Łojasiewicz inequalities near simple bubble trees (Q6619993)

Consider the \(H\)-functional for maps \(u: \Sigma \to \mathbb{R}^3\),\N\[\NE(u) := \frac{1}{2} \int_{\Sigma} |\nabla u|^2 dV_g - 2 V(u)\N\]\Nwhere \(\Sigma\) is a closed surface and \(V(u)\) is the enclosed volume the map \(u\). Critical points of this energy satisfy the \(H \equiv 1\)-surface equation\N\[\N-\Delta_x u = 2 u_{x_1} \wedge u_{x_2}\N\]\Nand \textit{conformal} solutions to the equation are (branched) immersed constant mean curvature surfaces in \(\mathbb{R}^3\).\N\NThe authors obtain Łojasiewicz-type inequalities for this energy functional under suitable assumptions on \(\Sigma\) (positive genus, Gauss curvature \(-1\) or \(0\)): If \(u\) is \(\dot{H}^1\)-close to an ``adapted'' bubble, then the distance of \(u\) to the bubble tree is controlled by \(\|dE(u)\|_{L^2}\), and so is, in a suitable way, the distance of the energy \(E(u)\) to the bubble energy.\N\NThe difficulty of this result lies in handling the change of topology of the bubble and the ``almost'' critical point \(u\) defined on \(\Sigma\).