A holomorphic map HH between real-analytic CR submanifolds MM and M′M' is locally rigid if every map sufficiently close to HH is obtained from HH by composing with automorphisms of MM and M′M'. Building on the 2017 sufficient criteria of della Sala, Lamel, and Reiter, their 2020 sequel characterized local rigidity for finitely nondegenerate maps by the triviality of all higher-order infinitesimal deformations, a proof that rests on the jet parametrization of nondegenerate maps by finite-dimensional analytic sets. The open problem is to extend this theory to maps that fail finite nondegeneracy: give sufficient, ideally necessary and sufficient, infinitesimal conditions for a finitely degenerate map to be locally rigid, and describe the local structure of the quotient of the mapping space by the automorphism-group action near such degenerate maps, where jet parametrization is unavailable and where even the model case of sphere maps carries open classification questions.

Background

A CR (Cauchy–Riemann) submanifold is a real submanifold M⊂CNM\subset\mathbb{C}^N whose complex tangent spaces TpM∩iTpMT_pM\cap iT_pM all have the same dimension; a real-analytic CR submanifold is called generic if it is locally defined by the smallest possible number of real-analytic equations, that is, its real codimension matches the number of independent complex-linear conditions it satisfies. The basic examples are real hypersurfaces, and among these the strictly pseudoconvex ones: hypersurfaces whose Levi form, the fundamental curvature invariant obtained by restricting the complex Hessian of a defining function to the complex tangent directions, is definite. The unit sphere S2n−1⊂Cn\mathbb{S}^{2n-1}\subset\mathbb{C}^n is the flat model of a strictly pseudoconvex hypersurface, and a hypersurface is called spherical if it is locally biholomorphic to a sphere.

For a CR submanifold MM, Aut(M)\mathrm{Aut}(M) denotes its group of CR automorphisms, realized as germs of biholomorphic maps of the ambient complex space carrying MM into itself; the corresponding infinitesimal automorphisms are the holomorphic vector fields whose real parts are tangent to MM (written hol0(M)\mathfrak{hol}_0(M) for germs fixing a base point). Given CR submanifolds M⊂CNM\subset\mathbb{C}^N and M′⊂CN′M'\subset\mathbb{C}^{N'}, one studies the space H(M,M′)\mathcal{H}(M,M') of germs of holomorphic maps HH with H(M)⊂M′H(M)\subset M', endowed with its natural inductive-limit topology. The product G=Aut(M)×Aut(M′)G=\mathrm{Aut}(M)\times\mathrm{Aut}(M') acts continuously on H(M,M′)\mathcal{H}(M,M') by H↦σ′∘H∘σ−1H\mapsto\sigma'\circ H\circ\sigma^{-1}. A map HH is locally rigid if its projection to the quotient H(M,M′)/G\mathcal{H}(M,M')/G is an isolated point, i.e. every map close enough to HH is equivalent to HH under this action.

The linearization of the rigidity question is the space hol(H)\mathfrak{hol}(H) of infinitesimal deformations of HH: holomorphic vector fields VV along HH whose real parts are tangent to M′M' along H(M)H(M). The subspace of trivial infinitesimal deformations, arising by differentiating one-parameter families of automorphisms of MM and M′M' composed with HH, is identified with the Lie algebra of the orbit G⋅HG\cdot H; nontrivial infinitesimal deformations are the infinitesimal obstruction to rigidity.

The hypothesis powering the known theory is finite nondegeneracy (Lamel, 2001): HH is finitely nondegenerate at the base point if the derivatives of HH obtained by applying products of the CR tangent vector fields of MM up to some finite order k0k_0 span the target space CN′\mathbb{C}^{N'}; the classes Fk0\mathcal{F}_{k_0} of such maps are open and invariant under GG. For such maps one has a jet parametrization: every k0k_0-nondegenerate map is determined by its finite jet (its derivatives up to a fixed finite order), and the set of jets that actually occur is a real-analytic subset of a finite-dimensional jet space, which reduces questions about the infinite-dimensional space H(M,M′)\mathcal{H}(M,M') and the GG-action to finite-dimensional real-analytic geometry.

In [1], della Sala, Lamel, and Reiter proved two sufficient criteria: (i) if MM is minimal (not locally contained in a proper CR submanifold of the same CR dimension) and generic real-analytic, M′M' generic real-analytic, and HH is finitely nondegenerate with dim⁡Rhol0(H)=0\dim_{\mathbb{R}}\mathfrak{hol}_0(H)=0, then HH is isolated in H(M,M′)\mathcal{H}(M,M'), hence locally rigid; and (ii) if MM and M′M' are strictly pseudoconvex real-analytic hypersurfaces, at least one of them non-spherical, and HH is 2-nondegenerate with dim⁡Rhol0(H)=dim⁡Rhol0(M′)\dim_{\mathbb{R}}\mathfrak{hol}_0(H)=\dim_{\mathbb{R}}\mathfrak{hol}_0(M'), then HH is locally rigid, the non-sphericity being used to obtain compactness of an isotropy factor and properness of the action. Their companion paper on hypersurface embeddings (Trans. Amer. Math. Soc., 2017) also showed by example that infinitesimal rigidity is strictly stronger than local rigidity. The sequel (Arkiv för Matematik, 2020) completed the nondegenerate theory: a sufficient condition hol(H)=aut(H)\mathfrak{hol}(H)=\mathfrak{aut}(H) valid for any class with the jet-parametrization property between real-analytic submanifolds, with no pseudoconvexity, sphericity, or properness assumptions, and a full characterization, namely that HH is locally rigid if and only if all its higher-order infinitesimal deformations, i.e. curves of maps tangent to H(M,M′)\mathcal{H}(M,M') to order kk for every kk, are trivial.

All of these results pass through jet parametrization and therefore exclude finitely degenerate maps, those lying in no class Fk\mathcal{F}_k, for which the derivatives along the CR tangent fields of MM never span CN′\mathbb{C}^{N'}. Degenerate maps exist and behave differently: for example, the linear embedding S3→S7\mathbb{S}^3\to\mathbb{S}^7 is degenerate, and it lies in the D'Angelo family (z,w)↦(z,sin⁡θ w,cos⁡θ zw,cos⁡θ w2)(z,w)\mapsto(z,\sin\theta\,w,\cos\theta\,zw,\cos\theta\,w^2) of pairwise inequivalent sphere maps, so degeneracy and non-rigidity are intertwined. The only general results in the degenerate direction concern spheres: the tangentially (2,N−3)(2,N-3)-degenerate smooth CR maps S3→S2N−1\mathbb{S}^3\to\mathbb{S}^{2N-1} (in the sphere-map notation used here and in every later sphere-map statement, the target is written S2N−1⊂CN\mathbb{S}^{2N-1}\subset\mathbb{C}^N, so the symbol NN denotes the target dimension, distinct from the source dimension NN of the general submanifold setup M⊂CNM\subset\mathbb{C}^N), i.e. those whose degeneracy is detected along the complex tangent plane and defined precisely in [6, Def. 3], were classified into explicit normal-form families, each locally rigid within the class of tangentially degenerate maps (J. Geom. Anal., 2024), while a complete list of sphere maps S3→S7\mathbb{S}^3\to\mathbb{S}^7 and the conjectured degree of ss-degenerate sphere maps, where a map is ss-degenerate when s=N−max⁡kdim⁡Ek′(p)s=N-\max_k\dim E_k'(p) is the codimension in CN\mathbb{C}^N of the span Ek′(p)E_k'(p) of its derivatives up to order kk, remain open. Nothing is known about infinitesimal rigidity criteria for degenerate maps between general CR manifolds, nor about the local structure of the quotient H(M,M′)/G\mathcal{H}(M,M')/G near a degenerate map.

Problem Statement

Let M⊂CNM\subset\mathbb{C}^N and M′⊂CN′M'\subset\mathbb{C}^{N'} be germs, at fixed base points, of generic real-analytic CR submanifolds, let H(M,M′)\mathcal{H}(M,M') be the space of germs of holomorphic maps sending MM into M′M', and let G=Aut(M)×Aut(M′)G=\mathrm{Aut}(M)\times\mathrm{Aut}(M') act by H↦σ′∘H∘σ−1H\mapsto\sigma'\circ H\circ\sigma^{-1}. Recall that HH is locally rigid if its projection to the quotient H(M,M′)/G\mathcal{H}(M,M')/G is isolated, and that HH is finitely degenerate if it belongs to none of the classes Fk\mathcal{F}_k of finitely nondegenerate maps, i.e. for every finite order kk the derivatives of HH along the CR tangent vector fields of MM up to order kk span a proper subspace of CN′\mathbb{C}^{N'}. For finitely nondegenerate maps, local rigidity is characterized by the triviality of all higher-order infinitesimal deformations, but the proof depends on jet parametrization and does not extend to finitely degenerate maps. The open problem, in two linked parts sharing the objective of a rigidity and moduli theory for CR maps outside the nondegenerate class:

(1) Rigidity criterion (mandatory). Establish sufficient conditions for local rigidity of a finitely degenerate holomorphic map H ⁣:M→M′H\colon M\to M', expressed through computable infinitesimal data of HH (its spaces of infinitesimal deformations of finite order kk, as kk ranges over the positive integers); necessary and sufficient conditions are the stronger, additional target. A sufficient criterion valid for a class of pairs (M,M′)(M,M') strictly larger than the tangentially (2,N−3)(2,N-3)-degenerate sphere maps S3→S2N−1\mathbb{S}^3\to\mathbb{S}^{2N-1} already classified counts as partial progress, and this part is resolved by a sufficient criterion valid for general generic real-analytic CR submanifolds, the case a complete answer covers, including where both source and target are strictly pseudoconvex hypersurfaces, spherical or not.

(2) Local structure of the moduli space (mandatory). Describe the local structure of the quotient H(M,M′)/G\mathcal{H}(M,M')/G near a finitely degenerate map: determine whether the degenerate maps form a finite-dimensional real-analytic or semi-analytic stratum of the quotient or of a suitable jet or parameter space, and determine its dimension, singularities, and orbit stratification near HH, the analogue, outside the nondegenerate class, of the finite-dimensional analytic description that jet parametrization provides inside it.

Concrete model instances of part (1), stated as open in the successor literature, are the complete classification of sphere maps S3→S7\mathbb{S}^3\to\mathbb{S}^7 and the conjecture that an ss-degenerate sphere map with linearly independent components into CN\mathbb{C}^N has degree N−s−1N-s-1; a solution of the general problem would in particular decide the rigidity status of such maps.

References

  1. Giuseppe Della Sala, Bernhard Lamel, Michael Reiter, "Infinitesimal and local rigidity of mappings of CR manifolds", Contemporary Mathematics 681 (2017), 71–83, DOI 10.1090/conm/681/13658, arXiv:1710.03963, https://arxiv.org/abs/1710.03963
  2. Giuseppe della Sala, Bernhard Lamel, Michael Reiter, "Local and infinitesimal rigidity of hypersurface embeddings", Transactions of the American Mathematical Society 369 (2017), no. 11, 7829–7860, DOI 10.1090/tran/6885, https://doi.org/10.1090/tran/6885
  3. Giuseppe della Sala, Bernhard Lamel, Michael Reiter, "Sufficient and necessary conditions for local rigidity of CR mappings and higher order infinitesimal deformations", Arkiv för Matematik 58 (2020), no. 2, 213–242, DOI 10.4310/arkiv.2020.v58.n2.a1, arXiv:1906.02584, https://arxiv.org/abs/1906.02584
  4. Michael Reiter, "The Reflection Map and Infinitesimal Deformations of Sphere Mappings", Journal of Geometric Analysis 31 (2021), no. 1, 772–800, DOI 10.1007/s12220-019-00298-w, arXiv:1906.02587, https://arxiv.org/abs/1906.02587
  5. Giuseppe Della Sala, Bernhard Lamel, Michael Reiter, "Deformations of CR maps and applications", Asian Journal of Mathematics 25 (2021), no. 5, 665–682, DOI 10.4310/AJM.2021.v25.n5.a3, arXiv:1906.02586, https://arxiv.org/abs/1906.02586
  6. Giuseppe della Sala, Bernhard Lamel, Michael Reiter, Duong Ngoc Son, "On Highly Degenerate CR Maps of Spheres", Journal of Geometric Analysis 34 (2024), no. 3, article 80, DOI 10.1007/s12220-023-01495-4, arXiv:2210.13525, https://arxiv.org/abs/2210.13525
  7. Michael Reiter, Duong Ngoc Son, "On CR maps from the sphere into the tube over the future light cone", Advances in Mathematics 410 (2022), article 108743, DOI 10.1016/j.aim.2022.108743, https://doi.org/10.1016/j.aim.2022.108743
  8. Nguyen Gia Hien, Michael Reiter, Duong Ngoc Son, "The classification of CR maps from hyperquadrics into tubes over null cones of symmetric forms", arXiv:2510.13252 (2025), https://arxiv.org/abs/2510.13252
  9. Bernhard Lamel, "Holomorphic maps of real submanifolds in complex spaces of different dimensions", Pacific Journal of Mathematics 201 (2001), no. 2, 357–387, DOI 10.2140/pjm.2001.201.357, https://doi.org/10.2140/pjm.2001.201.357