Existence of constant mean curvature surfaces with controlled topology in 3-manifolds

Theorem 1.1. Let (M 3 , g) be a smooth, closed, oriented Riemannian manifold with Heegaard genus h. For almost everywhere H > 0, there exists a closed Riemann surface Σ with genus g(Σ) ≤ h and a non-trivial branched H-constant-mean-curvature immersion u : Σ → M.

Here we say u is a branched H-CMC immersion if, in conformal coordinates, it satisfies ∆u + A u (∇u, ∇u) = Hu x × u y .

(1.1)

The image of such an immersion has constant mean curvature equal to H. H-CMC immersions are critical points to the following functional

where vol(f v ) roughly stands for the volume bounded by map v; see Remark 1.2 and Section 2 below.

A wide range of methods has been developed to prove the existence of constant mean curvature surfaces; see, for instance, [35,37,11,76,77,78,41,83,27,48,67,9,89,88,5,6,7]. We refer the reader to the survey by X. Zhou [87] for a comprehensive overview of the field. Despite this progress, the existence of H-CMC branched immersions with controlled genus in general three-manifolds has remained a challenging open problem. A significant advance in this direction was made by D. R. Cheng and X. Zhou [15], who proved that in any closed Riemannian three-manifold which is topologically a three-sphere (or more generally has non-trivial third homotopy group) and for almost every H > 0, there exists a nontrivial H-CMC branched immersed two-sphere. Their work, however, leaves open the case of general three-manifolds. Theorem 1.1 fills this gap. In particular, since the three-sphere has Heegaard genus zero, Theorem 1.1 provides an alternative proof to the result of D. R. Cheng and X. Zhou.

The result of D. R. Cheng and X. Zhou is based on a penalization of the Dirichlet energy of the form ε ´S2 |∆u| 2 . Using a min-max construction over sweep-outs of a spherical target manifold M by spheres, they obtain H-harmonic maps, that is, solutions of (1.1) which are not a priori required to be conformal. Since the two-sphere admits a unique conformal structure up to diffeomorphism, these maps are in fact H-CMC branched immersions.

For a general orientable, closed manifold M, there always exist a sweep-out of the form Σ × [0, 1] → M, where Σ is a Riemann surface. If Σ has positive genus, however, one cannot apply directly the methods of [15], as in general they would only produce H-harmonic maps.

To address this limitation, in the present work we adopt a different perturbation of the area functional, often referred to as a viscous penalization, developed by A. Pigati and T. Rivière [64,61,60]. More precisely, for σ > 0, we consider the family of functionals A σ (u) = Area(u) + σ 4 ˆ|II u | 4 dvol gu , for W 2,4 immersion u : Σ → M, where Σ is an arbitrary Riemann surface and II u denotes the second fundamental form of u(Σ).

Two key properties of these functionals are their invariance under reparameterizations of u, and the fact that-as observed by Langer [43] and Breuning [10]-for each fixed σ > 0, the conformal structures on Σ induced by maps lying in a sublevel set of A σ (u) remain in a compact subset of the moduli space of Σ. These properties are central to the work of Pigati and Rivière, and suggest that it may be possible to control the topology in our construction.

Pigati and Rivière showed that, given a sequence of critical (or almost critical) points {u k } k∈N ⊂ W 2,4 (Σ, M) for the functionals A σ k (in arbitrary codimension), with σ k → 0 and satisfying a natural entropy condition (Assumption (2) in Theorem 1.3), the associated varifolds converge to a parametrized stationary varifold. That is, the limit varifold admits a conformal parametrization for which the stationarity condition holds locally and is induced by a smooth branched minimal immersion from a Riemann surface Σ ′ to M, with g(Σ ′ ) ≤ g(Σ).

To prove Theorem 1.1, we employ similar ideas. Given a closed, connected 3-manifold M, smoothly embedded in R Q , there exists a family of sweep-outs f : Σ × [0, 1] → M, where Σ is a Riemann surface of genus h, equal to the Heegaard genus of M. Applying a min-max procedure to such a family of sweep-outs and to the perturbed H-CMC functional E H,σ defined below, for almost every H > 0 (and H = 0) we show that there exist a sequence σ k → 0 and a corresponding family of W 2,4 immersions u k such that, for every k ∈ N,

(1) u k is θ k -critical for the functional

Area(u k ) ≤ A.

Remark 1.2. Here α k is a sequence converging to zero, while the parameters θ k can be chosen to be arbitrarily small. In particular we will require that θ k ≤ σ 4 k ; we will fix the values of θ k in the proof of Theorem 4.5. The notion of θ k -almost criticality-a quantitative measure of proximity to a critical point-is introduced in Definition 2.4. Following [59], we work with almost critical points rather than exact critical points, thereby avoiding the need to establish Palai