First Conformal Steklov Eigenvalue

• The topic defines the first conformal Steklov eigenvalue as the supremum of the normalized first positive Steklov eigenvalue over all positive boundary densities in a fixed conformal class.
• It employs a weighted formulation and Rayleigh characterization, linking conformal covariance with metric deformation to optimize the boundary-scaled eigenvalue.
• Applications include rigidity results for annuli and Möbius bands, where the disk benchmark value of 2π plays a critical role in characterizing free boundary minimal surfaces.

The first conformal Steklov eigenvalue is the conformal-class supremum of the normalized first positive Steklov eigenvalue on a compact surface with boundary. For a compact Riemannian surface $(\Sigma,g)$ with nonempty boundary and a positive boundary density $\beta$, one considers the weighted Steklov problem

$$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$

the normalized quantity

$$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$

and then the conformal invariant

$$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$

By conformal covariance, this is equivalently the supremum of $\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$ over metrics $\tilde g$ in the conformal class $[g]$. The subject connects Steklov spectral geometry, moduli of bordered surfaces, and free boundary minimal surface theory (Petrides, 14 Aug 2025, Medvedev, 2020).

1. Definition and conformal meaning

The classical Steklov spectrum of a compact surface with boundary is defined by harmonic extension to the interior and a Robin-type spectral condition on the boundary. In the weighted formulation used in current conformal treatments, the boundary density $\beta$ is part of the datum, and the first positive eigenvalue is optimized after multiplying by the weighted boundary length $\int_{\partial\Sigma}\beta\,dL_g$. This is the precise invariant denoted $\beta$0 in the recent surface literature (Petrides, 14 Aug 2025).

A key point is that the weighted problem is equivalent to an unweighted problem after a conformal change of metric. If $\beta$1 is a smooth extension of $\beta$2 to $\beta$3, then

$\beta$4

Thus the optimization over boundary densities is exactly an optimization within a fixed conformal class. The associated topological supremum,

$\beta$5

is a different object: it ranges over all conformal classes rather than one fixed class (Petrides, 14 Aug 2025).

Conventions for indexing vary across the Steklov literature. In the surface papers centered on conformal optimization, $\beta$6 denotes the first positive eigenvalue. In some higher-dimensional papers the zero eigenvalue is indexed first, so the first positive Steklov eigenvalue appears as $\beta$7 instead (Cruz et al., 27 Jan 2026).

2. Variational structure and density formulation

The conformal invariant is fundamentally variational. In the density formulation developed for surfaces, the first weighted Steklov eigenvalue admits the Rayleigh characterization

$\beta$8

and the normalized quantity is

$\beta$9

The decisive conformal covariance is

$$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$0

and if $$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$1 extends $$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$2 to the interior then

$$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$3

On a surface, changing the boundary density is therefore equivalent to changing the metric conformally; this is the mechanism by which the “conformal” Steklov problem is encoded (Karpukhin, 2018).

This density viewpoint is also the natural compactness framework. The optimization problem can be enlarged to allow nonnegative $$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$4 boundary densities, and for uniformly bounded densities normalized in $$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$5, $$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$6-convergence implies convergence of the first Steklov eigenvalue. In Karpukhin’s formulation, this is the closest precise antecedent of the modern phrase “first conformal Steklov eigenvalue,” even though that paper is framed primarily as a global maximization problem over all metrics on a given topological surface rather than as a fixed-conformal-class invariant (Karpukhin, 2018).

3. The disk benchmark and rigidity on annuli and Möbius bands

The universal benchmark is the disk. The conformal Steklov invariant of the disk is

$$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$7

and equality is attained if and only if $$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$8 is Steklov-isometric to a Euclidean disk. Here “Steklov-isometric” means that a boundary-preserving diffeomorphism pulls back the comparison metric to $$\Delta u=0 \quad \text{in } \Sigma,\qquad \partial_\nu u=\sigma\,\beta\,u \quad \text{on } \partial\Sigma,$$9, which is the natural equivalence relation for the weighted conformal problem (Petrides, 14 Aug 2025).

A general comparison theorem implies

$$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$0

for every compact surface with boundary. This leads to the central conjectural picture: for every non-disk surface with boundary,

$$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$1

That strict inequality is proved for every conformal class of annulus and every conformal class of Möbius band. More precisely, if $$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$2 is an annulus or a Möbius band with a flat metric $$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$3, then there exists a smooth positive function $$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$4 on $$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$5 such that

$$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$6

Consequently,

$$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$7

for all such conformal classes (Petrides, 14 Aug 2025, Matthiesen et al., 2020).

Once the strict gap above $$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$8 is known, Petrides’ existence theorem yields an actual maximizer in the conformal class. For annuli and Möbius bands this means that there exists a smooth positive $$\bar{\sigma}_1(\Sigma,g,\beta)=\sigma_1(\Sigma,g,\beta)\int_{\partial\Sigma}\beta\,dL_g,$$9 with

$$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$0

and the extremal metric produces a free boundary harmonic map

$$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$1

by first eigenfunctions, with $$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$2 for annuli and $$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$3 for Möbius bands (Petrides, 14 Aug 2025).

4. Critical catenoid, critical Möbius band, and rigidity

The rigidity mechanism is geometric. If $$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$4 is a minimal free boundary immersion by first Steklov eigenfunctions, then Fraser–Schoen’s uniqueness theorem implies that an annulus must be homothetic to the critical catenoid and a Möbius band must be homothetic to the critical Möbius band. These are the only free boundary minimal models that can arise from first-eigenfunction extremals on those topologies (Matthiesen et al., 2020).

For annuli, one parametrizes conformal classes by

$$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$5

and considers

$$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$6

Fraser–Schoen showed that

$$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$7

At the same time, explicit thin-annulus asymptotics and the Matthiesen–Petrides gluing construction produce conformal classes near both ends of moduli space with value $$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$8. If there were some $$\sigma_1(\Sigma,[g])=\sup_{\beta\in \mathcal C^\infty_{>0}(\Sigma)}\bar{\sigma}_1(\Sigma,g,\beta).$$9 with $\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$0, compactness of maximizers above the threshold would force an interior local maximum $\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$1, hence an extremal metric and therefore a free boundary minimal immersion by first eigenfunctions. Fraser–Schoen rigidity would then force the conformal class to be that of the critical catenoid, contradicting the construction of another interior maximizer. The Möbius-band argument is parallel, with the critical Möbius band replacing the critical catenoid (Matthiesen et al., 2020).

This establishes a strong form of rigidity: the disk value $\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$2 occurs only as a degeneration limit for annuli and Möbius bands, never at an interior point of their moduli spaces. A later treatment states the same phenomenon in the appendix language of rigidity of the first conformal Steklov eigenvalue on annuli and Möbius bands (Petrides, 14 Aug 2025).

5. Degenerating conformal classes and the universal infimum

A complementary viewpoint studies the invariant as a function on moduli space. For a compact surface with boundary and a conformal class $\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$3, one writes

$\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$4

If $\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$5 is a degenerating sequence of conformal classes, Karpukhin established a precise limit formula for $\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$6 in terms of the boundary-bearing components of the compactified limit and copies of the disk arising from collapsing boundary components or collapsing boundary-crossing geodesics. Specializing that formula to $\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$7 gives a direct limit law for the first conformal Steklov eigenvalue: the limit is the maximum of the first conformal Steklov eigenvalues of the surviving boundary-bearing components and the disk value $\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$8 (Medvedev, 2020).

One consequence is the exact Friedlander–Nadirashvili-type invariant

$\sigma_1(\tilde g)L_{\tilde g}(\partial\Sigma)$9

for every compact surface with boundary. Thus $\tilde g$0 is a universal lower bound for the first conformal Steklov eigenvalue of any conformal class, and it is the sharp infimum over all conformal classes. For the cylinder and Möbius band, every degenerating sequence of conformal classes satisfies

$\tilde g$1

This clarifies the role of the disk benchmark: on non-disk topologies it is expected to be approached only through degeneration, not attained by a nondegenerate conformal class (Medvedev, 2020).

The degeneration theorem also explains why strict lower bounds above $\tilde g$2 are delicate. Fixed conformal classes can support values strictly larger than $\tilde g$3, but the moduli space contains sequences along which the conformal Steklov invariant collapses back to the disk threshold. This is the precise analogue, on bordered surfaces, of degeneration phenomena in conformal Laplace spectrum theory (Medvedev, 2020).

6. Relation to global Steklov optimization and adjacent problems

The first conformal Steklov eigenvalue should be distinguished from global topological maximization over all metrics. For an orientable surface $\tilde g$4 of genus $\tilde g$5 with $\tilde g$6 boundary components, Karpukhin proved that

$\tilde g$7

is achieved by a smooth metric for every $\tilde g$8. The proof uses exactly the density/conformal formulation described above, but the final variational problem ranges over all conformal classes rather than one fixed class. A corollary is that every compact orientable surface with boundary admits a free boundary branched minimal immersion into some Euclidean ball by first Steklov eigenfunctions (Karpukhin, 2018).

This broader optimization problem is adjacent to, but not identical with, the conformal-class invariant. In the fixed-conformal-class setting the main issues are conformal covariance, attainment inside one class, and degeneration in moduli space. In the global problem one must additionally control degeneration of conformal structures themselves. Karpukhin’s theorem resolves that topological maximization problem for orientable surfaces, extending the genus-zero existence theory of Fraser–Schoen (Karpukhin, 2018).

A further distinction concerns global domain optimization on closed surfaces. Through homogenization, it was shown that for every closed surface $\tilde g$9,

$[g]$0

and in the orientable genus-zero case

$[g]$1

These statements concern a different supremum, taken over perforated domains and metrics, and not the fixed-conformal-class invariant $[g]$2. They nevertheless indicate how closely Steklov extremal theory is intertwined with conformal Laplace eigenvalue theory (Girouard et al., 2020).

In this sense, the first conformal Steklov eigenvalue occupies a middle position between local conformal spectral geometry and fully global extremal geometry. Its defining feature is the optimization of the normalized first Steklov eigenvalue within a single conformal class; its characteristic benchmark is the disk value $[g]$3; its strongest current rigidity results concern annuli and Möbius bands; and its modern formulation is inseparable from the weighted boundary-density viewpoint that identifies conformal deformation with boundary measure deformation (Petrides, 14 Aug 2025, Medvedev, 2020).