Hyper-Torus Manifold

• Hyper-Torus manifolds are complete hyperkähler spaces defined by an effective tri-Hamiltonian Tⁿ action and rich quaternionic structures.
• They are constructed via hyperkähler quotients of flat Hilbert spaces, yielding both flat quaternionic tori and more complex infinite-topology examples.
• Embedded hypertori, such as constant mean curvature tori in spheres, demonstrate the application of computer-assisted proofs in geometric analysis.

A hyper-torus manifold, often termed a hypertoric manifold, is a complete hyperkähler manifold of real dimension $4n$ that admits an effective, tri-Hamiltonian action of a real $n$-dimensional torus $T^n$. In addition, the term “hypertorus” is used in geometric analysis to denote embedded or immersed $k$-dimensional tori in higher-dimensional spaces such as spheres, particularly when equipped with additional geometric properties like constant mean curvature. In both contexts, the structure and classification of such manifolds or submanifolds are governed by deep results in differential and symplectic geometry, complex and quaternionic structures, and moment-map theory. The subject interfaces with the study of hypercomplex tori, where the integrable complex structures satisfy quaternionic relations, and the only possibilities are flat hyperkähler geometries giving rise to flat quaternionic tori.

1. Hyper-Torus and Hypertoric Manifolds: Formal Definitions

A hypertoric manifold $M$ of real dimension $4n$ possesses a Riemannian metric $g$ and three integrable complex structures $I, J, K$ on $TM$, satisfying the quaternionic relations $I^2 = J^2 = K^2 = IJK = -\mathrm{Id}$, with $g$ Kähler with respect to each. The symplectic forms associated to these structures are $\omega_A(X, Y) = g(A X, Y)$, for $A \in \{I, J, K\}$. The torus action

• is tri-holomorphic (preserving $g$, $I$, $J$, $K$),
• and is tri-Hamiltonian if there exist equivariant moment maps $\mu_A : M \to \mathfrak t^*$ such that $d\langle \mu_A, X \rangle = \iota_{X_M} \omega_A$, $X \in \mathfrak{t} = \mathrm{Lie}(T^n)$.

A complete hyperkähler $4n$-manifold $M$ with effective, tri-Hamiltonian $T^n$-action is a hypertoric manifold (Dancer et al., 2016). When $M$ is a torus, every hypercomplex structure is in fact flat hyperkähler and determined by translation-invariant structures on quotient lattices, yielding so-called flat quaternionic tori (Federico et al., 22 Jun 2025).

2. Classification of Hypertoric and Hypercomplex Tori

The main classification for complete flat affine structures on real tori and their hypercomplex analogs is as follows:

• For a smooth $4n$-dimensional torus $T^{4n}$ carrying a hypercomplex structure $(I, J, K)$ with Obata connection $\nabla$, the connection is flat, complete, and has trivial holonomy (Federico et al., 22 Jun 2025).
• Every hypercomplex structure on a real torus $T^{4n}$ is a flat hyperkähler structure. Up to covering and translation, one has $$(\mathbb{H}^n, I_{\rm std}, J_{\rm std}, K_{\rm std})/\Lambda$$ for some rank-$4n$ lattice $\Lambda \subset \mathbb{H}^n$.
• The standard flat hyperkähler metric is preserved by the full quaternionic action.
• There are no exotic hypercomplex (i.e., non-hyperkähler) structures on tori; such structures are precluded by the classification of complete flat affine actions with unipotent abelian holonomy that must reduce to translations when quaternionic symmetry is enforced.

The general structure of a hypertoric manifold beyond tori is captured by the following classification (Dancer et al., 2016):

Setting	Main Classification Results
$n=1$ (dimension $4$)	Gibbons–Hawking metrics, multi–Taub–NUT, Anderson–Kronheimer–LeBrun with infinite topology (fibres of moment map connected)
$n>1$ (dimension $4n$)	Each arises as a hyperkähler quotient of a flat Hilbert space by Abelian Hilbert–Lie group; moment map surjective, fibres connected

3. Construction of Hypertoric Manifolds

Every complete hypertoric manifold arises as a hyperkähler quotient:

• Given a (finite or countable) index set $L$, primitive vectors $u_k \in \mathbb{Z}^n$ spanning $\mathbb{R}^n$, and weights $A_k \in \Im \mathbb{H} \cong \mathbb{R}^3$, one forms the Hilbert manifold $\mathcal{M}_\Lambda = \Lambda + \ell^2(\mathbb{H})$.
• The torus $T^L$ acts tri-Hamiltonianly, with a componentwise moment map.
• Using a surjection $B : \mathbb{R}^L \to \mathbb{R}^n$ with $B(e_k) = u_k$, and letting $N = \ker(\exp \circ B : T^L \to T^n)$, the hypertoric manifold is $M = \mu_\Lambda^{-1}(0) / N$, a smooth hyperkähler $4n$-manifold with effective $T^n$-action.
• The dense open part is the complement of the intersection of codimension $>1$ hyperplanes—the “hyperplane arrangement geometry” generalizes classic toric topology.

For tori, these methods reduce to the construction of flat quaternionic tori as quotients $\mathbb{H}^n/\Lambda$.

4. Rigidity and Obstructions: No Exotic Structures on Tori

For compact real tori, the imposition of a hypercomplex structure forces rigidity:

• The Obata connection for a hypercomplex structure is flat and complete.
• Any attempt to realize nontrivial linear holonomy compatible with the quaternionic structure yields only the trivial holonomy; all unipotent holonomy representations must be trivial under the required commutation with the quaternionic action (Federico et al., 22 Jun 2025).
• The only possible hypercomplex tori are flat hyperkähler, and no “exotic” structures (i.e., non-hyperkähler hypercomplex) exist on tori.

Consequences and corollaries:

• The twistor space $\mathrm{Tw}(T^{4n})$ is biholomorphic to $\mathrm{Tot}(\mathcal{O}(1)^{\oplus 2n})$ and has trivial canonical bundle.
• Topologically, every $T^{4n}$ is parallelizable with a flat quaternionic structure on its tangent bundle.

5. Embedded Hypertori in Spheres and Computer-Assisted Results

In geometric analysis, “hypertorus” refers also to embedded or immersed $k$-tori in higher-dimensional spaces, notably as compact constant mean curvature (CMC) hypersurfaces. A notable advance is the existence of an embedded CMC 3-torus (hypertorus) in the round $S^4$:

• The immersion $F: T^3 \to S^4 \subset \mathbb{R}^5$ is constructed using functions $r(t), \theta(t), \alpha(t)$ solving an ODE system ensuring CMC $H=-3$, with appropriate initial and boundary conditions (Perdomo, 24 Jun 2025).
• Using the Round–Taylor Method (RTM) with rational arithmetic and error control, and the Poincaré–Miranda theorem, rigorous existence of such an embedded CMC hypertorus is established.
• The resulting hypertorus $M = F(T^3) \cong S^1 \times S^1 \times S^1$ is the first known example of an embedded nonminimal hypertorus in $S^4$ not deformed from previously known minimal examples.

The computational framework rigorously controls truncation and rounding errors, demonstrating the viability of computer-assisted proofs for existence of compact CMC hypersurfaces in higher-dimensional spheres.

6. Infinite Topological Type and Non-Abelian Contrasts

Hypertoric manifolds of infinite topological type arise by taking infinite collections in the hyperkähler quotient construction:

• Examples include the Anderson–Kronheimer–LeBrun metrics in dimension $4$, where $b_2(M) = \infty$ (Dancer et al., 2016).
• The general criterion is that the collection of primitive vectors $u_k$ satisfies $\sum_k (1+\|u_k\|)^{-1} < \infty$.

In contrast, tri-Hamiltonian actions by non-Abelian groups (e.g., $SU(2)$ or $SU(3)$) on hyperkähler manifolds can fail to yield surjective moment maps or connected moment map fibres, pathologies forbidden in the hypertoric (abelian) case. For hypertoric manifolds, the hyperkähler moment map is always surjective with connected fibres.

7. Symplectic Cuts, Hyperkähler Modifications, and Additional Constructions

Modifications of hypertoric geometry exploit extensions of symplectic and hyperkähler cutting, and implosion techniques:

• Hyperkähler modifications: Replacing the auxiliary $\mathbb{C}$ factor in symplectic cutting with $\mathbb{H}$ and performing reduction via a tri-Hamiltonian $S^1$-action “twists” the manifold along the level set of the $S^1$-moment map, changing the metric in a controlled way by adding a Gibbons–Hawking style potential (Dancer et al., 2016).
• Implosion: Analogous to symplectic implosion, hyperkähler implosion replaces symplectic quotients with GIT-style complex-symplectic quotients, yielding universal imploded spaces with residual torus actions and connections to Kostant varieties via quiver constructions.

These operations lead to new families of hypertoric and related hyperkähler manifolds, sometimes of infinite topological type, thereby expanding the landscape beyond flat quaternionic tori.

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References:

• “Exotic hypercomplex structures on a torus do not exist” (Federico et al., 22 Jun 2025)
• “Existence of a constant-mean-curvature hypertorus in $S^4$ via computer assistance” (Perdomo, 24 Jun 2025)
• “Hypertoric manifolds and hyperKähler moment maps” (Dancer et al., 2016)