The contact Yamabe flow on \(K\)-contact manifolds
The contact Yamabe problem is an analog in contact geometry of the Yamabe problem for Riemannian manifolds and of the CR Yamabe problem on CR manifolds. Recall that a contact metric manifold endowed with a contact form \(\theta\) is called a \(K\)-contact manifold when the Reeb vector field \(\xi\) is a Killing vector field. The contact Yamabe problem consists in finding on a given contact metric manifold \((M, \theta_0, g_0, J_0)\) a contact form in the conformal class of \(\theta_0\) whose Webster scalar curvature is a constant. One defines the contact Yamabe functional by \[ Y(\theta) = {{\int_M W_\theta\, d\mu}\over {\left(\int_M \, d\mu\right)^{n/(n+1)}}}\,, \] where \(W_\theta\) is the Webster scalar curvature of the contact form \(\theta\) and \(d\mu = \theta \wedge (d\theta)^n\) is the natural volume form. In the present paper, the author uses the contact Yamabe flow, defined by \[ {{\partial \theta}\over{\partial t} }= (w - W)\theta, \] (where \(w\) is the average of the Webster scalar curvature) to solve this contact Yamabe problem on \(K\)-contact manifolds. The main result (Theorem 1.1) states that if \((M, \theta_0, J_0, g_0, \xi_0)\) is a \(K\)-contact manifold, then the contact Yamabe flow exists for all time and converges to a contact form \(\theta_\infty \in [\theta_0]\) with constant Webster scalar curvature. In particular, along the contact Yamabe flow, the conformal factor of \(\theta (t)\) is basic with respect to \((M, \theta_0, \xi_0)\). Moreover the set of solutions of the contact Yamabe problem on \((M, \theta_0, J_0, g_0)\) with \(\xi_0u = 0\) and fixed volume is compact.
scientific article; zbMATH DE number 5643480
scientific article