Non-isometric Steklov-isospectral metrics on balls in dimensions 3, 4, and 5

The Steklov spectrum of a compact Riemannian manifold with boundary is the spectrum of its Dirichlet-to-Neumann operator on the boundary. Gordon, Herbrich, and Webb adapted the two classical isospectrality constructions (Sunada's method and the torus action method) to the Steklov setting and, as a corollary of Schueth's torus-action examples, obtained non-isometric Steklov-isospectral metrics on the ball $B^n$ for every $n\ge 6$ (continuous families for $n\ge 8$, pairs for $n=6,7$) — the first Steklov-isospectral manifolds of dimension greater than 2 with connected boundary. The open question arising from these results is whether such metrics exist at all on $B^n$ for $n=3,4,5$, where no construction technique is currently known to apply; a negative answer would be the first spectral-rigidity statement of this type for simply connected manifolds in low dimensions.

Background

For a compact smooth Riemannian manifold $(M,g)$ with boundary $\partial M$, the Steklov problem seeks functions $u$ with $\Delta u=0$ in the interior and $\partial_\nu u=\sigma u$ on the boundary, where $\nu$ is the outward unit normal. Its eigenvalues $\sigma$ are the eigenvalues of the Dirichlet-to-Neumann operator $\Lambda$, which maps a smooth function $v$ on $\partial M$ to the normal derivative of its harmonic extension with boundary value $v$. Two metrics on the same manifold are Steklov isospectral if their Dirichlet-to-Neumann operators have the same eigenvalues with multiplicities; they are strongly Steklov isospectral if, in addition, the spectra agree at every frequency $\alpha$ (i.e., for the operator sending $v$ to the normal derivative of the solution of $\Delta V=\alpha V$ with $V|_{\partial M}=v$, for all $\alpha$ outside the Dirichlet spectrum), which is equivalent to isospectrality of the Robin boundary value problem for every Robin parameter.

In dimension two the Steklov spectrum is invariant under conformal changes of the metric away from the boundary, so surface examples are considered trivial unless one rules out this gauge freedom. In dimensions three and higher no such triviality exists, and until 2018 no pair of non-isometric Steklov-isospectral manifolds of dimension greater than 2 with connected boundary was known. Two general construction techniques exist. Sunada's method (Sunada 1985) exploits almost conjugate subgroups $H_1,H_2$ of a finite group $G$ of isometries of a common covering: the quotients $H_1\backslash M$ and $H_2\backslash M$ are isospectral for essentially every natural elliptic operator, and the construction is constrained by the available finite group actions. The torus action method (Gordon–Wilson, Schueth 2001) uses an effective isometric torus action with $T$-equivariant maps that are volume preserving on the manifold and on the boundary to produce metrics that are isospectral and typically not even locally isometric; on balls it underlies the deepest examples (Schueth's isospectral metrics on five-dimensional spheres yield, by Gordon–Herbrich–Webb, pairs of strongly Steklov-isospectral metrics on $B^6$ and $B^7$, and continuous families exist on $B^n$ for $n\ge 8$).

Gordon, Herbrich, and Webb (2018, published 2021) proved that both methods, appropriately adapted (the torus action method needs an extra boundary-volume-preserving hypothesis, automatic when the principal torus orbits are dense in the boundary), yield manifolds that are strongly Steklov-isospectral, Robin-isospectral for all parameters, and isospectral for mixed boundary problems including the sloshing problem of fluid oscillation in a container. Among their examples are Steklov-isospectral metrics on balls $B^n$ for every $n\ge 6$, which they highlight as the first examples of dimension greater than 2 whose boundary is connected (a sphere). The construction techniques, however, produce nothing on balls of dimensions 3, 4, and 5: the torus action method requires enough room for the torus orbits (all known torus-action ball examples live in dimension at least 6), and Sunada quotients of a ball by a finite linear group have a cone singularity at the center, hence are orbifolds rather than smooth balls. Meanwhile no rigidity phenomenon is known that would forbid such examples: on a simply connected low-dimensional ball the standard heat invariants of the Dirichlet-to-Neumann operator (boundary volume, integrated mean-curvature-type terms) are far from determining the metric.

Problem Statement

For which $n\in\{3,4,5\}$ do there exist two smooth Riemannian metrics $g_1,g_2$ on the closed $n$-dimensional ball $B^n$ such that $g_1$ and $g_2$ are not isometric but the Dirichlet-to-Neumann operators of $(B^n,g_1)$ and $(B^n,g_2)$ are isospectral, i.e., the two balls have the same Steklov spectrum with multiplicities? A complete answer settles existence for each of $n=3,4,5$; a positive answer may further ask for continuous families of mutually non-isometric Steklov-isospectral metrics on $B^n$ — a stronger target that, by the state of the art, is open even on $B^6$ and $B^7$, where only single pairs are known — and for the metrics to be strongly Steklov-isospectral (isospectral at every frequency), matching the property of the known examples in dimensions at least 6 (Gordon–Herbrich–Webb 2018/2021; Schueth 2001).

References

1. Carolyn Gordon, Peter Herbrich, David Webb, Steklov and Robin isospectral manifolds, Journal of Spectral Theory 11 (2021), 39–61, DOI: 10.4171/jst/335, https://doi.org/10.4171/jst/335 (preprint arXiv:1808.10741, https://arxiv.org/abs/1808.10741)
2. Bruno Colbois, Alexandre Girouard, Carolyn Gordon, David Sher, Some recent developments on the Steklov eigenvalue problem, Revista Matemática Complutense 37 (2024), no. 1, 1–161, DOI: 10.1007/s13163-023-00480-3, https://doi.org/10.1007/s13163-023-00480-3 (preprint arXiv:2212.12528, https://arxiv.org/abs/2212.12528)
3. Dorothee Schueth, Isospectral Metrics on Five-Dimensional Spheres, Journal of Differential Geometry 58 (2001), no. 1, 87–111, DOI: 10.4310/jdg/1090348283, https://doi.org/10.4310/jdg/1090348283
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