Extremal Metrics for Laplace Eigenvalues in Perturbed Conformal Classes on Products

Abstract

In this short note, we prove that conformal classes which are small perturbations of a product conformal class on a product with a standard sphere admit a metric extremal for some Laplace eigenvalue. As part of the arguments, we obtain perturbed harmonic maps with constant density.

1 Introduction

For a closed manifold M, we are interested in the eigenvalues of the Laplace operator considered as functionals of the metric.

We denote by

$$\begin{aligned} \mathcal {R}:=\{g \ : \ g \ \text {is a Riemannian metric on}\ M \text { with}\ {\text {vol}}(M,g)=1\} \end{aligned}$$

the space of all unit volume Riemannian metrics on M endowed with the $C^\infty$-topology, i.e. the smallest topology containing any $C^k$-topology. The group $C^\infty _+(M)$ of positive smooth functions acts via (normalized) pointwise multiplication on $\mathcal {R},$

$$\begin{aligned} \phi . g := {\text {vol}}(M,\phi g)^{-2/n} \phi g, \end{aligned}$$

(1.1)

so that ${\text {vol}}(M,\phi .g)=1.$ The quotient space

$$\begin{aligned} \mathcal {C}= C^\infty _+(M) \backslash \mathcal {R} \end{aligned}$$

is the space of all conformal structures on M.

Since M is compact, the spectrum of $\Delta _g$ consists of eigenvalues of finite multiplicity only for any $g \in \mathcal {R}.$ We list these as

$$\begin{aligned} 0=\lambda _0<\lambda _1\le \lambda _2\le \dots , \end{aligned}$$

(1.2)

where we repeat an eigenvalue as often as its multiplicity requires.

In recent years, there has been much interest in finding extremal metrics for eigenvalues $\lambda _k$ considered either as functionals

$$\begin{aligned} \lambda _k :\mathcal {R} \rightarrow \mathbb {R}\end{aligned}$$

(1.3)

or

$$\begin{aligned} \lambda _k :[g] \rightarrow \mathbb {R}, \end{aligned}$$

(1.4)

where

$$\begin{aligned}{}[g]=\{\phi g \ :\ \phi \in C_+^\infty (M)\} \end{aligned}$$

denotes the conformal class of a metric g, see for instance [4, 5, 7, 10, 11], and references therein. These functionals will not be smooth but only Lipschitz; therefore extremality has to be defined in an appropriate way, see below.

One reason to study these extremal metrics is their intimate connection to other classical objects from differential geometry. For (1.3), these are minimal surfaces in spheres, and for (1.4) these are sphere-valued harmonic maps with constant density, so called eigenmaps. There has been a lot of effort in the past to understand, which manifolds admit eigenmaps or even minimal isometric immersions into spheres, see for instance [14, Chap. 6] for a general overview over classical results for eigenmaps including the generalized Do Carmo–Wallach theorem, and [2, 8] to mention only the two most classical results.

Before we state our results, we have to introduce some notation. Let M be a smooth, closed manifold.

A smooth map $u :M \rightarrow S^\ell$ is called an eigenmap, if it is harmonic, i.e.

$$\begin{aligned} \Delta u = |\nabla u|^2 u, \end{aligned}$$

(1.5)

and has constant density $|\nabla u|^2=const.$ In other words, the components of u are all eigenfunctions corresponding to the same eigenvalue. Note that most Riemannian manifolds do not admit eigenmaps, since the spectrum is generically simple by [13, Theorem 8]. Even more, the spectrum of a generic metric in a conformal class is simple [1, 6, 13]. Moreover, we would like to point out that it is not clear at all whether eigenmaps exist in the presence of large multiplicity.

Theorem 1.6

Let (M, g) be a closed Riemannian manifold of dimension $\dim (M) \ge 3$, and assume

• (i) There is a non-constant eigenmap $u :(M,g) \rightarrow S^1,$

  or

• (ii) $(M,g)=(N \times S^\ell , g_N+ g_{st.}),$ where $g_{st.}$ denotes the round metric of curvature 1 on $S^\ell .$

Then there is a neighbourhood U of [g] in $\mathcal {C},$ such that for any $c \in U,$ there is a representative $h \in c,$ such that (M, h) admits a non-constant eigenmap to $S^1$ respectively $S^\ell .$

An obvious question is then, whether the set of conformal structures admitting non-constant eigenmaps is always non-empty. We answer this at least in the following case.

Corollary 1.7

Assume $\phi :M \rightarrow S^1$ is a submersion. Then the set $\mathcal {E} \subset \mathcal {C}$ of conformal structures admitting non-trivial eigenmaps to $S^1$ is open and non-empty.

Remark 1.8

It is not clear, whether $\mathcal {E}$ is also closed. This question is related to possible degenerations of n-harmonic maps, as it will become clear from the proof.

Not every manifold admits a submersion to $S^1.$ In fact, there are topological obstructions to the existence of such a map.

More precisely, since $S^1$ is a $K(\mathbb {Z},1)$, a submersion gives rise to a non-trivial element in $H^1(M,\mathbb {Z}).$ Moreover, the differentials of local lifts of the submersion to $\mathbb {R}$ give rise to a globally defined nowhere vanishing 1-form. In particular, M needs to have $\chi (M)=0.$

As mentioned above, the existence of an eigenmap $u:(M,h) \rightarrow S^\ell$ for a metric $h \in [g]$ implies that h is extremal for some of the functionals $\lambda _k$ on [g]. Therefore, Theorem 1.6 and Corollary 1.7 have the following consequences for the existence of extremal metrics.

Corollary 1.9

Under the assumptions of Theorem 1.6, there is a neighbourhood U of [g] in $\mathcal {C},$ such that for any $c \in U,$ there is a representative $h \in c,$ such that (M, h) is extremal for some eigenvalue functional on c.

Corollary 1.10

Under the assumptions of Corollary 1.7, the set $\mathcal {E} \subset \mathcal {C}$ of conformal structures admitting extremal metrics for some eigenvalue functional on conformal classes is open and non-empty.

The proof of Theorem 1.6 is rather simple once the correct conformally invariant formulation of the assertion is found.

This is as follows. Let n be the dimension of M. Then a smooth map into a sphere is called n-harmonic, if it is a critical point of the n-energy

$$\begin{aligned} E_n[u]= \int _M |\mathrm{d}u|^n \mathrm{d}V_g, \end{aligned}$$

which is a conformally invariant functional. These are precisely the solutions of the equation

$$\begin{aligned} -{\text {div}}(|\nabla u|^{n-2} \nabla u)= |\nabla u|^n u. \end{aligned}$$

(1.11)

From (1.5) and (1.11), it is evident that an eigenmap defines an n-harmonic map, which has $\nabla u \ne 0$ everywhere. The crucial observation is that also the converse holds up to changing the metric conformally, see Lemma 3.21.

Therefore, we will be concerned with n-harmonic maps with nowhere vanishing derivative.

In order to deduce Corollary 1.7 from Theorem 1.6, it suffices to find a single non-trivial eigenmap $u :(M,g) \rightarrow S^1$ for some metric g. This turns out to be very easy using that M is a mapping torus.

In Sect. 2, we discuss the necessary preliminaries on n-harmonic maps and Laplace eigenvalues. Sect. 3 contains the proofs.

2 Preliminaries

First, we explain the notion of extremal metrics and its connection to eigenmaps.

2.1 Extremal Metrics for Eigenvalue Functionals

In presence of multiplicity, the functionals $\lambda _k$ are not differentiable, but only Lipschitz. However, it turns out that for any analytic deformation, left and right derivatives exist. Using this, El Soufi–Ilias introduced a notion of extremal metrics for these functionals.

Definition 2.1

[3, Definition 4.1] A metric g is called extremal for the functional $\lambda _k$ restricted to the conformal class [g] of g,  if for any analytic family of metrics $(g_t) \subset [g],$ with $g_0=g,$ and ${\text {vol}}(M,g_0)={\text {vol}}(M,g_t),$ we have

$$\begin{aligned} \left. \frac{\mathrm{d}}{\mathrm{d}t} \right| _{t=0^-} \lambda _k(g_t) \cdot \left. \frac{\mathrm{d}}{\mathrm{d}t} \right| _{t=0+} \lambda _k(g_t) \le 0. \end{aligned}$$

We have

Theorem 2.2

[3, Theorem 4.1] The metric g is extremal for some eigenvalue $\lambda _k$ on [g] if and only if there is a eigenmap $u :(M,g) \rightarrow S^\ell$ given by $\lambda _k(g)$-eigenfunctions and either $\lambda _{k-1}(g)<\lambda _k(g)$, or $\lambda _k(g) < \lambda _{k+1}(g).$

2.2 Background on n-Harmonic Maps

First of all, we need some background on the existence of n-harmonic maps. We call a map $u \in W^{1,n}(M,S^\ell )$weaklyn-harmonic, if it is a weak solution of

$$\begin{aligned} -{\text {div}}(|\nabla u|^{n-2}\nabla u) = |\nabla u|^n u. \end{aligned}$$

(2.3)

We assume that we have fixed a CW-structure on M,  and denote by $M^{(l)}$ its l-skeleton. Let $v :M \rightarrow S^\ell$ be a Lipschitz map, where $l<n=\dim M.$ Denote by $v^{(l)}$ the restriction of v to the l-skeleton of M. The l-homotopy type ofv is the homotopy type of $v^{(l)}.$

Theorem 2.4

[16, Theorem 3.4] There exists a weakly n-harmonic map $u :M \rightarrow S^\ell ,$ with well-defined l-homotopy type, which agrees with the l-homotopy type of v. Moreover, u minimizes the n-energy among all such maps.

We do not elaborate here on how the l-homotopy type is defined for maps in $W^{1,n}(M,S^\ell ).$ For our purposes, this is not necessary, since the map u is actually continuous.

Theorem 2.5

Let $u \in W^{1,n}(M,S^\ell )$ be a weakly n-harmonic map, which is a minimizer for its own l-homotopy type. There is a constant C depending on an upper bound on the n-energy of u,  and on the bounds of the sectional curvature and injectivity radius of M,  such that $\Vert u \Vert _{C^{1,\alpha }} \le C.$

Proof

Let $x \in M$ and $r>0$ be small enough. If $v \in W^{1,n}(B(x,r),S^\ell )$ with $u=v$ on $\partial B(x,r)$, we can consider the map $w \in W^{1,n}(M,S^\ell )$ given by u in $M \setminus B(x,r)$ and by v in B(x, r). It is shown in [12, Theorem 2.8] that the l-homotopy type of w agrees with the l-homotopy type of u. In particular, we need to have

$$\begin{aligned} \int _{B(x,r)} |\mathrm{d}u|^n \mathrm{d}V_g \le \int _{B(x,r)} |\mathrm{d}v|^n \mathrm{d}V_g, \end{aligned}$$

which means that u is a minimizingn-harmonic map. Therefore, the assertion follows e.g. from [9, Theorem 2.19]. $\square$

In particular, these estimates are uniform as g varies over a compact set of $\mathcal {R},$ as long as the energy stays bounded.

At points, in which we do not have a lack of ellipticity, we actually get higher regularity.

Theorem 2.6

A weakly n-harmonic map $u \in C^{1,\alpha }$ is smooth near points with $\nabla u \ne 0.$

This follows from standard techniques for quasilinear elliptic equations. For completeness, we give a proof in Sect. 3.1.

The main reason for the restrictive assumptions in item (ii) of Theorem 1.6 is that the above results do not imply that for a sequence $g_k \rightarrow g$ we can find a sequence of n-harmonic maps $u_k$ (w.r.t. $g_k$), such that $u_k \rightarrow u,$ for a given n-harmonic map u.

In the case of maps to the circle, this problem does not appear, thanks to

Theorem 2.7

[15, Theorem A] Up to rotations of $S^1,$n-harmonic maps $u :M \rightarrow S^1$ are unique in their homotopy class.

3 Proofs

3.1 Higher Regularity of n-Harmonic Maps

In this section, we give a proof of Theorem 2.6. We start with $W^{2,2}$-regularity. The proof follows using standard techniques, since under our assumptions the equation is of the form

$$\begin{aligned} - (L u)(x) - b(x) u(x)=0, \end{aligned}$$

(3.1)

with L a quasilinear operator, which is elliptic at u (as demonstrated in Lemma 3.14 below) and $b \in L^\infty .$

Lemma 3.2

Let $U \subset M$ be open and $u :(U,g) \rightarrow S^\ell$ be weakly n-harmonic. Assume that $u \in C^{1,\alpha }(U,S^\ell )$ with $\nabla u \ne 0$ everywhere in u. Then we have $u \in W^{2,2}_{loc}(U,S^\ell ).$

Proof

For simplicity, we focus on the case $g_{ij}=\delta _{ij}$ and denote the usual differential of u in Euclidean Space by Du. The general case follows along the same lines but with some more notation.

Take open subsets $W\subset \subset V \subset \subset U,$ and a cut-off function $\eta$ which is 1 in W,  and has $\text {supp }\eta \subset V.$ We show that $u \in W^{2,2}(W,S^\ell ).$ We use the test functions given by $\phi ^k=-D_s^{-h}(\eta ^2D_s^h u^k),$ where $D_s^h$ denotes the difference quotient operator in coordinate direction s,

$$\begin{aligned} D_s^h \phi (x)=\frac{1}{h}(\phi (x+h e_s)-\phi (x)). \end{aligned}$$

(3.3)

To handle notation, let us write

$$\begin{aligned} F^\alpha _k(D u)= |D u|^{n-2} \partial _\alpha u^k, \end{aligned}$$

(3.4)

and

$$\begin{aligned} G_k(u,D u) = |D u|^n u^k. \end{aligned}$$

(3.5)

Then we have

$$\begin{aligned} - \int _U F^\alpha _k(D u) \partial _\alpha D_s^{-h}(\eta ^2 D_s^h u^k) = - \int _U G_k(u, Du) D_s^{-h}(\eta ^2 D_s^h u^k), \end{aligned}$$

(3.6)

Note that this is well defined thanks to Hölder’s inequality. For the left-hand side of (3.6), we have

$$\begin{aligned} - \int _U F_k^\alpha (D u) \partial _\alpha D_s^{-h}(\eta ^2 D_s^h u^k) = \int _U D_s^h (F^\alpha _k(D u)) \partial _\alpha (\eta ^2 D_s^h u^k). \end{aligned}$$

(3.7)

We can write

$$\begin{aligned} \begin{aligned} D_s^h F^\alpha _k(D u)&= \frac{1}{h} \int _0^1 \frac{\mathrm{d}}{\mathrm{d}t} F^\alpha _k(D u + t h D_s^h D u)\mathrm{d}t \\&= \frac{1}{h} \int _0^1 \frac{\partial F^\alpha _k}{\partial q_\beta ^l} (D u + t h D_s^h D u)hD_s^h \partial _\beta u^l \mathrm{d}t \\&=\int _0^1 \frac{\partial F^\alpha _k}{\partial q_\beta ^l} (D u + t h D_s^h D u)\mathrm{d}t\, D_s^h \partial _\beta u^l \mathrm{d}t \\&=: \theta _{kl}^{\alpha \beta }(D u) D_s^h \partial _\beta u^l. \end{aligned} \end{aligned}$$

(3.8)

Note that this is well defined pointwise, since $u \in C^{1,\alpha }.$ The condition $|Du|\ge c >0$ implies that $\theta ^{\alpha \beta }_{kl}$ are uniformly super strongly elliptic for $h \ll 1,$ as demonstrated below. Since the coefficients $\theta$ are uniformly super strongly elliptic, we have

$$\begin{aligned} \int _U \eta ^2 |D_s^h Du|^2 \le C \int _U \eta ^2 \theta _{kl}^{\alpha \beta }(D u) (D^h_s\partial _\alpha u^k) (D_s^h \partial _\beta u^l). \end{aligned}$$

(3.9)

Moreover, since $\theta$ and $|D \eta |$ are bounded, we can estimate

$$\begin{aligned} \begin{aligned} \left| \int _U \theta _{kl}^{\alpha \beta } (D_s^h \partial _\beta u^l) (D_s^h u^k) \eta \partial _\alpha \eta \right|&\le C \int _U |D_s^h Du| |D_s^h u|\eta \\&\le C \varepsilon \int _U \eta ^2 |D_s^h Du|^2 + \frac{C}{\varepsilon } \int _{V} |D_s^h u|^2 \\&\le C \varepsilon \int _U \eta ^2 |D_s^h Du|^2 + \frac{C}{\varepsilon } \int _U |Du|^2, \end{aligned} \end{aligned}$$

(3.10)

where we have used Young’s inequality and $u \in W^{1,2}.$ Combining the last two estimates with (3.6) and (3.7), we find that we can choose $\varepsilon$ sufficiently small so that

$$\begin{aligned} \int _U \eta ^2 |D_s^h Du|^2 \le C \int _U |Du|^2 + C \left| \int _U G_k(u, Du) D_s^{-h}(\eta ^2 D_s^h u^k)\right| . \end{aligned}$$

(3.11)

To estimate the last summand above, we note that $u,|D u| \in L^\infty ,$ implies $G_k(u,D u) \in L^\infty ,$ and hence

$$\begin{aligned} \begin{aligned} \left| \int _U G_k(u, Du) D_s^{-h}(\eta ^2 D_s^h u^k) \right|&\le C \int _U |D_s^{-h} (\eta ^2 D_s^h u^k)| \\&\le \frac{C}{\varepsilon } {\text {vol}}(U) + C \varepsilon \int _U |D_s^{-h}(\eta ^2 D_s^h u^k)|^2 \\&\le \frac{C}{\varepsilon } + C \varepsilon \ \int _U |D(\eta ^2 D_s^h u^k)|^2 \\&\le \frac{C}{\varepsilon }\left( 1 + \int _U |Du|^2\right) + C \varepsilon \int _U \eta ^2 |D_s^h Du^k|^2. \end{aligned} \end{aligned}$$

(3.12)

For $\varepsilon$ sufficiently small, we can absorb the last term, and find

$$\begin{aligned} \int _{V} |D_s^h Du|^2 \le \int _U \eta ^2 |D_s^h Du|^2 \le \frac{C}{\varepsilon }(1 + \int _U |Du|^2). \end{aligned}$$

(3.13)

Thus $u \in W^{2,2}_{loc}(U,S^\ell ).$$\square$

We still need to justify that the coefficients $\theta ^{\alpha \beta }_{kl}$ are uniformly super strongly elliptic.

Lemma 3.14

There is $h_0>0$ depending on $\Vert Du \Vert _{C^{0,\alpha }}$ such that we have $\theta _{kl}^{\alpha \beta } A^k_\alpha A^l_\beta \ge \nu |A|^2,$ for any h with $|h| \le h_0$ and $\nu =\nu (c),$ where $|Du|^2 \ge c.$

Proof

We have

$$\begin{aligned} \frac{\partial F_k^\alpha }{\partial q_\beta ^l}(q)=|q|^{n-4} \left( |q|^2 \delta ^{\alpha \beta }\delta _{kl} + (n-2) q_\alpha ^k q_\beta ^l\right) . \end{aligned}$$

(3.15)

Thus, it is not very hard to see that

$$\begin{aligned} \begin{aligned} \frac{\partial F_k^\alpha }{\partial q_\beta ^l}(q) A_\alpha ^k A_\beta ^l&= |q|^{n-4} \left( |q|^2 \delta ^{\alpha \beta }\delta _{kl} A_\alpha ^k A_\beta ^l + (n-2) q_\alpha ^k q_\beta ^l A_\alpha ^k A_\beta ^l\right) \\&\ge |q|^{n-2}|A|^2 \\&\ge 2 \nu |A|^2, \end{aligned} \end{aligned}$$

(3.16)

as long as $|q|^2 \ge (2 \nu )^{2/(n-2)}.$ Since $Du \in C^{0,\alpha },$ and $|D u|^2 \ge c,$ we can choose $h_0 \ll 1,$ such that $|(1-t)D u (x+he_s) + t D u(x)|^2 \ge c/2,$ for all x,  and $|h| \le h_0.$ Clearly, this implies

$$\begin{aligned} \begin{aligned} \theta _{kl}^{\alpha \beta } (D u)(x) A_\alpha ^k A_\beta ^l&= \int _0^1 \frac{\partial F_k^\alpha }{\partial q_\beta ^l}((1-t)D u (x+he_s) + t D u(x)) A_\alpha ^k A_\beta ^l \mathrm{d}t \\&\ge \int _0^1 \nu |A|^2 \mathrm{d}t \\&\ge \nu |A|^2, \end{aligned} \end{aligned}$$

for $\nu = c^{(n-2)/2}/2$. $\square$

In the next step, we derive the equation for $\partial _\alpha u^k$ and apply Schauder estimates to gain higher regularity. In particular, this completes the proof of Theorem 2.6

Lemma 3.17

Under the above assumptions, the function u is smooth.

Proof

Write

$$\begin{aligned} \vartheta _{kl}^{\alpha \beta } = \frac{\partial F^\alpha _k}{\partial q_\beta ^l}. \end{aligned}$$

(3.18)

By the calculation above, these coefficients are uniformly super strongly elliptic at u. We test the equation for $u^k$ with $\partial _\alpha \phi ^k$ for some test function $\phi$ and integrate by parts in order to find

$$\begin{aligned} \int _U \vartheta _{kl}^{\alpha \beta } (\nabla u) \partial _{\beta \gamma } u^l \partial _\alpha \phi ^k = \int _U \partial _\gamma G_k(u, \nabla u) \phi ^k. \end{aligned}$$

(3.19)

In other words, $v=\partial _\gamma u$ is a weak solution to

$$\begin{aligned} -{\text {div}}(\vartheta (Du) v) = \partial _\gamma G(u,Du). \end{aligned}$$

(3.20)

Since $|D u|^2 \ge c >0,$ the right-hand side of this equation is in $C^{k,\alpha },$ once we have $u \in C^{k+1,\alpha }.$ In this case, the left-hand side has coefficients in $C^{k,\alpha };$ thus, it follows that $v \in C^{k+1,\alpha }$ and thus $u \in C^{k+2, \alpha }.$ Since we know $u \in C^{1,\alpha },$ we can start this bootstrap argument at $k=0,$ and get $u \in C^{\infty }.$$\square$

3.2 Proofs of Main Results

We start with the following simple but crucial observation.

Lemma 3.21

Let $u :(M,g) \rightarrow S^\ell$ be a smooth n-harmonic map with $\mathrm{d}u \ne 0$ everywhere. Then there is metric $g'$ conformal to g,  such that $u :(M,g') \rightarrow S^\ell$ is an eigenmap.

Proof

Define $g'=|\mathrm{d}u|_g^2 g.$ Since we assumed $\mathrm{d}u \ne 0$ everywhere, this defines a smooth metric, which is conformal to g. Then $|\mathrm{d}u|_{g'}^2=|\mathrm{d}u_g|^{-2} |\mathrm{d}u|_g^2=1.$ Finally, u solves

$$\begin{aligned} -{\text {div}}_g (|\mathrm{d}u|_g^{n-2} \nabla u)= |\mathrm{d}u|_g^n u, \end{aligned}$$

which can also be written as

$$\begin{aligned} \Delta _{g'} u = -\frac{1}{|\mathrm{d}u|_g^n}{\text {div}}_g (|\mathrm{d}u|_g^{n-2} \nabla u)= u, \end{aligned}$$

and hence $u :(M,g') \rightarrow S^\ell$ is an eigenmap. $\square$

In order to prove Theorem 1.6, it now suffices to show that metrics close to the initial metric g on M also admit smooth n-harmonic maps with nowhere vanishing derivative.

Proof of Theorem 1.6 (i)

Let $u :(M,g) \rightarrow S^1$ be an eigenmap and assume that the assertion of the theorem was not correct. This means that any neighbourhood $U \subset \mathcal {C}$ of [g] contains a conformal class which does not contain any representative which admits an eigenmap to $S^1$. Let $U_k\subset \mathcal {R}$ be a sequence of open neighbourhoods of g with $\cap _{k \in \mathbb {N}} U_k = \{g\}$. (Such a sequence exists since the $C^\infty$-topology on $\mathcal {R}$ is first countable and Hausdorff.) Denote by $\pi :\mathcal {R} \rightarrow \mathcal {C}$ the quotient map and observe that this is an open map. In particular, the sets $\pi (U_k) \subset \mathcal {C}$ are open and we can find $g_k \in U_k$ such that no metric in $[g_k]$ admits an eigenmap to $S^1$. By Lemma 3.21, this implies that $g_k$ itself cannot admit a nowhere vanishing n-harmonic map to $S^1$.

We now plan to use Theorem 2.4 to obtain weakly n-harmonic maps $u_k :(M,g_k) \rightarrow S^1$ which are close to u for k sufficiently large. By assumption, the $u_k$ have some point $x_k$ with $\mathrm{d}u_k(x_k)=0$. This forces u to have a critical point as well, which gives the desired contradiction.

More precisely, we apply Theorem 2.4 to $u :(M,g_k) \rightarrow S^1$ and obtain n-harmonic representatives $u_k :(M,g_k) \rightarrow S^1$ of [u]. If $\mathrm{d}u_k \ne 0$ everywhere, Theorem 2.6 implies that $u_k$ is a smooth n-harmonic map from $(M,g_k)$ to $S^1$ with nowhere vanishing derivative contradicting the construction of $g_k$ in the preceding paragraph. Therefore, we can find $x_k \in M$ such that $\mathrm{d}u_k(x_k)=0$. Since $\dim (M) \ge 3$ and $S^1 \simeq K(\mathbb {Z},1),$ we have that $w \simeq u$ if and only if their l-homotopy type agrees for some $l \ge 2.$ In particular, we have that

$$\begin{aligned} \int _M |\mathrm{d}u_k|^n \mathrm{d}V_{g_k} \le \int _M |\mathrm{d}u|^n \mathrm{d}V_{g_k} \le C \int _M |\mathrm{d}u|^n \mathrm{d}V_g, \end{aligned}$$

so that we are in the position to apply Theorem 2.5.

By taking a subsequence if necessary, we may assume that $x_k \rightarrow x.$ Thanks to Theorem 2.5 and the compact embedding $C^{1,\alpha }(M) \hookrightarrow C^{1,\beta }(M)$ for $\beta <\alpha ,$ we can extract a further subsequence, such that $u_k \rightarrow v$ in $C^{1,\beta }(M,g).$ We have

$$\begin{aligned} \int _M |\mathrm{d}v|_g \mathrm{d}V_g&= \lim _{k \rightarrow \infty } \int _M |\mathrm{d}v|_{g_k} \mathrm{d}V_{g_k} \\&\le \lim _{k \rightarrow \infty } \left( \int _M |\mathrm{d}u_k|_{g_k} \mathrm{d}V_{g_k} + \int _M \left| |\mathrm{d}v|_{g_k} - |\mathrm{d}u_k|_{g_k} \right| \mathrm{d}V_{g_k} \right) \\&\le \lim _{k \rightarrow \infty }\left( \int _M |\mathrm{d}w|_{g_k} \mathrm{d}V_{g_k} + C d_{C^{1, \beta }(M,g_k)} (v,u_k) \right) \\&\le \lim _{k \rightarrow \infty } \int _M |\mathrm{d}w|_{g_k} \mathrm{d}V_{g_k} + \lim _{k \rightarrow \infty } C d_{C^{1, \beta }(M,g)} (v,u_k) \\&= \int _M |\mathrm{d}w|_g \mathrm{d}V_g, \end{aligned}$$

for any $w \simeq u.$ It follows that v is n-harmonic and homotopic to u. Thus, it follows from Theorem 2.7 that there is $A \in SO(2),$ such that $A\circ v=u.$ Then $A \circ u_k \rightarrow u$ in $C^{1,\beta }(M).$ It follows that

$$\begin{aligned} |\mathrm{d}u(x)| \le \limsup _{k \rightarrow \infty } C d(x,x_k)^\beta =0, \end{aligned}$$

contradicting the assumption on u. $\square$

In order to adapt the strategy from above for more general situations, we need to understand whether there exist eigenmaps $u :(M,g) \rightarrow S^\ell ,$ which can be approximated through n-harmonic maps for any sequence of metrics $g_k \rightarrow g.$

This is precisely what we do now for product metrics $g_{st.}+g_N$ on $S^\ell \times N.$ The natural candidate here is the projection map onto $S^\ell .$ In what follows n will denote the dimension of N,  so that the dimension of $N \times S^\ell$ is $n+l.$

Proposition 3.22

Let $g=g_N + g_{st.}$ be a product metric on $N \times S^\ell ,$ with $g_{st.}$ the round metric of curvature 1 on $S^\ell .$ The projection $u :N \times S^\ell \rightarrow S^\ell$ onto the second factor is the unique minimizer for the $(n+l)$-energy in its l-homotopy class up to rotations of $S^\ell .$

Proof

Let $v :N \times S^\ell \rightarrow S^\ell$ be a Lipschitz map whose restriction to the l-skeleton of $N \times S^\ell$ is homotopic to the restriction of the projection $N \times S^\ell \rightarrow S^\ell$ to the $\ell$-skeleton. We want to estimate

$$\begin{aligned} \int _{N \times S^\ell } |\mathrm{d}v|_g^{n+\ell } \mathrm{d}V_g \end{aligned}$$

(3.23)

from below.

We have

$$\begin{aligned} \begin{aligned} \int _{N \times S^\ell }&|\mathrm{d}v|_g^{n+\ell } \mathrm{d}V_g \\&= \int _N \int _{S^\ell } (|\nabla ^N v|^2 + |\nabla ^{S^\ell } v|^2)^{(n+\ell )/2} (x, \theta ) \mathrm{d}\theta \mathrm{d}x \\&\ge \int _N \int _{S^\ell } |\nabla ^{S^\ell }v|^{n+\ell }(x, \theta ) \mathrm{d}\theta \mathrm{d}x \\&\ge (\ell +1) \omega _{\ell +1})^{-n/\ell } \int _N \left( \int _{S^\ell } |\nabla ^{S^\ell }v|^\ell (x , \theta ) \mathrm{d}\theta \right) ^{(n+\ell )/\ell } \mathrm{d}x, \end{aligned} \end{aligned}$$

(3.24)

where we have used Hölder’s inequality in the last step. Equality holds in the above inequalities if and only if $|\nabla ^M v|^2=0$ and $|\nabla ^{S^\ell }v|^2= const.$

In order to estimate the remaining integral in the last line of (3.24), we use that the maps $v(x, \cdot ) :S^\ell \rightarrow S^\ell$ have degree 1. This can be seen by inspecting the l-homotopy type of v: If we endow $S^\ell$ with the CW-structure consisting of a single 0- and a single $\ell$-cell, we have $( N \times S^\ell )^{(\ell )} = N^{(\ell )} \times \{\theta _0\} \cup \{x_0\} \times S^\ell =N^{(\ell )} \vee S^\ell$ with $\theta _0 \in S^\ell$ and $x_0 \in N$ corresponding to the 0-cells. The projection onto $S^\ell$ restricts to the map $N^{(\ell )} \vee S^\ell \rightarrow S^\ell$ that collapses the first summand and is the identity on $S^\ell .$ In particular, we find that for any v,  such that $v^{(\ell )}$ is homotopic to the map described above, the degree of $v(x_0,\cdot ) :S^\ell \rightarrow S^\ell$ equals 1. Since N is connected, $v(x,\cdot ) \simeq v(x_0,\cdot )$ for any x,  thus $\deg v(x,\cdot )=1$ for any $x \in N.$

This implies that

$$\begin{aligned} \int _{S^\ell } |\nabla ^{S^\ell }v|^l(x , \theta ) \mathrm{d}\theta \ge (\ell +1)\omega _{\ell +1}\ell ^{\ell /2} |\deg v(x, \cdot ) | = (\ell +1)\omega _{\ell +1}l^{\ell /2}. \end{aligned}$$

(3.25)

Here, equality holds if and only if $v(x, \cdot )$ is conformal. Combining (3.24) and (3.25), we find

$$\begin{aligned} \int _{N \times S^\ell } |\mathrm{d}v|_g^{n+\ell } \mathrm{d}V_g \ge {\text {vol}}(N)(\ell +1)\omega _{\ell +1}l^{(n+\ell )/2}, \end{aligned}$$

(3.26)

with equality if and only if $|\nabla ^M v|^2=0,$ and $|\nabla ^{S^\ell }v|^2= const.,$ and $v(x, \cdot )$ is conformal. It follows in this case that $v(x,\theta )= \tilde{v} (\theta )$ with $\tilde{v} :S^\ell \rightarrow S^\ell$ of degree 1. Observe that $u :M \times S^\ell \rightarrow S^\ell$ realizes the equality in (3.26). Therefore,

$$\begin{aligned} \inf _v \int _{N \times S^\ell } |\mathrm{d}v|^{n+\ell } \mathrm{d}V_g = {\text {vol}}(N)(\ell +1)\omega _{\ell +1}l^{(n+\ell )/2}, \end{aligned}$$

(3.27)

where the infimum is taken over all Lipschitz maps v having the l-homotopy of u. In particular, by the equality discussion above, minimizers need to be $(n+l)$-harmonic maps $v(x,\theta )= \tilde{v} (\theta )$, with $|\nabla v|^2=const.$ Therefore, $\tilde{v}$ defines a harmonic selfmap of $S^\ell$ with constant density. Since $\tilde{v}$ is non-trivial, it follows that $|\nabla \tilde{v}|^2 \ge \lambda _1(S^\ell )=l.$ Consequently, equality in (3.26) is only achieved by maps of the form $A \circ u,$ with $A \in O(l+1).$$\square$

Using Proposition 3.22 instead of Theorem 2.7, assertion (ii) of Theorem 1.6 follows along the same lines as assertion (i).

Proof of Corollary 1.7

Let $f :M \rightarrow S^1$ be a submersion. Since M is compact this is a proper submersion. Moreover, f has to be surjective, since otherwise M would be contractible. It follows by Ehresmann’s lemma that $f :M \rightarrow S^1$ is a fibre bundle, $F \rightarrow M \rightarrow S^1,$ with F a smooth $(n-1)$-dimensional manifold. As a consequence, there is a diffeomorphism $\phi :F \rightarrow F,$ such that M is obtained as the mapping torus corresponding to $\phi ,$ i.e.

$$\begin{aligned} M \cong \left( F \times [0,1] \right) / (x,0) \sim (\phi (x),1). \end{aligned}$$

Choose a metric $g_0$ on F,  which is invariant under $\phi .$ We claim that the metric $g_1=g_0 + \mathrm{d}t^2$ defined on $F \times [0,1]$ descends to a smooth metric g on M. Clearly, $g_1$ descends to a metric g on M,  we only need to check that it is smooth. This is clear near all points (x, t) with $t \ne 0,1.$ We have coordinates with values in $F \times (- \varepsilon , \varepsilon )$ near the $t=0$-slice as follows:

$$\begin{aligned} (x,t) \mapsto {\left\{ \begin{array}{ll} (x,t-1) &{} \text {if}\ \ t \le 1 \\ (\phi (x),t) &{} \text {if}\ \ t >0. \end{array}\right. } \end{aligned}$$

(3.28)

In these coordinates, g is given by $g_0 + \mathrm{d}t^2,$ since $g_0$ is $\phi$-invariant.

It remains to show that (M, g) admits an eigenmap. Define $u :F \times [0,1] / (x,0) \sim (f(x),1) \rightarrow S^1$ by $(x,t) \mapsto t.$ With respect to g this is a Riemannian submersion. Moreover, it follows from (3.28) that u has totally geodesic fibres. Thus u is an eigenmap. $\square$