A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry
cites work: K-energy maps integrating Futaki invariants
cites work: Some symplectic geometry on compact Kähler manifolds. I
cites work: Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals
cites work: Sasaki-Einstein manifolds and volume minimisation
cites work: The Monge-Ampère operator and geodesics in the space of Kähler potentials
cites work: Test configurations for \(k\)-stability and geodesic rays
cites work: On differentiable manifolds with certain structures which are closely related to almost contact structure. I
cites work: Uniqueness of Sasaki-Einstein metrics
cites work: Complex Monge-Ampere and Symplectic Manifolds
cites work: THE SASAKI–RICCI FLOW
cites work: Partial differential equations. 1: Basic theory
cites work: On the ricci curvature of a compact kähler manifold and the complex monge-ampére equation, I
DOI: 10.1016/J.AIM.2011.12.002
review text: The space of Sasakian metrics of a compact \((2n+1)\)-dimensional Sasakian manifold \(M\), with Sasakian structure \((\xi,\eta, \Phi,g)\) given by a Riemannian metric \(g\), a Reeb vector field \(\xi\), a contact 1-form \(\eta=g(\xi, \cdot)\), and a transverse Kähler structure \(\Phi(X)=\nabla_X \xi\), can be seen as the open set \(\mathcal{H}\) of basic functions \(\varphi\in C_B^{\infty}(M)\) defining a nowhere vanishing measure \(d\mu_{\varphi}=\eta_{\varphi}\wedge (d\eta_{\varphi})^n\), where \(\eta_{\varphi}=\eta+ d^c_B\varphi\). Each \(\varphi\in \mathcal{H}\) deforms the initial Sasakian structure into a new one with the same Reeb vector, transverse holomorphic structure, and holomorphic structure on the cone \(C(M)\), and any Sasakian structure satisfying these conditions is defined by one of such function \(\varphi\). Moreover they define the same area of \(M\), and the corresponding transverse Kähler forms are in the same basic \((1,1)\) class \([d\eta]_B\), the basic Kähler class of the Sasakian manifold \(M\).NEWLINENEWLINENEWLINEIn a previous work of the authors [``A Geodesic equation in the space of Sasakian metrics'', in: Geometry and Analysis I, Advanced Lectures in Mathematics, International Press, 303--318 (2011)], they showed that defining a Weil-Peterson metric on \(\mathcal{H}\) by \((\psi_1,\psi_2)_{\varphi}=\int_M\psi_1\cdot \psi_2 d\mu_{\varphi}\), the corresponding geodesic equation for a path \(\varphi(t,\cdot)=\varphi_t(\cdot)\) from \(\varphi_0\) to \(\varphi_1\) is given by NEWLINE\[NEWLINE \frac{\partial^2\varphi}{\partial t^2} -\frac{1}{4}\left|d_B \frac{\partial \varphi}{\partial t} \right|^2_{g_{\varphi}}=0 \text{ on }M\times (0,1).NEWLINE\]NEWLINE NEWLINEThe authors show that this geodesic equation in \(\mathcal{H}\) is equivalent to the Dirichlet problem of a degenerate complex Monge-Ampère equation on the truncated cone \(M\times [1, \frac{3}{2}]\) of the Kähler cone \(C(M)\) of \(M\), given by NEWLINE\[NEWLINE (\Omega_{\psi})^{n+…
reviewed by: Isabel Salavessa
description: scientific article; zbMATH DE number 6029077