Minimal Lagrangian Connections on Compact Surfaces

6 The spherical case

The system of equations governing minimal Lagrangian connections are easy to analyse on the \(2\)-sphere \(S^2\):

Proposition 6.1

A connection on the tangent bundle of \(S^2\) is minimal Lagrangian if and only if it is the Levi-Civita connection of a metric of positive Gauss curvature.

Proof. Let \(\nabla\) be a minimal Lagrangian connection on \(TS^2\) with associated triple \((g,\beta,C)\). Since \(\beta\) is closed and \(H^1(S^2)=0\), the \(1\)-form \(\beta\) is exact and hence there exists a smooth real-valued function \(r\) on \(S^2\) such that \(\beta=\mathrm{d}r\). Hence we have \[\overline{\partial} C=\left(\mathrm{d}r-\mathrm{i}\star_g\mathrm{d}r\right)\otimes C.\] Observe that \(\mathrm{d}r-\mathrm{i}\star_g\mathrm{d}r=2\overline{\partial}r\), therefore, the cubic differential \(\mathrm{e}^{-2r}C\) is holomorphic. Since, by Riemann-Roch, there are no non-trivial cubic holomorphic differentials on the \(2\)-sphere, \(C\) must vanish identically. The connection form (5.5) of \(\nabla\) thus becomes \[\theta=\begin{pmatrix} -\mathrm{d}r & \star_g \mathrm{d}r-\varphi \\\varphi-\star_g \mathrm{d}r & -\mathrm{d}r\end{pmatrix},\] where \(\varphi\) denotes the Levi-Civita connection form of \(g\). We conclude that \(\nabla\) is a Weyl connection given by \[\nabla={}^g\nabla+g\otimes {}^g\nabla r-\mathrm{d}r \otimes \mathrm{Id}-\mathrm{Id}\otimes \mathrm{d}r,\] where \({}^g\nabla r\) denotes the gradient of \(r\) with respect to \(g\). Since the Levi-Civita connection of a Riemannian metric \(g\) transforms under conformal change as [2] \[{}^{\exp(2f)g}\nabla={}^g\nabla-g \otimes {}^g\nabla f+\mathrm{d}f \otimes \mathrm{Id}+\mathrm{Id}\otimes \mathrm{d}f,\] we obtain \(\nabla={}^{\exp(-2r)g}\nabla\), thus showing that \(\nabla\) is the Levi-Civita connection of a Riemannian metric. Moreover, since \(\mathrm{Ric}(\nabla)\) must be positive or negative definite, the Gauss curvature of the metric \(\mathrm{e}^{-2r}g\) cannot vanish and hence is positive by the Gauss–Bonnet theorem. Finally, Example 4.6 shows that conversely the Levi-Civita connection of a Riemannian metric of positive Gauss curvature defines a minimal Lagrangian connection, thus completing the proof.