Free Boundary Willmore Immersions

• Free boundary Willmore immersions are surfaces in R³ that are critical for the Willmore energy under variations with fixed boundaries and free conormal directions.
• The Euler–Lagrange equations yield natural third-order boundary conditions balancing curvature derivatives with geometric constraints along the fixed boundary.
• Recent studies establish existence, stability, and precise energy bounds for these immersions, linking variational methods with applications in membrane bending and minimal surface theory.

A free boundary Willmore immersion is an oriented surface immersed in $\mathbb{R}^3$ that is critical for the Willmore energy under variations that constrain only its boundary locus—while leaving the orthogonal directions (conormal) along the boundary curve unconstrained except for possible geometric conditions such as orthogonality to a fixed surface. The Willmore energy, given by $W(\Sigma) = \int_\Sigma H^2\, dA$ for mean curvature $H$ and area form $dA$, arises naturally in the bending theory of membranes, the calculus of variations for curvature functionals, and the analysis of geometric evolution equations. In the free boundary context, surfaces are considered whose boundary is fixed in $\mathbb{R}^3$ (or constrained to a support submanifold), with criticality captured by natural third-order boundary conditions intimately related to the geometry of the support and the immersion.

1. Willmore Energy and Free Boundary Framework

Given a smooth, oriented, immersed surface $\Sigma\subset\mathbb{R}^3$ with nonempty boundary $\partial\Sigma \cong \Gamma$, the Willmore energy is

$W(\Sigma) = \int_\Sigma H^2\, dA$

with $H$ the scalar mean curvature (trace of the second fundamental form $II$ with respect to the induced metric $g$). The "free boundary" condition refers to the setup where $\Gamma$ is fixed as a set in $\mathbb{R}^3$, but the conormal field—the orthogonal direction to the surface along $\Gamma$—is left unconstrained (Navier-type), except possibly for geometric restrictions such as orthogonality to a support surface $S$. Typical instances include the case where $\Sigma$ meets a supporting surface $S$ orthogonally (i.e., $\langle\nu_f, N^S\circ f\rangle = 0$ along $\partial\Sigma$, where $N^S$ is the unit normal of $S$) or is required only to fix its boundary curve.

This setting admits classes of variations: either all variations that leave $f(\partial\Sigma)\subset S$ and possibly $\langle\nu_f, N^S\circ f\rangle=0$ fixed, or only minimal restrictions on the conormal, leading to different boundary conditions in the Euler–Lagrange equations. The paradigm recovers and generalizes classical problems from minimal surface and elastic membrane theory to critical points for fourth-order geometry functionals under natural geometric boundary constraints (Pozzetta, 2018, Dall'Acqua et al., 26 Jan 2026, Kuwert et al., 2019, Alessandroni et al., 2014).

2. Euler–Lagrange Equations and Natural Boundary Conditions

The interior criticality condition for a Willmore immersion is governed by the Willmore equation:

$$\Delta_g H + (|II|^2 - \tfrac12 H^2) H = 0 \quad \text{on}\ \Sigma\setminus\partial\Sigma$$

where $\Delta_g$ is the Laplace–Beltrami operator associated to $g$. For immersions into $\mathbb{R}^3$, $|II|^2$ denotes the squared Hilbert–Schmidt norm of $II$.

The free boundary setup introduces additional boundary conditions, derived from boundary terms in the first variation. For basic free boundary (Navier) problems where the conormal is unconstrained except the boundary is fixed, the natural boundary conditions are:

$$\partial_\nu H + \tfrac12 H\, II(\tau,\tau) = 0,\qquad II(\nu,\nu) = 0 \quad \text{along}\ \partial\Sigma,$$

where $\nu$ is the unit conormal to $\Sigma$ along $\partial\Sigma$, and $\tau$ the unit tangent to $\Gamma$. These conditions encode that the surface meets its fixed boundary with vanishing normal–normal curvature and a balance on the boundary mean curvature derivatives (Pozzetta, 2018).

In problems where the boundary is constrained to a support surface $S$ with unit normal $N^S$, the orthogonality condition $\langle\nu_f, N^S\circ f\rangle=0$ is imposed, and the corresponding third-order boundary condition is

$$\partial_\eta H + h^S(\nu_f, \nu_f) H = 0 \quad \text{on}\ \partial\Sigma,$$

where $\eta$ is the outward conormal of $(\Sigma, g)$, and $h^S$ the second fundamental form of $S$ (Dall'Acqua et al., 26 Jan 2026, Alessandroni et al., 2014). The presence of such higher-order conditions directly reflects the variational constraints and is central to the geometric analysis of stability and regularity.

3. Existence, Nonexistence, and Sharp Energy Bounds

A central problem is the minimization of Willmore energy among free boundary immersions with prescribed boundary $\Gamma$ and genus $g\geq1$. For a planar boundary curve $\Gamma$ (including the unit circle $S^1$), the following holds:

• For $\Gamma=S^1$ and $g\geq1$, no embedded minimizer for the Willmore energy exists among immersions with $\partial\Sigma=S^1$. The infimum is

$$\inf_{\substack{\Sigma:\ \partial\Sigma=S^1\ \text{genus}(\Sigma)=g}} W(\Sigma) = \beta_g - 4\pi,$$

where $$\beta_g = \min\{ W(S) : S\subset\mathbb{R}^3\ \text{closed, genus}\ g\}$$ is the classic Willmore minimum for closed genus-$g$ surfaces (Pozzetta, 2018). The nonexistence is a consequence of sharp lower bounds for surfaces with high boundary turning or multiplicity. The result also extends to asymptotically flat surfaces with straight-line boundary: again, no minimizer exists and the infimum is $\beta_g-4\pi$.

• For $g=1$ and certain classes of convex planar $\Gamma$, minimizers exist in the sense of integer rectifiable varifolds with fixed genus, fixed boundary $\Gamma$, bounded $L^2$ norm of the second fundamental form, and no prescribed conormal. The minimizing varifold is induced by a possibly branched immersion, smooth up to the boundary except at interior branch points. This establishes existence for special topologies and boundary conditions in the varifold category.

The key technical ingredient is the multiplicity lower bound: if a competitor surface $\Sigma\rightarrow \mathbb{R}^3$ with $\partial\Sigma=S^1$ either has a point of multiplicity at least two or insufficient boundary turning, then $W(\Sigma)\geq 4\pi$. Comparison with closed Willmore minimizers yields the precise infimum and shows that no smooth minimizer can exist except in highly restricted settings (Pozzetta, 2018).

4. Local Existence and Construction of Free Boundary Willmore Disks

For a smooth domain $\Omega\subset \mathbb{R}^3$ with boundary $S = \partial\Omega$, one can construct Willmore disks as critical points in the class of immersed disks meeting $S$ orthogonally (i.e., $\langle \nu, \eta\rangle = 0$ on $\partial D^2$ for the disk $D^2$). By prescribing small area and using a Lyapunov–Schmidt reduction and rescaling techniques, for each small $\lambda>0$ there exist at least two immersed disks $f_\lambda\colon D^2 \to \mathbb{R}^3$ meeting $S$ orthogonally, satisfying the Willmore equation with area constraint

$$\Delta H + |h^\circ|^2 H = \Lambda_\lambda H,\quad \langle \nu, \eta \rangle = 0,\quad \partial_\eta H + h^S(\nu, \nu) H = 0,$$

where $h^\circ$ is the trace-free second fundamental form. For small area, these solutions concentrate near distinct critical points of the boundary mean curvature function $H^S$, yielding at least two distinct Willmore disks for generic $S$ (Alessandroni et al., 2014).

The analytic framework involves splitting the linearized operator into dilation and translation kernels (dealt with via area and barycenter constraints), yielding precise energy expansions of the form

$$W(a,\lambda) = 2\pi + \pi\lambda^2 H^S(a) + O(\lambda^3).$$

5. Regularity, Reflection, and Boundary Structure

Boundary regularity theory for free boundary Willmore immersions leverages reflection methods to extend local solutions across the boundary and bootstrap regularity from $W^{2,2}$ to smooth. For conformal immersions $f$ meeting a plane $\Pi$ orthogonally (i.e., $f(\partial\Sigma)\subset\Pi$, $T\Sigma\perp\Pi$ on $\partial\Sigma$), reflected extensions are constructed so that the extended maps are weakly Willmore and thus regular up to the boundary. Analogous constructions hold when the boundary is constrained to a line and the mean curvature vanishes at the boundary. These techniques establish smoothness up to and across the free boundary in the fundamental cases (Kuwert et al., 2019).

6. Stability, Quantitative Rigidity, and Gradient Flow

Quantitative stability and convergence phenomena for free boundary Willmore immersions have been established through adapted Łojasiewicz–Simon inequalities for the Willmore energy on Banach manifolds of free boundary immersions. For a $C^4$-immersed surface $\bar{f}$ in a real analytic domain $S \subset \mathbb{R}^3$ that is a local minimizer of $W$, small $C^{4,\alpha}$ perturbations evolve under the free boundary Willmore flow:

$$\langle \partial_t f, \nu_f \rangle = -(\Delta_{g_f} H_f + |A^0_f|^2 H_f)$$

subject to the natural free boundary conditions. Solutions with initial data sufficiently close to $\bar{f}$ exist for all time and converge $C^\infty$ to a (possibly reparametrized) free boundary Willmore immersion with the same Willmore energy (Dall'Acqua et al., 26 Jan 2026).

Quantitative stability is reflected in the inequality

$$\inf_{f\in\mathcal{C},\, \psi\in \operatorname{Diff}^4(\Sigma)} \|f_0\circ\psi - f\|_{C^4} \leq C [W(f_0) - W(\bar{f})]^\gamma,$$

where $\mathcal{C}$ denotes the class of free boundary Willmore immersions. Local rigidity near free boundary minimal immersions is established: for any sufficiently small neighborhood in $W^{4,2}$, the only free boundary Willmore immersions are themselves minimal (i.e., $H\equiv0$).

These results are grounded in analytic and PDE–Fredholm techniques, parabolic regularity, and Banach manifold methods, representing a sharp extension of stability theory to higher-order geometric flows with nonlinear boundary conditions.

7. Broader Context and Notable Examples

Prototypical free boundary Willmore immersions include the stereographic projection of a Clifford torus in $S^3$ meeting the equatorial sphere along a circle, with one end removed and inverted to yield a planar-boundary torus of genus one. Existence of minimizers is highly sensitive to both boundary geometry and topological constraints: for disks and, in certain convex planar boundary cases for $g=1$, smooth minimizers exist; for higher genus or highly symmetric boundaries (e.g., $S^1$), minimizers fail to exist and the infimum is explicitly determined via comparison to closed Willmore minimizers.

The study of free boundary Willmore immersions connects variational problems in geometry, the structure of singularities in geometric flows, higher-order elliptic PDE with nonlinear boundary conditions, and advances questions regarding compactness, boundary regularity, and energy gap phenomena (Pozzetta, 2018, Dall'Acqua et al., 26 Jan 2026, Kuwert et al., 2019, Alessandroni et al., 2014).

Table: Key Boundary Conditions in Principal Free Boundary Willmore Settings

Setting	Boundary Condition	Reference
Free boundary, planar support (Navier)	$\partial_\nu H + \tfrac12 H\,II(\tau,\tau)=0$, $II(\nu,\nu)=0$ on $\partial\Sigma$	(Pozzetta, 2018)
Orthogonal to fixed surface $S$	$f(\partial\Sigma)\subset S$, $\langle\nu_f, N^S\circ f\rangle=0$; $\partial_\eta H + h^S(\nu_f, \nu_f)H=0$	(Dall'Acqua et al., 26 Jan 2026, Alessandroni et al., 2014)
Meets fixed plane orthogonally	$f(\partial\Sigma)\subset\Pi$, $T\Sigma \perp \Pi$; $\partial_n H=0$	(Kuwert et al., 2019)
Boundary mapped into line	$f(\partial\Sigma)\subset\ell$, $H=0$ on $\partial\Sigma$	(Kuwert et al., 2019)

These geometric and analytic structures underpin the landscape of free boundary Willmore immersions and their interaction with boundary value problems for critical curvature functionals.