[Paper Review] Harmonic self-maps of cohomogeneity one manifolds

Thomas Puettmann, Anna Siffert|arXiv (Cornell University)|Aug 30, 2016

Geometric Analysis and Curvature Flows6 references3 citations

TL;DR

This paper constructs new harmonic self-maps of compact Lie groups such as SO(4ℓ+2), SO(8), SO(10), SO(14), and SO(26) by solving non-linear singular boundary value problems for equivariant maps on cohomogeneity one manifolds. It introduces a geometric framework using $(k,r)$-maps and proves that the tension field vanishes when the normal and tangential components both vanish, leading to explicit harmonic maps of degrees -3, -5, -7, and -11.

ABSTRACT

We develop the theory of equivariant harmonic self-maps of compact cohomogeneity one manifolds and construct new harmonic self-maps of the compact Lie groups SO(4L+2), L >= 1, with degree -3, of SO(8), SO(14) and SO(26) with degree -5 each, of SO(10) with degree -7, and of SO(14) with degree -11 by exhibiting linear solutions to non-linear singular boundary value problems.

Motivation & Objective

• To develop a theory of equivariant harmonic self-maps on compact cohomogeneity one manifolds with group actions.
• To extend Urakawa's results by allowing more general invariant metrics and geometric constructions beyond homogeneous spaces.
• To construct new topologically non-trivial harmonic self-maps of compact Lie groups such as SO(4ℓ+2), SO(8), SO(10), SO(14), and SO(26).
• To solve non-linear singular boundary value problems for the normal component of the tension field and verify vanishing of the tangential component via Lie algebraic conditions.
• To establish a framework where linear solutions to non-linear problems yield harmonic maps with prescribed degrees.

Proposed method

• Define $(k,r)$-maps as equivariant maps $g \cdot \gamma(t) \mapsto g \cdot \gamma(r(t))$ where $r: [0,L] \to \mathbb{R}$ is smooth with $r(0)=0$, $r(L)=kL$.
• Derive explicit formulas for the normal and tangential components of the tension field using parallel transport $\Pi_t^{r(t)}$ and the action field homomorphism $J_t^{r(t)}$.
• Express the normal component of the tension field as $\tau^{\perp}_{|\gamma(t)} = \ddot{r}(t) - \dot{r}(t)\operatorname{trace} S_{|\gamma(t)} + \operatorname{trace}\left( (J_t^{r(t)})^* (\Pi_t^{r(t)})^{-1} S_{|\gamma(r(t))} \Pi_t^{r(t)} J_t^{r(t)} \right)$.
• Express the tangential component as $\tau^{\tan}_{|\gamma(t)} = -\sum_{\mu,\nu=1}^n \langle [E_\mu, F_\nu]^*, E_\mu^* \rangle_{|\gamma(r(t))} F_\nu^*_{|\gamma(r(t))}$ using orthonormal bases of fundamental vector fields.
• Establish that the tangential component vanishes when the Lie bracket $[\tilde{E}_\mu, \tilde{F}_\nu]$ is orthogonal to $\tilde{E}_\mu$ in the metric, verified via representation-theoretic decomposition of $\mathfrak{so}(n)$.
• Use trigonometric identities (Lemma 8.1 and 8.2) to evaluate the normal component in specific cases like $(g,m)$- and $(g,m_0,m_1)$-actions on spheres and $\mathrm{SO}(n+2)$.

Experimental results

Research questions

• RQ1 Under what geometric conditions do $(k,r)$-maps on cohomogeneity one manifolds become harmonic?
• RQ2 Can non-linear singular boundary value problems for the normal tension field be solved explicitly to yield harmonic self-maps?
• RQ3 When does the tangential component of the tension field vanish, and what Lie-theoretic conditions ensure this?
• RQ4 What are the possible degrees of harmonic self-maps of compact Lie groups such as SO(n) constructed via equivariant cohomogeneity one maps?
• RQ5 How can trigonometric identities be used to verify the vanishing of the normal tension field in symmetric settings?

Key findings

• The paper constructs new harmonic self-maps of $\mathrm{SO}(4\ell+2)$ for $\ell \geq 1$ with degree $-3$ by solving the tension field equations.
• It exhibits harmonic self-maps of $\mathrm{SO}(8)$, $\mathrm{SO}(14)$, and $\mathrm{SO}(26)$ with degree $-5$ each, using linear solutions to the non-linear boundary value problem.
• A harmonic self-map of $\mathrm{SO}(10)$ with degree $-7$ is constructed, and another of $\mathrm{SO}(14)$ with degree $-11$ is produced.
• The tangential component of the tension field vanishes when the Lie bracket $[\tilde{E}_\mu, \tilde{F}_\nu]$ is orthogonal to $\tilde{E}_\mu$, which holds due to skew-symmetry of $\operatorname{ad}_{\tilde{F}_\nu}$ and decomposition of $\mathfrak{so}(n)$ into $\tilde{\mathfrak{m}}_i$ and $\mathfrak{so}(6)$.
• The normal component vanishes when $r(t)$ satisfies the non-linear ODE $\ddot{r}(t) - \dot{r}(t)\operatorname{trace} S_{|\gamma(t)} + \operatorname{trace}\left( (J_t^{r(t)})^* (\Pi_t^{r(t)})^{-1} S_{|\gamma(r(t))} \Pi_t^{r(t)} J_t^{r(t)} \right) = 0$ with boundary conditions $r(0)=0$, $r(L)=kL$.
• Trigonometric identities (Lemmas 8.1 and 8.2) are used to verify the vanishing of the normal tension field in symmetric cases, such as for $(g,m)$-actions on $\mathbb{S}^{n+1}$ and $\mathrm{SO}(n+2)$.