Embedded Two-Spheres with Prescribed Mean Curvature

• The paper establishes existence and multiplicity results for embedded two-spheres satisfying a prescribed mean curvature in spaces like S³ and ℝ³.
• Variational approaches and min–max theory are employed to ensure smooth embeddings under specific pinching conditions on the curvature function.
• Key challenges include preventing branch points and extending the framework to higher codimension and other ambient geometries.

Embedded two-spheres with prescribed mean curvature are fundamental objects in differential geometry, characterized as smooth embeddings $\phi: S^2 \hookrightarrow M^3$ whose images $\Sigma = \phi(S^2)$ have scalar mean curvature $H_\Sigma$ determined pointwise by a function $h: M^3 \to \mathbb{R}$. This geometric constraint, which generalizes the classical constant mean curvature (CMC) case, leads to variational, topological, and analytic challenges. Recent advances provide rigorous existence, multiplicity, and classification results for such embeddings in various ambient spaces, notably the unit round sphere $S^3$ and Euclidean space $\mathbb{R}^3$.

1. Definitions and Prescribed Mean Curvature Equation

Given a three-dimensional Riemannian manifold $(M^3, g)$ (such as $S^3$ or $\mathbb{R}^3$), a two-sphere embedding $\phi: S^2 \hookrightarrow M^3$ is said to satisfy the prescribed mean curvature equation if

$H_\Sigma(x) = h(x), \quad x \in \Sigma,$

where $H_\Sigma$ is the mean curvature computed as the divergence of the unit normal along $\Sigma$, and $h: M^3 \to \mathbb{R}$ is a given smooth function. In special cases, $h$ may depend on geometric quantities such as the Gauss map, i.e., $h = H(\nu(p))$ for $p \in M^2$ and unit normal $\nu$.

Local analysis near a graphical point $z = u(x, y)$ (for $\Sigma$ expressed as a graph in $\mathbb{R}^3$) leads to a quasilinear elliptic PDE: $$\operatorname{div} \left(\frac{Du}{\sqrt{1+|Du|^2}}\right) = 2 H\left(\frac{(-Du, 1)}{\sqrt{1+|Du|^2}}\right).$$

2. Existence Theorems and Multiplicity in $S^3$

Recent results in the unit round three-sphere $(S^3, g_0)$ establish the existence and multiplicity of embedded two-spheres $\Sigma \subset S^3$ solving $H_\Sigma = h$ under explicit pinching conditions. The paper "Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature" (Mazurowski et al., 11 Nov 2025) introduces the pinching constant

$h_0 \approx 0.547, \quad \text{where %%%%24%%%%},$

and proves:

• If $|h(x)| < h_0$ for all $x \in S^3$—for positive or suitably sign-changing functions—there exist at least two geometrically distinct smoothly embedded two-spheres $\Sigma_1, \Sigma_2 \subset S^3$ each satisfying $H_\Sigma = h$.
• The existence proof utilizes a min–max theory for the functional

$$\mathcal{A}^h(\Sigma) = \operatorname{Area}(\Sigma) - \int_{\Omega} h\, dV,$$

where $\Omega\subset S^3$ is the region bounded by $\Sigma$.

• The nonexistence of "double-sheeted" blow-ups for varifolds under the pinching condition ensures that the min–max construction yields smooth embedded spheres.
• Extension to sign-changing $h$ holds for generic zero sets, provided the same pinching bound.

The min–max and interpolation arguments rely on the topology of the space of oriented embedded two-spheres $\mathcal{E} \simeq S^3$ (via the Smale conjecture). Disjointness of critical values in the sweepout parameter space and homological considerations guarantee multiplicity.

3. Variational and Analytic Frameworks

Variational approaches underpin existence proofs. For prescribed mean curvature in $\mathbb{R}^3$, one seeks critical points of the functional

$$E^\omega(u) = \frac{1}{2} \int_{S^2} |\nabla u|^2 + \int_{S^2} u^* \omega,$$

where $u: S^2 \to N \subset \mathbb{R}^k$, $\omega$ encodes the mean curvature information, and $N$ is a closed Riemannian manifold.

For constant or prescribed $H$, the Dirichlet energy

$D(u) = \frac{1}{2}\int_{S^2} |\nabla u|^2$

is perturbed by higher-order regularizations (e.g., biharmonic terms) to ensure compactness (Palais–Smale sequences), as in (Cheng et al., 2020).

Min–max theory is then applied to sweepouts of $S^2$ embeddings parametrized by topological data (homotopy classes, cubical complexes). Index estimates, volume bounds, and bubbling analysis ensure regularity and rule out energy loss in the "neck" regions.

4. Embeddedness, Branch Points, and Regularity

Embeddedness does not follow automatically from variational methods; branching and self-intersections can occur. The pinching condition $|h| < h_0$ in $S^3$ provides an a priori area and density estimate preventing the formation of singular varifolds or double-sheeted spheres (Mazurowski et al., 11 Nov 2025).

A general conclusion—substantiated by blow-up and compactness analysis—is:

• Under appropriate pinching and regularity, the min–max critical varifold is everywhere of density one, smooth, and embedded.
• In higher codimension or absence of pinching, critical points are, at best, branched immersions; removing branch points is a deep open problem.

For spheres in ambient manifolds $N$ of higher dimension, (Gao et al., 16 Jul 2024) demonstrates existence of branched immersed 2-spheres with prescribed mean curvature for generic choices of the mean curvature vector, and for all possible choices under additional Ricci or isotropic curvature conditions.

5. Classification, Uniqueness, and Symmetry

In $\mathbb{R}^3$, the classification of embedded H-spheres is nearly complete under symmetry hypotheses. If $H \in C^2(S^2)$ is positive and invariant under isometries without fixed points (e.g., $H(x) = H(-x)$), there exists a unique strictly convex embedded H-sphere $S_H$ (Bueno et al., 2018). The uniqueness for genus-zero immersed H-surfaces (Hopf-type theorem) states: $$\text{Any immersed H-surface of genus zero is a translate of } S_H.$$

Alexandrov reflection and Codazzi-pair holomorphic differentials enforce symmetry and uniqueness of the embedded model in the presence of additional reflection invariance. A priori curvature and diameter bounds further constrain the geometry in terms of $H$ and positivity.

Recent work in $S^3$ ensures embeddedness and multiplicity only up to the respective pinching thresholds and topological arguments as above (Mazurowski et al., 11 Nov 2025).

6. Generalizations and Open Problems

The techniques and results for prescribed mean curvature spheres extend, with modifications, to settings such as hyperbolic space $\mathbb{H}^3$ (Cora et al., 2020) and to arbitrary ambient Riemannian manifolds of higher dimension and codimension (Gao et al., 16 Jul 2024). Key difficulties persist:

• For generic prescribed mean curvature function $h$, existence of embedded rather than merely branched immersed spheres is generally not assured without pinching estimates or symmetry.
• The removal of branch points for higher-index solutions remains open except in strong symmetry or low-index scenarios.
• The homotopy problem concerning representatives within prescribed mean curvature vector fields is partially resolved only under specific hypotheses.

A plausible implication is that the interplay of topological, variational, and geometric analysis will continue to produce further existence and uniqueness results for embedded two-spheres with prescribed mean curvature, but removal of branching in general remains elusive.

7. Key Formulas and Summary Table

Key Equations

Notation/Functional	Definition/Statement	Context
$H_\Sigma(p) = h(p)$	Prescribed scalar mean curvature at $p\in\Sigma$	$S^3$, $\mathbb{R}^3$
$\mathcal{A}^h(\Sigma)$	$\operatorname{Area}(\Sigma) - \int_\Omega h\, dV$	Min–max existence
$h_0$	$\pi h_0^3 + 2h_0^2 + 4\pi h_0 - 8 = 0$	Pinching threshold
Dirichlet energy $D(u)$	$\frac{1}{2}\int_{S^2}|\nabla u|^2$	Variational analysis
Sacks–Uhlenbeck $E_\alpha$	$\frac{1}{2}\int_{S^2}(1+|\nabla u|^2)^\alpha dV$	Regularization

Together, these results establish existence and in some cases multiplicity or classification for embedded two-spheres with prescribed mean curvature in $S^3$, $\mathbb{R}^3$, and more general ambient spaces, conditional upon geometric, analytic, and topological constraints.