Stationary Curves of the Length Functional

• Stationary curves are defined as critical points of length functionals in settings like Riemannian, sub-Riemannian, and symplectic manifolds, generalizing classical geodesics.
• Their analysis employs Euler–Lagrange equations, Morse theory, and calibration methods to assess stability, regularity, and variational properties.
• Applications span geodesic computations, interface network modeling, and image registration, highlighting practical impacts in differential geometry and computational anatomy.

A stationary curve of the length functional is a curve which is a critical point (in the sense of first variation) of an associated length or energy functional, typically in the context of a geometric space with additional structure—such as a Riemannian or symplectic manifold, a metric space of interfaces, or a graded manifold endowed with substructures. Stationary curves naturally generalize classical geodesics, the locally length-minimizing curves of a metric space, but their precise regularity and analytic properties depend strongly on the functional’s structure, the manifold’s geometry, and the admissible class of variations. The study of stationary curves, encompassing weak geodesics, Ustilovsky geodesics, and curves of fixed degree, forms a core component of modern differential geometry, calculus of variations, and metric geometry, with significant connections to Morse theory, Floer theory, the theory of calibrated geometries, and computational applications such as image registration.

1. The Length Functional and Admissible Curves

Let $(M,g)$ be a smooth, finite-dimensional Riemannian manifold. For a continuous curve $\gamma: [a,b] \to M$ satisfying suitable regularity—piecewise $C^1$ with $g(\dot{\gamma},\dot{\gamma})>0$ almost everywhere—the (Riemannian) length functional is defined by

$$L[\gamma] = \int_a^b \sqrt{g_{ij}(\gamma(t))\,\dot{\gamma}^i(t)\,\dot{\gamma}^j(t)}\,dt$$

where $x^i(t)$ are local coordinates. The domain of the length functional is the set of (absolutely) continuous curves for which the integrand is well-defined almost everywhere (Starke, 9 Jan 2026).

In sub-Riemannian and graded manifold contexts, as in the theory for fixed-degree curves, the length functional is adapted to account for horizontal directions and a grading structure—see

$L_d(\gamma;[a,b]) = \int_a^b \theta_d(t)dt$

where $\theta_d$ encodes the contributions of the degree-$d$ components along the adapted frame to the tangent vector (Citti et al., 2019).

For geometric variational problems on networks or partitions, the length functional may measure the total interface length: $$E[\Omega] = \sum_{1\leq i<j\leq P} \sigma_{i,j}\,\mathcal{H}^1(S_{i,j})$$ where $S_{i,j}$ is the interface between distinct phases, and $\sigma_{i,j}$ encodes the surface tension (Fischer et al., 2022).

In symplectic geometry, the (positive) Hofer length functional on contractible loops in the Hamiltonian group $\Omega\Ham(M,\omega)$ is defined by

$L^+(\gamma) = \int_0^1 \max_{x\in M} H_t\, dt$

where $H_t$ is the normalized generating Hamiltonian for the path $\gamma$ (Savelyev, 2013).

2. Variational Characterization and Euler–Lagrange Equations

Stationary curves are those for which the first variation of the length functional vanishes for all admissible variations vanishing at endpoints. For the classical Riemannian case, the variation yields the Euler–Lagrange equations: $$\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}^i} \right) - \frac{\partial L}{\partial x^i} = 0$$ Explicitly, under arclength parameterization ($g_{ij}\dot{x}^i\dot{x}^j=1$), this reduces to the geodesic ODE: $$\ddot{x}^k + \Gamma^k_{ij}(x)\dot{x}^i\dot{x}^j = 0$$ or invariantly,

$\nabla_{\dot{\gamma}}\dot{\gamma} = 0$

characterizing geodesics as stationary curves (Starke, 9 Jan 2026). Weak geodesics require only piecewise $C^1$ regularity, and satisfy the ODE almost everywhere or in the distributional sense.

On metric spaces of interfaces or partitions, the first variation takes the form: $$| \nabla E |(\chi) = \max\left\{0, \limsup_{d_H(\chi,\tilde{\chi})\to 0} \frac{E[\chi] - E[\tilde{\chi}]}{d_H(\chi,\tilde{\chi})} \right\}$$ A partition is stationary if and only if this metric slope vanishes. The corresponding Euler–Lagrange equation involves vector-valued surface divergence on the network (Fischer et al., 2022).

In symplectic and contact geometry, the first variation of the Hofer length functional leads to Ustilovsky geodesics: loops generated by time-dependent Hamiltonians having a unique nondegenerate maximum at each time. These satisfy a variational condition

$\frac{d}{ds}\Big|_{s=0}L^+(\gamma_s) = 0$

forcing the generator to be Morse at $x_{\max}(t)$ for every $t$ (Savelyev, 2013).

For length functionals of fixed-degree curves in graded manifolds, admissible variations are constrained so the variational vector field $V$ satisfies a degree-preservation ODE. The stationarity leads to an Euler–Lagrange-type system, generalizing geodesic equations, dependent on the surjectivity of a holonomy map and the associated structure constants (Citti et al., 2019).

3. Regularity, Weak Geodesics, and Generalizations

Stationary curves can exhibit weak regularity depending on the functional and admissible class. In the Riemannian case, classical geodesics are $C^2$ solutions; weak geodesics may be only piecewise-$C^1$, but are still stationary for all compactly supported variations. On interfaces, stationary points (for the planar length functional under appropriate tensions and partition regularity) are exactly finite Steiner networks: unions of straight segments meeting at triple junctions with prescribed angles (the Herring condition) (Fischer et al., 2022). The structure theorem excludes higher-order junctions and non-straight interfaces for stationary points.

In infinite-dimensional manifolds, such as diffeomorphism groups modeling image matching, stationary curves of the energy or length functional generalize to solutions of infinite-dimensional Euler–Lagrange systems (e.g., EPDiff equations). Numerically, these are relevant for shooting methods in registration problems (Starke, 9 Jan 2026).

Further generalizations are found in sub-Riemannian and graded settings. For curves of fixed degree, the ability to form arbitrary admissible variations depends crucially on the surjectivity of the holonomy map along the curve—a condition that distinguishes regular from singular stationary curves (Citti et al., 2019).

4. Morse Theory, Index, and Stability

The Morse index of a stationary curve is the maximal dimension of a subspace of variations on which the second variation of the length is negative definite. In Riemannian geometry, this index measures the instability and multiplicity of conjugate points.

For the Hofer length functional, the Morse index at an Ustilovsky geodesic is computed via Floer theory: $$\text{Index}(\gamma) = |\mathrm{CZ}(o_{\max})-\mathrm{CZ}([M])|$$ where $\mathrm{CZ}$ is the Conley–Zehnder index, $o_{\max}$ is the constant orbit at the maximizer, and $[M]$ is the fundamental class under the PSS isomorphism (Savelyev, 2013). The index formula provides strict algebraic constraints: in monotone symplectic manifolds, robust Ustilovsky geodesics (with nontrivial Floer homology) give rise to nontrivial Morse cycles, with their existence or nonexistence governing the topology of the Hamiltonian group and minimizers of the Hofer length. Stable (index-zero) robust Ustilovsky geodesics are globally length-minimizing in their homotopy class.

In planar interface networks, minimality and stability are addressed using localized paired calibrations: vector field constructions in a neighborhood of the stationary network yield relative energy identities ensuring local minimality for regular flat partitions (Fischer et al., 2022).

5. Stationary Networks and Interface Geometry

For networks or partitions of domains in $\mathbb{R}^2$, stationary points of the length (or perimeter) functional are precisely regular flat partitions. These are characterized by networks of straight segments meeting in triple junctions, with the Herring force-balance condition: $$\sigma_{i,j} n_{i,j}(p) + \sigma_{j,k} n_{j,k}(p) + \sigma_{k,i} n_{k,i}(p) = 0$$ at every junction $p$ (Fischer et al., 2022). For equal surface tension ($\sigma_{i,j}=1$), this enforces 120° junction angles. Monotonicity arguments and competitor constructions rule out higher-order junctions; the local minimality of these configurations is established via localized versions of the Lawlor–Morgan calibration argument.

The classification result states that all stationary and locally minimal configurations in the planar length functional, with suitable regularity, are finite unions of straight edges meeting three at a time at 120°, with boundary edges meeting the domain orthogonally as required by boundary conditions.

6. Examples and Applications

Representative cases include:

• Euclidean space: All straight lines are stationary curves, and achieve strict minimality for the length functional (Starke, 9 Jan 2026).
• Spheres ($S^2$): Great circles are stationary curves; their geometric characterization arises from both the geodesic equation and variational properties.
• Symplectic geometry: Rotational Hamiltonian flows on $S^2$ yield robust Ustilovsky geodesics of every even index, constructing nontrivial classes in the topology of Hamiltonian path spaces (Savelyev, 2013).
• Heisenberg group and graded structures: Stationary curves of the degree-2 length functional coincide with vertical lines (for certain energy gradings) (Citti et al., 2019).
• Interface networks in planar domains: Minimal partitions for multiclass perimeter functionals correspond to finite Steiner trees, relevant in models of polycrystalline structure and grain boundary networks (Fischer et al., 2022).
• Computational anatomy: Geodesics in the infinite-dimensional diffeomorphism group, understood as stationary curves of a right-invariant metric, underlie LDDMM algorithms for high-dimensional image registration (Starke, 9 Jan 2026).

7. Integrability, Holonomy, and Further Structure

The existence of admissible variations and the validity of first variation formulas depend on integrability and holonomy properties:

• In Riemannian geometry, classic theorems ensure that all endpoint-fixed variations exist for smooth curves.
• For fixed-degree curves on graded manifolds, the surjectivity of the holonomy map

$H^{a,b}_\gamma: G(\cdot) \mapsto F(b)$

provides the crucial criterion for regularity, integrability of variations, and applicability of Euler–Lagrange equations (Citti et al., 2019). Sufficient conditions for regularity include pointwise full row rank of the ODE coefficient matrices associated to the structure.

• In contact and sub-Riemannian settings, this machinery yields generalized geodesic equations and helps classify "singular" (non-regular) extremals.

The interplay between analytic stationarity, topological invariants (via Morse and Floer theory), calibration techniques, and geometric constraint problems demonstrates the depth and interconnectedness of stationary curve theory across differential and metric geometry, variational problems, symplectic topology, and applications to image and interface modeling.