Regularity of the geodesic equation in the space of Sasakian metrics
degenerate complex Monge-Ampère equation
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\[NEWLINE (\Omega_{\psi})^{n+1}=0\text{ on } M\times \left(1, \frac{3}{2}\right),NEWLINE\]NEWLINE with \(\psi(\cdot, 1)=\psi_1\) and \(\psi(\cdot,\frac{3}{2})=\psi_{\frac{3}{2}}\), where \(\psi(\cdot,r)=\varphi_{2(r-1)}(\cdot) + 4\log r\) and NEWLINE\[NEWLINE\Omega_{\psi}= \bar{\omega}+ \frac{r^2}{2}\sqrt{-1} (\partial \bar{\partial}\psi -\frac{\partial \psi}{\partial r} \partial \bar{\partial} r),NEWLINE\]NEWLINE where \(\bar{\omega}\) is the Kähler form of the Kähler metric \(\bar{g}=r^2g + dr^2\) on \(C(M)\).NEWLINENEWLINENEWLINEThe main aim of the present paper is to answer a question posed in the above paper [op. cit.] showing that any two points \(\varphi_0,\varphi_1\in \mathcal{H}\) can be connected by a geodesic path, by defining this path in a suitable normed space, and, with analogy to the Kähler case, to conclude on the uniqueness of a constant scalar curvature transverse metric and its minimizing properties for a \(\mathcal{K}\)-energy functional defined on \(\mathcal{H}\).NEWLINENEWLINENEWLINEUsing the equivalent formulation via the above degenerated Monge-Ampère equation, the authors prove the existence of a unique \(C^2_{\omega}\) geodesic path, \(\varphi(t)\), with \(\varphi(0)=\varphi_0\), and \(\varphi(1)=\varphi_1\), lying in the completion \(\bar{\mathcal{H}}\) of \(\mathcal{H}\) with respect to the norm on \(C^2(M \times [1, \frac{3}{2}])\), given by NEWLINE\[NEWLINE \|\psi\|_{C^2_{\omega}(M\times [1, \frac{3}{2}])}= \|\psi\|_{C^1(M\times [1, \frac{3}{2}])}+ \sup_{M \times [1, \frac{3}{2}]}|\Delta \psi|.NEWLINE\]NEWLINE The method of continuity is used and a suitable perturbed equation of the degenerated Monge-Ampère equation, with perturbation defined by each positive basic function \(f\) and constant \(0<\epsilon \leq 1\), obtaining a unique smooth solution \(\psi\), that turns out to be basic and with \(C^2_{\omega}\)-norm bounded from above by a constant that only depends of the \(C^2\) norm of \(f^{1/n}\) and the initial functions \(\psi_1\), \(\psi_{\frac{3}{2}}\), and on the Sasakian structure of \(M\). The geodesic path \(\varphi(t)\) is a \(C^2_{\omega}\) weak limit of solutions \(\varphi_{\epsilon}\) of a related perturbed geodesic equation on \(M\times (0,1)\). Uniqueness of solutions to the degenerated Monge-Ampère equation under given boundary conditions is settled.NEWLINENEWLINENEWLINETaking \(d(\varphi_0,\varphi_1)\) as the length of the geodesic path \({\varphi}_t\) w.r.t. \(d\mu_{\varphi(t)}\), the authors prove that \(\mathcal{H}\) is a metric space. They define a \(\mathcal{K}\)-energy map \(\mu:\mathcal{H}\to \mathbb{R}\), where \(\mathcal{K}\) is the space of transverse Kähler forms in the basic (1,1) class \([d\eta]_B\), identified with a subset \(\mathcal{H}_0\) of \(\mathcal{H}\), by considering any path \(\varphi_t\in \mathcal{H}\) from \(0\) to \(\varphi\) and setting NEWLINE\[NEWLINE \mu(\varphi)=-\int_a^b\int_M \dot{\varphi}_t(S_t^T-\bar{S}) (d \eta_t)^n\wedge \eta_t ~dt,NEWLINE\]NEWLINE where \( S_t^T\) is the transverse scalar curvature of the Sasakian structure \((\xi, \eta_{\varphi_t}, \Phi_{\varphi_t}, g_{\varphi_t})\) defined by \(\varphi_t\) and \(\bar{S}\) is its \(d\mu_{\varphi_t}\)-average. Under the condition on the basic first Chern class \(C_1^B(M)\leq 0\), where \(C_1^B(M)\) is the basic cohomology class defined by \(\frac{1}{2\pi}\rho^T\), with \(\rho^T\) the transverse Ricci form of the initial Sasakian structure, if \(\varphi\in \mathcal{K}\) defines a constant scalar transverse Kähler metric, then the authors show that it is a global minimum for \(\mu\) on \(\mathcal{K}\), and that it is unique if \(C_1^B(M)\) is either strictly zero or negative.
scientific article; zbMATH DE number 6029077
scientific article