Abstract:We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $\Sigma$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The genus of the surface $\Sigma$ is bounded from above by the Heegaard genus $h$ of $\mathcal{M}$.
Starting from a family of sweep-outs of $\mathcal{M}$ by surfaces of genus $h$, we apply a min-max construction for a family $\{E_{H,\sigma}\}_\sigma$ of perturbations of the energy involving the second fundamental form of the immersions to produce almost-critical points $u_k$ of $E_{H,\sigma}$. We then show, following ideas introduced by Rivière and developed by Pigati and Rivière, that the maps $u_k$ converge to a "CMC-parametrized varifold". This limiting object is then shown to be a smooth, branched immersion with the prescribed mean curvature $H$.
From: Filippo Gaia [view email]
[v1]
Wed, 18 Feb 2026 17:26:30 UTC (77 KB)
[v2]
Thu, 19 Feb 2026 18:56:54 UTC (76 KB)
[v3]
Wed, 17 Jun 2026 05:34:15 UTC (85 KB)
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