Spherical Minimal Immersions Research Articles

Lagrangian Homothetic Solitons for the Inverse Mean Curvature Flow

We study solutions to the inverse mean curvature flow which evolve by homotheties of a given submanifold with arbitrary dimension and codimension. We first show that the closed ones are necessarily spherical minimal immersions and so we reveal the strong rigidity of the Clifford torus in this setting. Mainly we focus on the Lagrangian case, obtaining numerous examples and uniqueness results for some products of circles and spheres that generalize the Clifford torus to arbitrary dimension. We also characterize the pseudoumbilical ones in terms of soliton curves for the inverse curve shortening flow and minimal Legendrian immersions in odd-dimensional spheres. As a consequence, we classify the rotationally invariant Lagrangian homothetic solitons for the inverse mean curvature flow.

On the space of orthogonal multiplications in three and four dimensions and Cayley’s nodal cubic

The study of orthogonal multiplications is more than 100 years old and goes back to the works of Hurwitz and Radon. Yet, apart from the extensive literature on admissibility of domain and range dimensions near the Hurwitz-Radon range (in codimension \(\le 8\)), only sporadic and fragmentary results are known about full classification (in large codimension), more specifically, about the moduli space \(\mathfrak {M}_m\) of orthogonal multiplications \(F:\mathbb {R}^m\times \mathbb {R}^m\rightarrow \mathbb {R}^n\) (for various n), even for \(m\le 4\). In this paper we give an insight to the subtle geometries of \(\mathfrak {M}_3\) and \(\mathfrak {M}_4\). Orthogonal multiplications are intimately connected to quadratic eigenmaps between spheres via the Hopf-Whitehead construction. The 9-dimensional moduli space \(\mathfrak {M}_3\) lies on the boundary of the 84-dimensional moduli of quadratic eigenmaps of \(S^5\) into spheres. Similarly, the 36-dimensional moduli space \(\mathfrak {M}_4\) is on the boundary of the 300-dimensional moduli of quadratic eigenmaps of \(S^7\) into spheres. We will show that \(\mathfrak {M}_3\) is the \(SO(3)\times SO(3)\)-orbit of a 3-dimensional convex body bounded by Cayley’s nodal cubic surface with vertices in a real projective space \(\mathbb {R}P^3\), the latter imbedded equivariantly and minimally in an 8-sphere of the space of quadratic spherical harmonics on \(S^3\). For \(\mathfrak {M}_4\), we show that it possesses two orthogonal 18-dimensional slices each of which is an \(SO(4)\times SO(4)\)-orbit of a 6-dimensional polytope \(\mathcal {P}\subset \mathbb {R}^6\). This polytope itself is the convex hull of two orthogonal regular tetrahedra. The corresponding orthogonal multiplications are explicitly constructible. Finally, we give an algebraic description of the 24-dimensional space of diagonalizable elements in \(\mathfrak {M}_4\). The crucial fact in \(\mathbb {R}^4\) is the splitting of the exterior product \(\Lambda ^2(\mathbb {R}^4)\) into self-dual and anti-self-dual components. The techniques employed here can be traced back to the work of Ziller and the author (Toth and Ziller 1999) in describing the 18-dimensional moduli for quartic spherical minimal immersions of \(S^3\) into spheres. As a new feature, we point out the importance of multiplicity of the zeros of the polynomials that define the boundary \(\partial \,\mathfrak {M}_m\) as a determinantal variety.

Simplicial slices of the space of minimal $$SU(2)$$ -orbits in spheres

Let \((\mathcal{M }_3^k)^{SU(2)}\) denote the DoCarmo–Wallach moduli space of \(SU(2)\)- equivariant spherical minimal immersions of the three sphere \(S^3\) of degree \(k\). Although the complexity of these moduli increases rapidly with \(k\) (for example, \(\dim (\mathcal{M }_3^k)^{SU(2)} = \mathcal{O}(k^2)\)), we show here that they possess linear slices that are simplices of dimension \(\mathcal{O}(k)\). The construction of these simplicial slices depend on the DeTurck–Ziller classification of 3-dimensional spherical space forms imbedded into spheres as minimal \(SU(2)\)-orbits. The existence of these slices enables us to give asymptotically sharp estimates on a sequence of Grunbaum type measures of symmetry of these moduli.

A measure of symmetry for the moduli of spherical minimal immersions

A Grunbaum type of measure of symmetry is calculated and estimated for the DoCarmo-Wallach moduli spaces for eigenmaps and spherical minimal immersions. The DeTurck-Ziller classification of minimal imbeddings of 3-dimensional space forms is used to obtain exact determination of the measure for the SU(2)-equivariant moduli.

On the shape of the moduli of spherical minimal immersions

The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold $M$ is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of $M$. In this paper we define and study a sequence of metric invariants $\sigma _m$, $m\geq 1$, associated to a compact convex body $\mathcal {L}$ with base point $\mathcal {O}$ in the interior of $\mathcal {L}$. The invariant $\sigma _m$ measures how lopsided $\mathcal {L}$ is in dimension $m$ with respect to $\mathcal {O}$. The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate $\sigma _m$ for convex polytopes.

Spherical Minimal Immersions with Prescribed Codimension

We describe a general construction of manufacturing new spherical minimal immersions between round spheres out of old ones. The new immersions have higher domain dimension and degree and the construction has a precise control on the codimension. Applied to classified and recent examples, the construction gives an abundance of new spherical minimal immersions with prescribed codimensions.

Simplicial intersections of a convex set and moduli for spherical minimal immersions

Let H be a Euclidean vector space. Let S2 0(H) denote the space of symmetric endomorphisms of H with vanishing trace; S2 0(H) is a Euclidean vector space with respect to the natural scalar product 〈C,C ′ 〉 = trace(CC ′), C,C ′ ∈ S2 0(H). We define the (reduced) moduli space [7] as K0 = K0(H) = {C ∈ S 0(H) | C + I ≥ 0}, where ≥ means positive semidefinite. We observe that K0 is a convex body in S2 0(H). The interior of K0 consists of those C ∈ K0 for which C + I > 0, and the boundary of K0 consists of those C ∈K0 for which C + I has nontrivial kernel. The eigenvalues of the elements in K0 are contained in [−1, dimH−1]. Hence K0 is compact. Finally, an easy argument using GL(H)-invariance of K0 shows that the centroid of K0 is the origin. Let M be a compact Riemannian manifold and H = Hλ the eigenspace of the Laplacian M (acting on functions of M) corresponding to an eigenvalue λ. The DoCarmo–Wallach moduli space that parameterizes spherical minimal immersions f : M → SV of M into the unit sphere SV of a Euclidean vector space V, for various V, is the intersection K0 ∩ Eλ, where Eλ is a linear subspace of S2 0(Hλ). Here f is an isometric minimal immersion of dimM/λ times the original metric of M. (For further details, see [3; 6; 8].) Intersecting K0 further with suitable linear subspaces of Eλ, we obtain moduli that parameterize spherical minimal immersions with additional geometric properties (such as higher-order isotropy, equivariance with respect to an acting group of isometries of M, etc.). A result of Moore [4] states that a spherical minimal immersion f : S → S n with n ≤ 2m − 1 is totally geodesic; in particular, the image of f is a great m-sphere in S . An important example showing that the upper bound is sharp is provided by the tetrahedral minimal immersion f : S3 → S 6 (see [2; 6]). Here f is SU(2)-equivariant and non–totally geodesic. The name comes from the fact that the invariance group of f is the binary tetrahedral group T∗ ⊂ S3 = SU(2), so that f factors through the canonical projection S3 → S3/T∗ and gives a minimal imbedding f : S3/T∗ → S 6 of the tetrahedral manifold S3/T∗ into S 6. Let M = S3 and let Hλp be the pth eigenspace of the Laplacian on S3 corresponding to the eigenvalue λp = p(p + 2). According to a result in [5; 6] there

Spherical minimal immersions of the 3-sphere

In 1966 Takahashi [11] proved that a minimal isometric immersion \(f:S^m(1) \to S^N(r)\) of round spheres exists iff \(r=\sqrt{m/\lambda_p}\), where \(\lambda_p\) is the p-th eigenvalue of the Laplacian on S m in this case, the components of f are spherical harmonics on S m of order p. This immersion is unique up to congruence on the range and agrees with the generalized Veronese map if m = 2 as was shown in 1967 by Calabi [1]. In 1971 DoCarmo and Wallach [3] proved that the same rigidity holds for p = 2,3. The main aim of their work, however, was to show that, for \(m\geq 3\) and \(p\geq 4\), unicity fails, and, indeed, the set of (congruence classes of) minimal isometric immersions \(f:S^m\to S^N(\sqrt{m/\lambda_p})\) can be parametrized by a moduli space \({\Cal M}^p_m\), a compact convex body in a representation space \({\Cal F}^p_m\) of SO(m + 1) of dimension \(\geq 18\). In 1994, the first author [14] determined the exact dimension of the moduli, and with Gauchman [5] in 1996, revealed intricate connections beween the irreducible components of \({\Cal F}^p_m\) and the geometry of the immersions these components represent. The purpose of the present paper is to provide a complete geometric description of the fine details of the (boundary of the) 18-dimensional space \({\Cal M}^4_3\), the first nontrivial moduli. This is made possible by several reductions that make use of the splitting \(SO(4)=SU(2)\cdot SU(2)'\) as well as rely on the structure of SU(2) equivariant minimal isometric immersions treated in the work of DeTurck and the second author [2] in 1992. The equivariant imbedding theorem [14] asserts that the structure of \({\Cal M}^4_3\) reappears in the moduli \({\Cal M}^p_m\) for \(m\geq 3\) and \(p\geq 4\).

Universal constraints on the range of eigenmaps and spherical minimal immersions

The purpose of this paper is to give lower estimates on the range dimension of spherical minimal immersions in various settings. The estimates are obtained by showing that infinitesimal isometric deformations (with respect to a compact Lie group acting transitively on the domain) of spherical minimal immersions give rise to a contraction on the moduli space of the immersions and a suitable power of the contraction brings all boundary points into the interior of the moduli space.

New Construction for Spherical Minimal Immersions

We describe a general method of manufacturing new minimal immersions between round spheres out of old ones. The resulting spherical minimal immersions are given analytically in terms of the harmonic projection operator and have higher source dimensions. Applied to classical examples, this gives an abundance of new minimal immersions of even-dimensional spheres.

Fine structure of the space of spherical minimal immersions

The space of congruence classes of full spherical minimal immersions f : S m → S n f:S^m\to S^n of a given source dimension m m and algebraic degree p p is a compact convex body M m p \mathcal {M}_m^p in a representation space F m p \mathcal {F}_m^p of the special orthogonal group S O ( m + 1 ) SO(m+1) . In Ann. of Math. 93 (1971), 43–62 DoCarmo and Wallach gave a lower bound for F m p \mathcal {F}_m^p and conjectured that the estimate was sharp. Toth resolved this “exact dimension conjecture” positively so that all irreducible components of F m p \mathcal {F}_m^p became known. The purpose of the present paper is to characterize each irreducible component V V of F m p \mathcal {F}_m^p in terms of the spherical minimal immersions represented by the slice V ∩ M m p V\cap \mathcal {M}_m^p . Using this geometric insight, the recent examples of DeTurck and Ziller are located within M m p \mathcal {M}_m^p .