Min-max minimal surfaces, horizons and electrostatic systems (Q6633906)

The index of a minimal surface, seen as a critical point of the area functional, is a non-negative integer that measures the number of distinct deformations that decrease the area to second order. The paper explores the role of minimal surfaces of index one in general relativity.\N\NLet \((M,g)\) be a Riemannian \(3\)-manifold, \(E\) a tangent vector field and \(V\) a positive smooth function on \(M\). We say \((M, g, V, E)\) is an electrostatic system if \((M, g)\) is oriented and the following equations are satisfied\N\begin{align*}\N\mathrm{Hess}_{g}V &=V(\mathrm{Ric}_{g}-\Lambda g +2E^{b}\otimes E^{b} - |E|^{2}g), \\\N\Delta_{g}V&=(|E|^{2}-\Lambda)V, \\\N\mathrm{div}_{g}E&=0, \\\N\mathrm{curl}_{g}(VE)&=0, \\\N\end{align*}\Nfor some constant \(\Lambda \in \mathbb{R}\). Moreover, the system is called complete, if \((M, g)\) is complete.\N\NThe authors prove the following results that link index-one minimal surfaces and min-max problems in complete electrostatic systems:\N\NTheorem. Consider a complete electrostatic system \((M,g,V,E)\) such that \(\Lambda+|E|^{2}>0\). Let \(\Omega_1, \Omega_2\) be connected maximal regions where \(V\neq 0\), such that \(\Omega_i\) is compact and let \(\Sigma=\partial\Omega_1 \cap \partial\Omega_2 \subset V^{-1}(0)\) be unstable. Suppose \(\partial\Omega_i \setminus \Sigma\) is either empty or strictly stable, for \(i = 1, 2\). Then \(\Sigma\) is the solution of a one-parameter min-max problem for the area functional in \((M, g)\) and has index equal to one.\N\NTheorem. Consider a complete electrostatic system \((M, g, V, E)\), such that \(\Lambda + |E|^2 > 0\). Let \(\Omega_1, \Omega_2\) be connected maximal regions where \(V\neq 0\), such that \(\Omega_i\) is compact and let \(\Sigma=\partial\Omega_1 \cap \partial\Omega_2 \subset V^{-1}(0)\) be unstable. Suppose that \(\partial(\Omega_1 \cup \Omega_s)\) is non empty and stable, and at least one of its components is degenerate. Then \(\Sigma\) has index one.