Higher-Dimensional Smale Conjecture for 2-Convex Spheres

Definitions and mathematical statement

The supplied record calls the following assertion the higher-dimensional Smale conjecture: "the moduli space of 2-convex embedded $n$-spheres in ${\mathbb{R}}^{n+1}$ is contractible for every $n$." It does not specify the dimension range, the equivalence relation defining the moduli space, or its topology. The formulation below makes one conventional interpretation explicit; these choices require confirmation against the source.

For an integer $n\geq 2$, let $S^n=\{x\in{\mathbb{R}}^{n+1}:\lVert x\rVert=1\}$. For a smooth embedding $f:S^n\hookrightarrow{\mathbb{R}}^{n+1}$, put $\Sigma_f=f(S^n)$. Let $\nu$ be the outward unit normal of the bounded region enclosed by $\Sigma_f$, and define the principal curvatures $\lambda_1(p)\leq\cdots\leq\lambda_n(p)$ to be the eigenvalues of $\mathrm d\nu_p:T_p\Sigma_f\to T_p\Sigma_f$. Thus round spheres have positive principal curvatures. Say that $f$ is strictly two-convex if \[ \lambda_1(p)+\lambda_2(p)>0\qquad\text{for every }p\in\Sigma_f. \] Let $\mathcal E_n^{(2)}$ be the space of these embeddings, with the $C^\infty$ topology, and define the candidate unparametrized moduli space \[ \mathcal M_n^{(2)}=\mathcal E_n^{(2)}/\operatorname{Diff}^{\infty}(S^n), \qquad f\sim f\circ\phi, \] with the quotient topology, where $\phi$ ranges over all smooth diffeomorphisms of $S^n$. This interpretation does not additionally identify hypersurfaces by ambient Euclidean motions or scaling.

Conjectural statement under these conventions. For every integer $n\geq2$, the space $\mathcal M_n^{(2)}$ is contractible: there exist $m_n\in\mathcal M_n^{(2)}$ and a continuous map \[ H_n:\mathcal M_n^{(2)}\times[0,1]\longrightarrow\mathcal M_n^{(2)} \] such that $H_n(m,0)=m$ and $H_n(m,1)=m_n$ for every $m\in\mathcal M_n^{(2)}$.

The restriction $n\geq2$, strict curvature inequality, smooth category, and quotient topology are explicit interpretive choices, not additional hypotheses established by the supplied record. Its literal phrase "for every $n$" does not explain the lower dimensions, where the displayed two-curvature condition is undefined.

Short English statement

Under an explicit conventional interpretation, the conjecture asks whether the space of smooth, unparametrized, strictly two-convex embedded spheres is contractible in every dimension $n\geq2$. The source leaves the dimension range and precise meaning of the moduli space unspecified. Its supplied dated assessment reports path-connectedness and labels the stronger contractibility question open; that assessment has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003155$, source problem code catalogue entry, title Higher-Dimensional Smale Conjecture for 2-Convex Spheres; collection: Oberwolfach Reports Open Problems. The record gives proposed year $2016$ and names no proposer.
• Supplied source citation: Geometrie (2017), Oberwolfach Reports, pp. 1577--1634, DOI 10.4171/owr/2016/28. https://doi.org/10.4171/owr/2016/28. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports. The literature-source list also identifies this reference as Oberwolfach Report 28.
• Supplied literature reference: Buzano--Haslhofer--Hershkovits, The moduli space of two-convex embedded spheres. https://doi.org/10.4310/jdg/1622743139.
• Literature assessment dated exactly 2026-08-21: the supplied curation states that Buzano, Haslhofer, and Hershkovits proved path-connectedness of the moduli space of two-convex embedded spheres, while contractibility remains open. The supplied status is partially_solved. This is supplied dated curation derived from the OWR record, not independently verified literature or a current-status determination. The record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement label is corrected_verified; its accompanying qualification says that only safe notation, typography, and TeX-delimiter cleanup was applied, preserving hypotheses, quantifiers, and subquestions. This label does not establish mathematical correctness or resolve the formulation choices above.