Rigidity of low index solutions on $\mathbf{S}^{3}$
via a Frankel theorem
for the Allen–Cahn equation

Let $(\epsilon_{j}\mid j\in\mathbf{N})$ be a positive sequence with $\epsilon_{j}\to 0$ as $j\to\infty$, and $(u_{j}\mid j\in\mathbf{N})$ be a corresponding sequence with $u_{j}\in\mathcal{Z}_{\epsilon_{j}}$ for all $j$. We assume uniform bounds for the energy and that the Morse index of these solutions is fixed, namely there are $C>0$ and $k\in\mathbf{Z}_{\geq 0}$ so that for all $j\in\mathbf{N}$,

A subsequence of the $V_{j}=V(\epsilon_{j},u_{j})$ (which we extract without relabelling) converges to a stationary integral varifold, as was shown by Hutchinson–Tonegawa [HT00]. Relying on the regularity theory developed by Wickramasekera [Wic14], the combined work of Tonegawa–Wickramasekera [TW12] and Guaraco [Gua18] established that this limit is supported in a smooth embedded minimal surface $\Sigma\subset\mathbf{S}^{3}$, and

with integer multiplicity $Q\in\mathbf{Z}_{>0}$. (Here and throughout we write $\lvert\Sigma\rvert$ for the unit density varifold associated to $\Sigma$.) Using different methods, Hiesmayr [Hie18] and Gaspar [Gas20] both proved that

the former building on previous work of Tonegawa [Ton05] in the stable case, the latter by adapting computations of Le [Le11, Le15].

As the three-sphere $\mathbf{S}^{3}$ has positive Ricci curvature, the results of Chodosh–Mantoulidis [CM20] give that the convergence is with multiplicity one, meaning $Q=1$ and $V_{j}\to\lvert\Sigma\rvert$ as $j\to\infty$. They also show that

Combining this with the lower semicontinuity of the Morse index, we find that for large enough $j$,

We are especially interested in the situation where the Morse index of the $u_{j}$ is low, specifically where it is five at the most. In this range the classification of minimal surfaces of Urbano [Urb90] shows that for the limit minimal surface $\Sigma$,

• •

  either $\operatorname{index}\Sigma=1$ and $\Sigma=S$ is an equatorial sphere,

• •

  or $\operatorname{index}\Sigma=5$ and $\Sigma=T$ is a Clifford torus.

By [HL71] they respectively have $\operatorname{nullity}S=3$ and $\operatorname{nullity}T=4$.

The orthogonal group $SO(4)$ acts on the space of minimal surfaces in $\mathbf{S}^{3}$, and preserves their index and nullity. Given a surface $\Sigma\subset\mathbf{S}^{3}$, let as usual $\operatorname{Stab}\Sigma=\{P\in SO(4)\mid P(\Sigma)=\Sigma\}$ and $\operatorname{Orb}\Sigma=\{P(\Sigma)\subset\mathbf{S}^{3}\mid P\in SO(4)\}$. The former is a closed Lie subgroup of $SO(4)$ denoted $G_{\Sigma}$, and $\dim\operatorname{Stab}\Sigma+\dim\operatorname{Orb}\Sigma=6$, the dimension of $SO(4)$.

Let $\Sigma$ be minimal and $L_{\Sigma}$ be the Jacobi operator on $\Sigma$. We write $\operatorname{index}\Sigma\in\mathbf{Z}_{\geq 0}$ for the number of strictly negative eigenvalues of $L_{\Sigma}$, counted with multiplicity. The kernel of $L_{\Sigma}$ is spanned by the so-called Jacobi fields. Those that stem from the action of $SO(4)$ are also called Killing Jacobi fields. They span a linear subspace of $L^{2}(\Sigma)$ of dimension $\nu(\Sigma)=\dim\operatorname{Orb}\Sigma$; this is the Killing nullity. (Here we follow the conventions of Hsiang–Lawson [HL71].) For any minimal surface $\Sigma\subset\mathbf{S}^{3}$,

We are especially interested in surfaces for which $\nu$ equals the full nullity. Surfaces with this property are sometimes said to be integrable through rotations. The equatorial sphere and the Clifford torus both have this property, and quoting [HL71] they respectively have

Let $\Sigma\subset\mathbf{S}^{3}$, with isotropy subgroup $G_{\Sigma}\subset SO(4)$. The surface is called homogeneous if $G_{\Sigma}$ acts transitively on it. This is equivalently expressed by saying that $\Sigma$ is an orbit of the action of $G_{\Sigma}$ on $\mathbf{S}^{3}$. The equatorial sphere and the Clifford torus both satisfy this property; in fact they are the only homogeneous minimal surfaces in $\mathbf{S}^{3}$ [HL71].

These notions have natural analogues in the setting of the Allen–Cahn equation. The group $SO(4)$ gives a right action on functions defined on $\mathbf{S}^{3}$ by pre-composition, where $P\in SO(4)$ sends a function $u$ to $u\circ P$. Given a function $u$ on $\mathbf{S}^{3}$ we write $\operatorname{Stab}u=\{P\in SO(4)\mid u\circ P=u\}$ and $\operatorname{Orb}u=\{u\circ P\mid P\in SO(4)\}$. Again we have $\dim\operatorname{Stab}u+\dim\operatorname{Orb}u=6$. (To see this fix any $\alpha\in(0,1)$; by elliptic regularity any critical point has $u\in C^{2,\alpha}(\mathbf{S}^{3})$. Consider the orbit map $P\in SO(4)\mapsto u\circ P\in C^{2,\alpha}(\mathbf{S}^{3})$. Upon quotienting by the stabiliser $G_{u}=\operatorname{Stab}u$ this defines a homeomorphism onto its image. In particular $SO(4)/G_{u}$ and $\operatorname{Orb}u$ have the same dimension, namely $6-\dim G_{u}$. Note that a rigorous rederivation of the analogous orbit-stabiliser identity we quoted above for minimal surfaces $\Sigma\subset\mathbf{S}^{3}$ would go along the same lines, working in a suitable Banach space of embeddings into $\mathbf{S}^{3}$.)

Let $M_{\epsilon}$ be the semilinear operator corresponding to (5). The linearisation of $M_{\epsilon}$ around a function $u$ is (up to multiplication by $\epsilon$) the linear operator $L_{\epsilon,u}=\Delta-\epsilon^{-2}W^{\prime\prime}(u)$. (This describes the second variation of $-E_{\epsilon}$ at $u$ via integration by parts.) This operator has finite-dimensional kernel, whose dimension is denoted $\operatorname{nullity}u\in\mathbf{Z}_{\geq 0}$. We also write $\operatorname{index}u\in\mathbf{Z}_{\geq 0}$ for the number of strictly negative eigenvalues of $L_{\epsilon,u}$ counted with multiplicity. The action of $SO(4)$ preserves the Allen–Cahn functional, $E_{\epsilon}(u\circ P)=E_{\epsilon}(u)$ for all $P\in SO(4)$. It also maps $\mathcal{Z}_{\epsilon}$ to itself, and for all $P\in SO(4)$, $\operatorname{index}u\circ P=\operatorname{index}u$ and $\operatorname{nullity}u\circ P=\operatorname{nullity}u$. The invariance of $E_{\epsilon}$ under $SO(4)$ also means that the action generates functions in the kernel of $L_{\epsilon,u}$, which span a space of dimension $\nu(u)\in\mathbf{Z}_{\geq 0}$, the Killing nullity of $u$. As above we have $\nu(u)=\dim\operatorname{Orb}u$ and

(The fact that $\nu(u)=\dim\operatorname{Orb}u=6-\dim G_{u}$ is not hard to see. To derive this rigorously requires working again with the orbit map $P\in SO(4)\to u\circ P\in C^{2,\alpha}(\mathbf{S}^{3})$ we used above for the orbit-stabiliser identity. This differentiates to the map $A\in\mathfrak{so}(4)\mapsto-\langle\nabla u,\xi_{A}\rangle$, where $\xi_{A}$ is the Killing vector field corresponding to $A$. Writing $\mathfrak{g}_{u}\subset\mathfrak{so}(4)$ for the Lie algebra corresponding to $G_{u}$, we obtain a linear isomorphism between $\mathfrak{so}(4)/\mathfrak{g}_{u}$ and the tangent space to the orbit of $u$. In particular there are precisely $\nu=6-\dim\mathfrak{g}_{u}=6-\dim G_{u}$ Killing vector fields so that $\langle\nabla u,\xi_{1}\rangle,\dots,\langle\nabla u,\xi_{\nu}\rangle$ forms a linearly independent family.)

Let $\epsilon_{j}\to 0$ and $(u_{j}\mid j\in\mathbf{N})$ be a sequence of critical points in $\mathbf{S}^{3}$ with $V(\epsilon_{j},u_{j})\to\lvert\Sigma\rvert$, where $\Sigma\subset\mathbf{S}^{3}$ is an embedded minimal surface. Using either a simple calculation as in [CG19] or Lemma 2 as justification, one finds that for large $j$,

Assume additionally that $\Sigma$ is integrable through rotations, that is $\nu(\Sigma)=\operatorname{nullity}\Sigma$. Stringing together (11), (14) and (15), we find that eventually

We now specialise this to the setting of low index in $\mathbf{S}^{3}$, and show that if $\operatorname{index}u_{j}\leq 5$ then either $\operatorname{index}u_{j}=1$ or $5$, provided $j$ is large enough. To this end assume that $E_{\epsilon_{j}}(u_{j})\leq C$ and $\operatorname{index}u_{j}\leq 4$ along the sequence. By (9) and Urbano’s classification, $V(\epsilon_{j},u_{j})$ converges to an equatorial sphere $S\subset\mathbf{S}^{3}$; moreover by [CM20] the convergence is with multiplicity one, that is $V(\epsilon_{j},u_{j})\to\lvert S\rvert$ as $j\to\infty$. From the above we have that for large $j$, $\operatorname{nullity}u_{j}=\nu(u_{j})=3$. At the same time $4\geq\operatorname{index}u_{j}+\operatorname{nullity}u_{j}\geq\operatorname{index}u_{j}+3$ by (10). As $u_{j}$ is unstable, we find $\operatorname{index}u_{j}=1$.

We apply this observation to the context of Corollary 2, where no assumption is made about the index of the solutions. Instead there only imposes that the $u_{j}\in\mathcal{Z}_{\epsilon_{j}}$ have $E_{\epsilon_{j}}(u_{j})=c_{\epsilon_{j}}(5)$. This notwithstanding, critical points obtained through five-parameter min-max arguments as in [GG18] have index at most five. Taking the conclusions of Theorem 1 for granted, and as $c_{\epsilon_{j}}(4)<c_{\epsilon_{j}}(5)$, the functions obtained from min-max methods must be symmetric solutions vanishing on a Clifford torus. Moreover $c_{\epsilon_{j}}(5)\to 2\pi^{2}$ as $j\to\infty$. By [MN14], the Clifford tori are the only minimal surfaces in $\mathbf{S}^{3}$ whose area has this value (or a fraction thereof), so that $V(\epsilon_{j},u_{j})\to\lvert T\rvert$, after extracting a subsequence if necessary. Arguing as above one obtains index bounds for the $u_{j}$, which prove that they too must be of the rigid form described in Theorem 1.

The Lie group $SO(4)$ can be endowed with a bi-invariant metric, which induces a distance function $d$. Write $\mathcal{G}$ for the set of closed subgroups of $SO(4)$. When endowed with the topology induced by the Hausdorff distance, the couple $(\mathcal{G},d_{H})$ forms a compact metric space. Moreover, we have the following useful result. (We state this for an arbitrary compact Lie group $G$, although for our purposes $G=SO(4)$ or $SO(n+2)$ would be sufficient. Moreover given a closed subgroup $H\subset G$, we let $(H)_{\tau}=\{P\in G\mid\operatorname{dist}(P,H)<\tau\}$ be its open tubular neighbourhood of size $\tau>0$.)

Let $\epsilon_{j}\to 0$ be a sequence of positive scalars and $(u_{j}\mid j\in\mathbf{N})$ be a sequence of solutions of (5), with respectively $u_{j}\in\mathcal{Z}_{\epsilon_{j}}$. We assume that they additionally have uniformly bounded Allen–Cahn energy and $\operatorname{index}u_{\epsilon_{j}}=5$ for all $j$, whence $V(\epsilon_{j},u_{j})\to\lvert T\rvert$ as $j\to\infty$. We abbreviate

Moreover, we may assume throughout that $j$ is large enough that (11) holds, ensuring that $\operatorname{nullity}u_{j}\leq 4$. We use Lemma 1 to show that eventually the stabilisers $G_{j}$ and $G_{T}$ are conjugate subgroups of $SO(4)$.

First we point that out that $SO(4)$ defines a natural action on the space of two-varifolds $\operatorname{\mathbf{V}}_{2}(\mathbf{S}^{3})$. A rotation $P\in SO(4)$ maps $V\in\operatorname{\mathbf{V}}_{2}(\mathbf{S}^{3})$ to its push-forward $P_{\#}V$. (Here we implicitly identify $P\in SO(4)$ with the isometry it induces on $\mathbf{S}^{3}$, as we have been doing.) This action is continuous in the varifold topology. To see this, let $(P,V)\in SO(4)\times\operatorname{\mathbf{V}}_{2}(\mathbf{S}^{3})$ be arbitrary, and consider two sequences $P_{k}\to P$ and $V_{k}\to V$, the latter being in the varifold topology. Given $\varphi\in C(Gr_{2}(\mathbf{S}^{3}))$, one has $(P_{k\#}V_{k}-P_{\#}V)\varphi=P_{k\#}(V_{k}-V)\varphi+(P_{k\#}V-P_{\#}V)\varphi$, both of which tend to zero as $k\to\infty$. We define the stabiliser and orbit of $V$ in the usual way. For an embedded surface $\Sigma\subset\mathbf{S}^{3}$ and all $P\in SO(4)$, it holds that $P_{\#}\lvert\Sigma\rvert=\lvert P(\Sigma)\rvert$ and we need not distinguish between the stabiliser and orbit of $\Sigma$ as an embedded surface or as a varifold.

Let $u\in\mathcal{Z}_{\epsilon}$, and the varifold $V(\epsilon,u)$ be defined as in (6). Via a simple change of variable we find

To see this, let $X\in\mathbf{S}^{3}$ be arbitrary and $Y=P(X)$. Assume without loss of generality that $\lvert\nabla u\rvert(X)=\lvert\nabla(u\circ P^{-1})\rvert(Y)>0$, ensuring that the level sets $\{u=u(X)\}$ and $\{u\circ P^{-1}=(u\circ P^{-1})(Y)\}$ are regular near $X$ and $Y$ respectively. The map $P$ induces on $Gr_{2}(\mathbf{S}^{3})$ sends $(X,T_{X}\{u=u(X)\})$ to $(Y,T_{Y}\{u\circ P^{-1}=(u\circ P^{-1})(Y)\}$. Given any $\varphi\in C(Gr_{2}(\mathbf{S}^{3}))$, we may thus compute $[P_{\#}V(\epsilon,u)](\varphi)=V(\epsilon,u)(\varphi\circ P)$ to be equal $V(\epsilon,u\circ P^{-1})(\varphi)$:

Although the compared actions are on the right and left respectively,it follows from (20) that $\operatorname{Stab}u\subset\operatorname{Stab}V(\epsilon,u)$, and specialised to our sequence $G_{j}\subset\operatorname{Stab}V(\epsilon_{j},u_{j})$ for all $j$.

Therefore eventually $\dim\operatorname{Stab}u_{j}\leq 2$ and $\nu(u_{j})\geq 4$. Combining this with (11), we find that for large $j$

Thus $G_{j}$ is conjugate to $G_{T}$; further given any $\tau>0$ there is $J(\tau)\in\mathbf{N}$ so that for $j\geq J(\tau)$, there is $P_{j}\in SO(4)$ with $d(P_{j},I)<\tau$ and $P_{j}^{-1}G_{j}P_{j}=G_{T}$.

Applying the results of [BO86] in the present setting, one obtains the following; see also [CGGM20] for an alternative construction of the symmetric critical point.