# Refined asymptotics of the Teichmüller harmonic map flow into general targets

- 633 Downloads
- 1 Citations

## Abstract

The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. Given a weak solution of the flow that exists for all time \(t\ge 0\), we find a sequence of times \(t_i\rightarrow \infty \) at which the flow at different scales converges to a collection of branched minimal immersions with no loss of energy. We do this by developing a compactness theory, establishing no loss of energy, for sequences of almost-minimal maps. Moreover, we construct an example of a smooth flow for which the image of the limit branched minimal immersions is disconnected. In general, we show that the necks connecting the images of the branched minimal immersions become arbitrarily thin as \(i\rightarrow \infty \).

## Mathematics Subject Classification

53C44 53C43 58E20 53A10## 1 Introduction

*E*(

*u*) of a map

*u*from a smooth closed oriented Riemannian surface (

*M*,

*g*) of genus \(\gamma \) to a smooth compact Riemannian manifold \(N=(N,G)\) of any dimension. Here the energy

*E*(

*u*) is defined by

*g*to vary, and restrict

*g*to have fixed constant curvature 1, 0 or \(-1\) (and fixed area), then we end up with the Teichmüller harmonic map flow introduced in [11]. This is the flow given, for fixed parameter \(\eta >0\), by

*u*, \(P_g\) is the \(L^2\)-orthogonal projection from the space of quadratic differentials on (

*M*,

*g*) onto the space of

*holomorphic*quadratic differentials, and

Assume for the moment that *M* has genus \(\gamma \ge 2\), so *g* flows within the space \({\mathcal M}_{-1}\) of hyperbolic metrics (i.e. Gauss curvature everywhere \(-1\)). In the case that the length \(\ell (g(t))\) of the shortest closed geodesic in the domain (*M*, *g*(*t*)) is bounded from below by some \(\varepsilon >0\) uniformly as \(t\rightarrow \infty \) (which corresponds to no collar degeneration, even as \(t\rightarrow \infty \)) it was proved in [11] that the maps *u*(*t*) subconverge, after reparametrisation, to a branched minimal immersion (or a constant map) with the same action on \(\pi _1\) as the initial map \(u_0\). The same result also applies in the case of *M* being a torus with the length \(\ell (g(t))\) of the shortest closed geodesic bounded away from zero, as follows from the work in [2] when combined with the Poincaré inequality on nondegenerate tori proved in [14] (see also Remark 3.4).

On the other hand, on surfaces of genus \(\gamma \ge 2\) and in the case that \(\liminf _{t\rightarrow \infty }\ell (g(t))=0\), an initial description of the asymptotics of the flow was given in [13]. Loosely speaking, it was shown that the surface (*M*, *g*(*t*)) will degenerate into finitely many lower genus surfaces, with the map *u*(*t*) subconverging (modulo bubbling) to branched minimal immersions (or constant maps) on each of these components. It is convenient for us here to split the main result in [13] into an initial result extracting a sequence of times \(t_i\rightarrow \infty \) at which the maps \((u_i,g_i)=(u(t_i),g(t_i))\) are a * sequence of almost-minimal maps*, and a separate compactness result for such sequences.

### Definition 1.1

*M*, a closed Riemannian manifold (

*N*,

*G*), and a pair of sequences \(u_i:M\rightarrow N\) of smooth maps and \(g_i\) of metrics on

*M*with fixed constant curvature and fixed area, we say that \((u_i,g_i)\) is a

*sequence of almost-minimal maps*if \(E(u_i,g_i)\) is uniformly bounded and

For a justification of this terminology, note that using the construction in [11]—i.e. viewing the harmonic map energy *E* as a functional on the space of maps and constant curvature metrics modulo diffeomorphisms isotopic to the identity, equipped with the natural analogue of the Weil–Petersson metric—the gradient of *E* at points represented by \((u_i,g_i)\) converges to zero, while critical points are branched minimal immersions (see [11]). We are primarily concerned with the case that *M* has genus at least two, in which case each \(g_i\) will lie in the space \({\mathcal M}_{-1}\) of hyperbolic metrics.

### Proposition 1.2

Given an oriented closed surface *M*, a closed Riemannian manifold (*N*, *G*), and a smooth flow (*u*, *g*) solving (1.1) for which \(\liminf \nolimits _{t\rightarrow \infty }\ell (g(t))=0\), there exists a sequence \(t_i\rightarrow \infty \) such that \(\lim \nolimits _{i\rightarrow \infty }\ell (g(t_i))=0\) and \((u(t_i),g(t_i))\) is a sequence of almost-minimal maps.

This standard observation is proved, for completeness, in Sect. 3. The sequence found in this proposition can be analysed with the following result, which is effectively what is proved in [13] (modulo minor adjustments to Sect. 3 of that paper).

### Theorem 1.3

(Content from [13]) Suppose we have an oriented closed surface *M* of genus \(\gamma \ge 2\), a closed Riemannian manifold (*N*, *G*), and a sequence \((u_i,g_i)\) of almost-minimal maps in the sense of Definition 1.1, for which \(\lim _{i\rightarrow \infty }\ell (g_i)=0\).

*k*punctures (i.e. a closed Riemann surface \(\hat{\Sigma }\) with complex structure \(\hat{c}\), possibly disconnected, that is then punctured 2

*k*times to give a Riemann surface \(\Sigma \) with

*c*the restricted complex structure, which is then equipped with a conformal complete hyperbolic metric

*h*) such that the following holds.

- 1.The surfaces \((M,g_i,c_i)\) converge to the surface \((\Sigma ,h,c)\) by collapsing
*k*simple closed geodesics \(\sigma ^{j}_{i}\) in the sense of Proposition A.2 from the appendix; in particular there is a sequence of diffeomorphisms \( f_i:\Sigma \rightarrow M{\setminus }\cup _{j=1}^k \sigma ^{j}_{i}\) such thatwhere \(c_i\) denotes the complex structure of \((M,g_i)\).$$\begin{aligned} f_i^*g_i\rightarrow h \text { and } f_i^*c_i\rightarrow c \text { smoothly locally, } \end{aligned}$$ - 2.
The maps \(U_i:=u_i\circ f_i\) converge to a limit \(u_\infty \) weakly in \(W^{1,2}_{loc}(\Sigma )\) and weakly in \(W_{loc}^{2,2}(\Sigma {\setminus } S)\) as well as strongly in \(W_{loc}^{1,p}(\Sigma {\setminus } S)\), \(p\in [1,\infty )\), away from a finite set of points \(S\subset \Sigma \) at which energy concentrates.

- 3.
The limit \(u_\infty :\Sigma \rightarrow N\) extends to a smooth branched minimal immersion (or constant map) on each component of the compactification \((\hat{\Sigma },\hat{c})\) of \((\Sigma ,c)\) obtained by filling in each of the 2

*k*punctures.

In this paper, we take this analysis of the asymptotics of \((u_i,g_i)\) and we refine it in several ways. First, after passing to a further subsequence, we extract all bubbles that can develop. What is well understood is that we can extract bubbles at each of the points in *S* (where possibly multiple bubbles can develop). In what follows we will call these bubbles \(\{\omega _k\}\). Our first task is to isolate a new set of bubbles, called \(\{\Omega _j\}\) below, that are disappearing into the 2*k* punctures found in Theorem 1.3, or equivalently (as we describe below), being lost down the one or more collars that degenerate in the domain \((M,g_i)\) as \(i\rightarrow \infty \).

Having extracted the complete set of bubbles, we show that the chosen subsequence enjoys a no-loss-of-energy property in which the limit \(\lim \nolimits _{i\rightarrow \infty }E(u_i,g_i)\) is precisely equal to the sum of the energies (or equivalently areas) of the branched minimal immersions found in Theorem 1.3 and the new branched minimal immersions obtained as bubbles. A special case of what we prove below in Theorem 1.11, combined with existing theory, is the following result. (Recall that \(\delta \text {-thick}(M,g)\) consists of all points in *M* at which the injectivity radius is at least \(\delta \). Its complement is \(\delta \text {-thin}(M,g)\).)

### Theorem 1.4

Showing that no loss of energy occurs in intermediate regions around the bubbles developing at points in *S* is standard, following in particular the work of Ding and Tian [3] we describe in a moment, although one could also use energy decay estimates of the form we prove in this paper. However, showing that no energy is lost near the 2*k* punctures, away from where the bubbles develop, is different, and a key ingredient is the Poincaré estimate for quadratic differentials discovered in [14], which is applied globally, not locally where the energy is being controlled. In this step we exploit the smallness of \(P_g(\Phi (u,g))\) that holds for almost-minimal maps. That this is essential is demonstrated by the work of Parker [8] and Zhu [23], which established that energy *can* be lost along ‘degenerating collars’ in general sequences of harmonic maps from degenerating domains.

The following is the foundational compactness result when the domain is fixed, cf. [3, 7, 9, 18, 19, 20].

### Theorem 1.5

Suppose \((\Upsilon ,g_0)\) is a fixed surface, possibly noncompact, possibly incomplete, and let \(u_i\) be a sequence of smooth maps into (*N*, *G*) from either \((\Upsilon ,g_0)\), or more generally from a sequence of subsets \(\Upsilon _i\subset \Upsilon \) that exhaust \(\Upsilon \). Suppose that \(E(u_i)\le E_0\) and that \(\Vert \tau _{g_0}(u_i)\Vert _{L^2}\rightarrow 0\) as \(i\rightarrow \infty \). Then there is a subsequence for which the following holds true.

*S*, a bubble tree develops in the following sense. After picking local isothermal coordinates centred at the given point in

*S*, there exist a finite number of nonconstant harmonic maps \(\omega _j:S^2\rightarrow (N,G)\), for \(j\in \{1,\ldots , J\}\), \(J\in {\mathbb N}\) (so-called bubbles) which we view as maps from \({\mathbb R}^2\cup \{\infty \}\) via stereographic projection, and sequences of numbers \(\lambda _i^j\downarrow 0\) and coordinates \(a^j_i\rightarrow 0\in {\mathbb R}^2\), such that

### Remark 1.6

We note that the proof of the first part of Theorem 1.4 (virtually) immediately follows from Theorems 1.3 and 1.5: Away from *S* we can combine the strong \(W^{1,2}\)-convergence of the maps with the convergence of the metrics. To analyse the maps \(U_i=u_i\circ f_i\) near points in *S* we then apply Theorem 1.5 on small geodesic balls \(B_{r}^{f_i^*g_i}(p)\subset (\Sigma , f_i^*g_i)\), which are of course isometric to one another provided \(r>0\) is chosen sufficiently small as the metrics \(g_i\) are all hyperbolic. Finally, the convergence of the metrics allows us to relate the \(\delta \text {-thick}\) part of \((\Sigma ,f_i^*g_i)\) to the \(\delta \text {-thick}\) part of \((\Sigma ,h)\), compare [13, Lemma A.7], as well as the geodesic balls \(B_{r}^{f_i^*g_i}(p)\) in \((\Sigma , f_i^*g_i)\) to geodesic balls in \((\Sigma ,h)\). This completes the proof of the first part of Theorem 1.4.

### Remark 1.7

To do more, we must recall more about the structure of sequences of degenerating hyperbolic metrics, and in particular we need the precise description of the metrics \(g_i\) near to the geodesics \(\sigma _i^j\) of Theorem 1.3 given by the Collar Lemma A.1 in the appendix. In particular, for \(\delta \in (0,\mathop {\mathrm {arsinh}}\nolimits (1))\) sufficiently small, the \(\delta \text {-thin}\) part of \((M,g_i)\) is isometric to a finite disjoint union of cylinders \(\mathcal {C}^{\delta ,j}_i:=(-X_\delta (\ell ^j_i),X_\delta (\ell ^j_i))\times S^1\) with the metric from Lemma A.1; each cylinder has a geodesic \(\sigma _i^j\) at the centre, with length \(\ell ^j_i\rightarrow 0\) as \(i\rightarrow \infty \). These initial observations motivate us to analyse in detail almost-harmonic maps from cylinders.

### Definition 1.8

When we apply Theorem 1.5 in the case that \((\Upsilon ,g_0)={\mathbb R}\times S^1\) is the cylinder with its standard flat metric, then we say that the maps \(u_i\) *converge to a bubble branch*, and extract bubbles \(\{\Omega _j\}\) as follows. First we add all the bubbles \(\{\omega _j\}\) to the list \(\{\Omega _j\}\). In the case that \(u_\infty :{\mathbb R}\times S^1\rightarrow N\) is nonconstant, we view it (via a conformal map of the domain) as a harmonic map from the twice punctured 2-sphere, remove the two singularities (using the Sacks–Uhlenbeck removable singularity theorem [17]) to give a smooth nonconstant harmonic map from \(S^2\), and add it to the list \(\{\Omega _j\}\). We say that \(u_i\) converges to a *nontrivial* bubble branch if the collection \(\{\Omega _j\}\) is nonempty.

We use the term ‘bubble branch’ alone to informally refer to the collection of bubbles together with the limit \(u_\infty \).

In the present paper we prove a refinement of the above convergence to a bubble branch. To state this result, we shall use the following notations: For \(a<b\), define \({\mathscr {C}}(a,b):=(a,b)\times S^1\) to be the finite cylinder which will be equipped with the standard flat metric \(g_0=ds^2+d\theta ^2\) unless specified otherwise. For \(\Lambda >0\) we write for short \({\mathscr {C}}_\Lambda ={\mathscr {C}}(-\Lambda ,\Lambda )\). Furthermore, given sequences \(a_i\) and \(b_i\) of real numbers we write \(a_i\ll b_i\) if \(a_i<b_i\) for all \(i\in {\mathbb N}\) and \(b_i-a_i\rightarrow \infty \) as \(i\rightarrow \infty \).

### Theorem 1.9

*i*, there exist a finite number of sequences \(s_i^m\) (for \(m\in \{0,\ldots ,\bar{m}\}\), \(\bar{m}\in {\mathbb N}\)) with \(-X_i=: s_i^0\ll s_i^1\ll \cdots \ll s_i^{\bar{m}}:= X_i\) such that the following holds true.

- 1.
For each \(m\in \{1,\ldots ,\bar{m}-1\}\) (if nonempty) the translated maps \(u_i^m(s,\theta ):=u_i(s+s_i^m,\theta )\) converge to a nontrival bubble branch in the sense of Definition 1.8.

- 2.The
*connecting cylinders*\({\mathscr {C}}(s_i^{m-1}+\lambda ,s_i^m-\lambda )\), \(\lambda \) large, are mapped near curves in the sense thatfor each \(m\in \{1,\ldots ,\bar{m}\}\).$$\begin{aligned} \lim _{\lambda \rightarrow \infty }\limsup _{i\rightarrow \infty }\sup _{s\in (s_i^{m-1}+\lambda ,s_i^m-\lambda )}\mathop {{\mathrm {osc}}}\limits (u_i;\{s\}\times S^1)=0, \end{aligned}$$(1.8) - 3.If we suppose in addition that the Hopf-differentials tend to zerothen there is no loss of energy on the connecting cylinders \({\mathscr {C}}(s_i^{m-1},s_i^m)\) in the sense that for each \(m\in \{1,\ldots ,\bar{m}\}\) we have$$\begin{aligned} \Vert \Phi (u_i)\Vert _{L^1({\mathscr {C}}_{X_i})}\rightarrow 0 \end{aligned}$$(1.9)$$\begin{aligned} \lim _{\lambda \rightarrow \infty }\limsup _{i\rightarrow \infty } E(u_i; {\mathscr {C}}(s_i^{m-1}+\lambda ,s_i^m-\lambda ))=0. \end{aligned}$$(1.10)

### Definition 1.10

In the setting of Theorem 1.9, we abbreviate the conclusions of parts 1 and 2 by saying that the maps \(u_i\) *converge to a full bubble branch*. In the case that (1.10) also holds (i.e. the conclusion of part 3) we say that the maps \(u_i\) *converge to a full bubble branch with no loss of energy*.

Returning to the observations of Remark 1.7, we note that the length of each of the cylinders \(\mathcal {C}^{\delta ,j}_i\) is converging to infinity, and that any fixed length portion of either end of any of these cylinders will lie within the \(\hat{\delta }\text {-thick}\) part of \((M,g_i)\) for some small \(\hat{\delta }\in (0,\delta )\), and thus be captured by the limit \(u_\infty \) from Theorem 1.3. Our main no-loss-of-energy result can therefore be stated as the following result about the limiting behaviour on the middle of the collars, which constitutes our main theorem.

### Theorem 1.11

In the setting of Theorem 1.3, we fix \(j\in \{1,\ldots ,k\}\) in order to analyse the *j*th collar surrounding the geodesic \(\sigma ^j_i\). Now that *j* is fixed, we drop it as a label for simplicity. Thus we consider the collar \(\mathcal {C}(\ell _i)=(-X(\ell _i),X(\ell _i))\times S^1\), with its hyperbolic metric, where \(X(\ell _i)\rightarrow \infty \). Then after passing to a subsequence, the restrictions of the maps \(u_i\) to the collars \(\mathcal {C}(\ell _i)\) converge to a full bubble branch with no loss of energy in the sense of Definition 1.10.

We will furthermore analyse almost-minimal maps from degenerating tori in order to obtain the refined asymptotics for the flow (1.1) also in case that the genus of *M* is one.

### Remark 1.12

*g*is isometric to the quotient of \(({\mathbb R}\times S^1, \frac{1}{2\pi b}g_{eucl})\), with \((s,\theta )\) and \((s+ b, \theta +a)\) identified. The length \(\ell (g)\) of the shortest closed geodesic of \((T^2,g)\) is then given by

Given a pair of sequences of maps \(u_i:T^2\rightarrow N\) and of flat unit area metrics \(g_i\) on \(T^2\), we let \((a_i,b_i)\) be as above so that \((T^2, g_i)\) is isometric to the quotient of \(({\mathbb R}\times S^1, \frac{1}{2\pi b_i}g_{eucl})\), with \((s,\theta )\) and \((s+ b_i, \theta +a_i)\) identified. We then lift the maps \(u_i\) to maps defined on the full cylinder \({\mathbb R}\times S^1\), also called \(u_i\) and extend the notion of convergence to a full bubble branch as follows.

### Definition 1.13

*converge to a nontrivial full bubble branch*if there exists a sequence \(s_i^0\) so that the maps \(u_i(s_i^0+\cdot , \cdot )\) converge to a nontrivial bubble branch and so that the restrictions of \(u_i(s_i^0+\tfrac{b_i}{2}+\cdot ,\cdot )\) to \({\mathscr {C}}_{\frac{b_i}{2}}\) converge to a full bubble branch in the sense of Definition 1.10.

### Theorem 1.14

*N*, with \(g_i\) flat metrics of unit area, for which \(\ell (g_i)\rightarrow 0\). Then there is a subsequence such that either (i) the maps \(u_i\) converge to a nontrivial full bubble branch with no loss of energy or (ii) \(E(u_i,g_i)\rightarrow 0\) and the whole torus is mapped near an i-dependent curve in the sense that \(\limsup _{i\rightarrow \infty }\sup _{s\in [0,b_i]}\mathop {{\mathrm {osc}}}\limits (u_i;\{s\}\times S^1)=0.\) In particular, we always have that for this subsequence

As usual, this theorem can be applied to the flow (1.1), thanks to Proposition 1.2. For a discussion of existing claims concerning no-loss-of-energy for \(M=T^2\) we refer to Remark 3.4.

Both Theorems 1.11 and 1.14 indirectly describe the map \(u_i\) on ‘connecting cylinders’ as being close to an *i*-dependent curve, thanks to (1.8). We are not claiming that this curve has zero length in the limit, as is the case in some similar situations, e.g. for necks in harmonic maps [8] and the harmonic map flow from fixed domains [9]. We are also not claiming that in some limit the curve should satisfy an equation, for example that it might always be a geodesic as would be the case for sequences of harmonic maps from degenerating surfaces, see [1]. The following construction can be used to show that these claims would be false in general.

### Proposition 1.15

*N*,

*G*), a \(C^2\) unit-speed curve \(\alpha :[-L/2,L/2]\rightarrow N\), and any sequence of degenerating hyperbolic collars \({\mathscr {C}}_{X_i}\), \(X_i\rightarrow \infty \), equipped with their collar metrics \(g_i\) as in the Collar Lemma A.1, the maps \(u_i:{\mathscr {C}}_{X_i}\rightarrow N\) defined by

We give the computations in Sect. 4. The proposition can be used to construct a sequence of almost-minimal maps with nontrivial connecting curves. For example, one can take any curve \(\alpha \) as above, and any sequence of hyperbolic metrics \(g_i\) with a separating collar degenerating, and then take the maps \(u_i\) to be essentially constant on either side of this one degenerating collar where the map is modelled on that constructed in the proposition. A slight variation of the construction would show that the *i*-dependent connecting curve need not have a reasonable limit as \(i\rightarrow \infty \) in general, whichever subsequence we take, and indeed that its length can converge to infinity as \(i\rightarrow \infty \).

*i*, the restriction of \(u(t_i)\) to the connecting cylinders \({\mathscr {C}}(s_i^{m-1}+\lambda ,s_i^m-\lambda )\) is close in \(C^0\) to curves \(\gamma _i(s)\) connecting the end points

Now that we have restricted to the particular case in which our theory is applied to the Teichmüller harmonic map flow, one might hope to rule out or restrict necks from developing. However, these necks do exist, and we do not have to have \(p^{m-1}_+= p^m_-\), as we now explain.

### Theorem 1.16

*M*of genus at least two, there exists a smooth solution of the Teichmüller harmonic map flow into \(S^1\) that develops a nontrivial neck as \(t\rightarrow \infty \). More precisely, if we extract a sequence of almost-minimal maps \((u(t_i),g(t_i))\) as in Proposition 1.2, then we can analyse it with Theorem 1.11, and after passing to a further subsequence we obtain

The simplest way of constructing an example as required in the theorem is to arrange that there can be no nonconstant branched minimal immersions in the limit, while preventing the flow from being homotopic to a constant map. The flow then forces a collar to degenerate in the limit \(t\rightarrow \infty \), and maps it to a curve in the target as we describe in Theorem 1.11. The precise construction will be given in Sect. 4. A key ingredient is the regularity theory for flows into nonpositively curved targets developed in [15].

### Remark 1.17

It would be interesting to prove that in a large class of situations the connecting curves of Theorems 1.11 and 1.14, when applied to the Teichmüller harmonic map flow, will have a limit, and that that limit will necessarily be a geodesic. In the case that \(M=T^2\), and under the assumption that the total energy converges to zero as \(t\rightarrow \infty \), Ding et al. [2] proved that the image of the torus indeed converges to a closed geodesic.

The conclusion of the theory outlined above is a much more refined description of how the flow decomposes an arbitrary map into a collection of branched minimal immersions from lower genus surfaces.

As in [15], although we state our flow results for compact target manifolds, the proofs extend to some noncompact situations, for example when *N* is noncompact but the image of *u*(0) lies within the sublevel set of a proper convex function on *N*.

The paper is organised as follows: In the next section we derive bounds on the angular part of the energy of almost harmonic maps on long Euclidean cylinders. The main results about almost-minimal maps are then established in Sect. 3, where we first prove Theorem 1.9, which then allows us to show Theorem 1.11, and as a consequence to complete the proof of Theorem 1.4, and to prove Theorem 1.14. In Sect. 4 we prove the results on the images of the connecting cylinders stated in Proposition 1.15 and Theorem 1.16. In the appendix we include the statements of two well-known results for hyperbolic surfaces, the Collar lemma and the Deligne–Mumford compactness theorem, the statements and notations of which are used throughout the paper.

## 2 Angular energy decay along cylinders for almost-harmonic maps

*u*is given by

*A*(

*u*) denotes the second fundamental form of the target \(N \hookrightarrow \mathbb {R}^{N_0}\).

### Lemma 2.1

*N*and \(E_0\), such that if

We require a standard ‘small-energy’ estimate, very similar to e.g. [3, Lemma 2.1] or [20, Lemma 2.9].

### Lemma 2.2

*N*such that any map \(u \in W^{2,2}({\mathscr {C}}(-1,2),N))\) which satisfies \(E(u; {\mathscr {C}}(-1,2)) < \delta _0\) must obey the inequality

Applying the Sobolev Trace Theorem gives the following (cf. [20]):

### Corollary 2.3

*C*, again only depending on

*N*.

We now establish a differential inequality for \(\vartheta (s)\). This is similar to [7, Lemma 2.1], but without requiring a bound on \(\sup |\nabla u|\). It is proved analogously to [20, Lemma 2.13], working on cylinders instead of annuli and considering a general target *N*.

### Lemma 2.4

*N*such that for \(u \in C^\infty ({\mathscr {C}}(-1,2),N)\) satisfying \(E(u;{\mathscr {C}}(-1,2))<\delta \) and \(\Vert \tau \Vert ^2_{L^2({\mathscr {C}}(-1,2))} < \delta \), and for any \(s\in (0,1)\), we have the differential inequality

### Proof of Lemma 2.4

*C*is a constant only depending on

*N*, that is revised at each step. Summing gives

*N*, we can improve (2.5) to

Lemma 2.4 can be applied all along a long cylinder \({\mathscr {C}}_\Lambda \) as arising in Lemma 2.1, and we can analyse the resulting differential inequality as in the next lemma to deduce bounds on \(\vartheta \).

### Lemma 2.5

### Proof

We now apply the estimate from Lemma 2.5 to establish decay of angular energy.

### Proof of Lemma 2.1

## 3 Proofs of the main theorems; convergence to full bubble branches

Our main initial objective in this section is to prove Theorem 1.9, giving convergence of almost-harmonic maps to full bubbles branches. This will then be combined with the Poincaré estimate for quadratic differentials of [14] to give Theorem 1.11. Theorem 1.9 will also lead us to a proof of the analogous result for tori, Theorem 1.14. But we begin with a proof of Proposition 1.2.

### Proof of Proposition 1.2

*i*, and claim that \(\ell (g(t))\le C/i\) for \(t\in [\tilde{t}_i,\tilde{t}_i+1/i]\), with

*C*depending only on the genus \(\gamma \), the coupling constant \(\eta \) and an upper bound \(E_0\) for the energy. If this claim were true then we would be able to pick our sequence of times \(t_i\) from the set \(\cup _i [\tilde{t}_i,\tilde{t}_i+1/i]\), which has infinite measure, in the usual way. When the genus of

*M*is at least 2, the claim follows from [15, Lemma 2.3], which implies in particular the Lipschitz bound

*M*is a torus, we shall see in Appendix A.2 that

### Proof of Theorem 1.9

Let \(u_i:{\mathscr {C}}_{X_i}\rightarrow N\) be a sequence of smooth almost harmonic maps as considered in Theorem 1.9. The first task is to construct sequences \(s_i^m\) as in the statement of the theorem. We would like to apply (2.3) on the regions \({\mathscr {C}}(s_i^{m-1} + \lambda , s_i^{m} - \lambda )\) to the maps \(u_i\) for large *i*, so we let \(\delta > 0\) be as in Lemma 2.1, which will be independent of *i*, of course.

We proceed to construct auxiliary sequences \(\hat{s}_i^m\), where \(m \in \{0, \dots , \hat{m}+1\}\) for some \(\hat{m} \ge 0\). For each *i*, consider the overlapping chunks of length 3 of the form \((k-1,k+2)\times S^1\subset {\mathscr {C}}_{X_i}\) for \(k \in \mathbb {Z}\), i.e. for integral *k* such that \(-X_i< k - 1< k+2 < X_i\). These chunks cover \({\mathscr {C}}_{X_i}\) except possibly for cylinders of length no more than 1 at the ends.

For each *i*, we initially choose the numbers \(\hat{s}_i^m\), for \(m=1,2,\ldots ,m_i\), to be the increasing sequence of integers so that \((\hat{s}_i^m-1,\hat{s}_i^m+2)\times S^1\) are precisely the chunks above that have energy at least \(\frac{\delta }{2}\).

Note that by the bound on the total energy, there is a uniform bound on the number \(m_i\) of such chunks, depending only on *N* and \(E_0\). Finally, we add in \(\hat{s}_i^0 = -X_i\) and \(\hat{s}_i^{m_i + 1} = X_i.\) By passing to a subsequence of the \(u_i\) we can assume that for each *i*, we have the same number of sequence elements \(\hat{s}_i^m\), i.e. \(m_i=\hat{m}\) for each *i*. Note also that for any region \((s-1,s+2)\times S^1\subset {\mathscr {C}}_{X_i}\) with energy at least \(\delta \) there is some associated overlapping integer chunk \((k-1,k+2)\times S^1\subset {\mathscr {C}}_{X_i}\) of energy at least \(\frac{\delta }{2}\) which is assigned a label in the above construction, except possibly for regions very close to the ends of the cylinder in the sense that \(s-1<-X_i+1\) or \(s+2>X_i-1\).

From this auxiliary sequence we form \(s_i^m\). Set \(s_i^0 = -X_i=\hat{s}_i^0\), and consider the difference \(\hat{s}_i^1-s_i^0\). If this has a subsequence converging to infinity, pass to that subsequence and take \(s_i^1 = \hat{s}_i^1\); if not, discard \(\hat{s}_i^1\). Proceed iteratively to define \(s_i^m\) (i.e. \(s_i^2\) is the next \(\hat{s}_i^m\) such that the respective difference \(\hat{s}_i^m-s^1_i\) diverges for some subsequence, after having passed to that subsequence). This process will terminate with the selection of \(s_i^{\bar{m}}\), for some \(\bar{m}\). Whatever sequence \(s_i^{\bar{m}}\) was chosen, redefine it as \(s_i^{\bar{m}} = X_i\), which can only change it by an amount that is uniformly bounded in *i*. This finishes the construction.

For each \(m\in \{1,2,\ldots , \bar{m}-1\}\), consider the shifted maps \(u_i^m(s, \theta ) := u_i(s + s^m_i, \theta )\). These maps have uniformly bounded energy and \(\tau (u_i^m)\rightarrow 0\) in \(L^2\). Theorem 1.5 applied in the case of Definition 1.8 gives, for a subsequence, convergence of each sequence \(u_i^m\) to a nontrivial bubble branch with associated bubbles \(\{\Omega _j\}\). This completes the proof of Part 1 of the theorem.

*i*(otherwise we would not have discarded the respective \(\hat{s}_i^m\)). Now let

*i*, giving

*i*) for \(s \in (-\Lambda ^m_i + \lambda , \Lambda ^m_i - \lambda )\) given by

The principle of the above proof also allows us to estimate the loss of energy on the connecting cylinders for sequences \(u_i\) of almost harmonic maps that do not satisfy (1.9), as we shall require in the case \(M=T^2\).

### Lemma 3.1

We remark that the reason for obtaining a weaker estimate in the case of \(m=1\) and \(m=\bar{m}\) is that the corresponding cylinder does not connect two bubble branches but rather connects the ends of the cylinder (if \(\bar{m}=1\)) or an end of the cylinder to a bubble branch (if \(\bar{m}>1\)). To prove this lemma we will use

### Remark 3.2

### Proof of Lemma 3.1

The key step needed to derive Theorem 1.11 from Theorem 1.9 is to use the Poincaré inequality for quadratic differentials to get control on the Hopf differential.

### Lemma 3.3

### Proof

*M*,

*g*), and in particular for the Hopf differential \(\Phi \), we have

*C*depends only on the genus \(\gamma \ge 2\) of

*M*and is thus in particular

*independent*of

*g*. By (1.4), as the area of \((M,g_i)\) is fixed, we know that

Based on Lemma 3.3 and Theorem 1.9 we can now give the

### Proof of Theorem 1.11

*Euclidean*cylinders \({\mathscr {C}}_{X_i}\), \(X_i=X(\ell _i)\), which are almost harmonic (with respect to \(g_0\)). We first remark that \(E(u_i; {\mathscr {C}}_{X_i})\) is bounded uniformly thanks to the conformal invariance of the energy and the assumed uniform bound on \(E(u_i,g_i)\). We then note that the conformal factors of the metrics \(\rho ^2(s)(ds^2+d\theta ^2)\) of the hyperbolic collars \((\mathcal {C}(\ell ),\rho ^2 g_0)\), \(\ell \in (0,2\mathop {\mathrm {arsinh}}\nolimits (1))\), described in Lemma A.1 are bounded uniformly by

### Proof of Theorem 1.4

### Proof of Theorem 1.14

*nontrivial*bubble branch in the sense of Definition 1.8. We can then analyse the restriction of the maps \( u_i(s_i^0+\tfrac{1}{2} b_i+\cdot ,\cdot )\) to the cylinder \({\mathscr {C}}_{\frac{b_i}{2}}\) with Theorem 1.9 and, after passing to a further subsequence, obtain that the maps \(u_i\) converge to a nontrivial full bubble branch as described in Definition 1.13.

*M*being a torus this issue does not arise and so we obtain (3.8) also for \(m=1\) and \(m=\bar{m}\). To be more precise, if we extend \(u_i\) periodically to a larger cylinder, say \({\mathscr {C}}_{s_i^0-b_i,s_i^0+2b_i}\), and repeat the above argument then the cylinders \({\mathscr {C}}(s_i^{0},s_i^1)\) and \({\mathscr {C}}(s_i^{\bar{m}-1},s_i^{\bar{m}})\) obtained above appear as connecting cylinders between two bubble branches, so Lemma 3.1 implies that (3.2) is valid also for \(m=1\) and \(m=\bar{m}\).

*nontrivial*bubble branch. In this case we apply Theorem 1.9 twice on \( {\mathscr {C}}_{\frac{1}{2}b_i}\), once for \(u_i\) itself and once for the shifted map \(\hat{u}_i:=u_i(\cdot + \frac{1}{2}b_i,\cdot )\). The assumption means that the set of points \(s_i^m\), \(m=1,\ldots ,\bar{m}-1\), where nontrivial bubble branches develop, is empty, i.e. \(\bar{m}=1\). Part 2 of Theorem 1.9 thus implies that the whole torus is mapped close to an

*i*-dependent curve as described in the theorem. But Theorem 1.5 also implies that, after passing to a subsequence, the maps \(u_i\) converge to a bubble branch, which must be trivial by assumption, and thus that \(E(u_i;{\mathscr {C}}_\lambda )\rightarrow 0\) for every \(\lambda >0\). Combined with Lemma 3.1 and (3.7) we thus obtain

### Remark 3.4

When we apply Theorem 1.14 to the flow (1.1), using Proposition 1.2, we obtain a no-loss-of-energy result from (1.12) for *M* a torus. An assertion of this form was already made in [2], where a claim is made that the Hopf differential converges to zero in \(L^1\), based on the knowledge that its *‘average’* converges to zero. (Note that taking our viewpoint, the fact that the average of the Hopf differential converges to zero would be interpreted as the projection of the Hopf differential onto the space of holomorphic quadratic differentials converging to zero.) However, our take on the genus one case, in the absence of a uniform Poincaré estimate available in the higher genus case, is that we can only deduce that the \(L^1\) norm of the Hopf differential converges to zero as a *consequence* of no-loss-of-energy, and along the way we need to appeal to the \(L^1\) smallness of the Hopf differential of bubble branches from Remark 3.2.

## 4 Construction of a nontrivial neck

The main purpose of this section is to prove Theorem 1.16, but we first record the following elementary computations.

### Proof of Proposition 1.15

*i*for the following computations. We also simplify matters by embedding (

*N*,

*G*) isometrically in some Euclidean space and composing

*u*with that embedding. The energy is conformally invariant, thus we calculate with respect to the flat metric

The remainder of this section is devoted to the proof of Theorem 1.16, constructing a flow that develops a nontrivial neck. We opt for a general approach, although essentially explicit constructions are also possible. To this end, consider any closed oriented surface *M* of genus at least 2, and take the target *N* to be \(S^1\). Choose a smooth initial map \(u_0:M \rightarrow S^1\) that maps some closed loop \(\alpha \) on *M* exactly once around \(S^1\), and take any hyperbolic metric \(g_0\) on *M*. We claim that the subsequent flow (1.1) develops a nontrivial neck.

The first key point is that since \(S^1\) has nonpositive sectional curvature, the regularity theory from [15, Theorems 1.1, 1.2] applies, so the flow exists for all time.

The second key point is that because the target is \(S^1\), there do not exist any branched minimal immersions, except if one allows constant maps. In particular, no bubbles can form. If no collar degenerated in this flow, i.e. if there were a uniform positive lower bound for the lengths of all closed geodesics in (*M*, *g*(*t*)), then by the results in [11], the map \(u_0\) would be homotopic to the constant map, which is false by hypothesis.

Therefore there are degenerating collars, and we can analyse them with Theorems 1.3 and 1.11 (using Proposition 1.2). We next demonstrate that a neck forms that is nontrivial in the sense that (1.15) holds. If not, then after passing to a subsequence, the maps from each degenerating collar would become \(C^0\) close to constant maps. By [13, Theorem 1.1] this would imply that \(u_0\) would be \(C^0\) close to a constant map, and thus in particular it would be homotopic to a constant map, which again is false by hypothesis.

We have proved that our flow develops a neck in the sense that (1.15) holds for some degenerating collar, and some *m*. By Theorem 1.11 the image of the subcollar \({\mathscr {C}}(s_i^{m-1}+\lambda ,s_i^m-\lambda )\) will be close to a curve for each *i*. However, the limiting endpoints (1.13) and (1.14) of the curves may not be distinct, i.e. (1.16) might fail in general.

To make a construction in which (1.16) must hold for some collar and some *m*, it suffices to adjust our construction so that again all extracted branched minimal immersions must be constant, but so that the union of the images is not just one point. By our theory, the connecting cylinders will thus be mapped close to curves connecting these distinct image points so a nontrivial neck with distinct end points must develop.

To achieve this, we will lift the whole flow to a finite cover \(\overline{M}\) of *M*. Given such a cover, we need to check that the lifted flow still satisfies (1.1). Locally, the lifting will not affect the tension field of *u*, so the lifted flow will satisfy the first equation of (1.1). However, care is required with the second equation since when we pass to the cover, new holomorphic quadratic differentials arise in addition to the lifts of the original holomorphic quadratic differentials. The following lemma will establish that this causes no problems.

### Lemma 4.1

*g*is a metric on

*M*, and \(\overline{g} := q^*g\). Then for all quadratic differentials \(\Psi \) on (

*M*,

*g*), we have

*M*,

*g*), respectively.

As a consequence, given a solution (*u*, *g*) to (1.1) on *M*, and a covering map *q* as in the lemma, the lifted pair \((u\circ q,q^* g)\) will be a solution to (1.1) on \(\overline{M}\).

### Proof

*q*is an

*n*-fold cover, then the linear map from quadratic differentials on (

*M*,

*g*) to quadratic differentials on \((\overline{M},\overline{g})\) given by

*M*,

*g*). Therefore to establish (4.1), it remains to prove that if \(\Phi \) is a holomorphic quadratic differential on \((\overline{M},\overline{g})\), then \(\langle \Phi ,q^*\Theta \rangle =0\) for every quadratic differential \(\Theta \in \mathcal {H}^\perp \). To see this, we consider the adjoint \(\Upsilon \) to the map \(\Psi \mapsto q^*\Psi \), which pushes down a quadratic differential on \((\overline{M},\overline{g})\) to (

*M*,

*g*), adding up the

*n*preimages. In particular, \(\Upsilon \) maps

*holomorphic*quadratic differentials to

*holomorphic*quadratic differentials, and so

Returning to our construction, we may assume that all the branched minimal immersions are mapping to the same limit point *p* (or we are done already). The aim is now to use a lifting construction, justified by the above, to obtain a lifted flow with the images of the corresponding branched minimal immersions being the two *different* lifts of *p* in a double cover of the target.

To this end, fix an arbitrary base point \(x_0\) on M. Now consider the index 2 subgroup *H* of \(\pi _1(M,x_0)\) consisting of loops whose images under the initial map \(u_0\) go round the target \(S^1\) an even number of times. We can pass to a (double) cover \(q:\overline{M} \rightarrow M\) of the domain satisfying \(q_*\left( \pi _1\left( \overline{M},\overline{x}_0 \right) \right) = H\) (see e.g. [5, Prop. 1.36]) and lift \(u_0\) to the map \(\overline{u}_0=u_0\circ q: \overline{M} \rightarrow S^1\). By the choice of *H* we can further lift \(\overline{u}_0\) to a map \(\tilde{u}_0\) into a (connected) double cover of the target \(S^1\) (e.g. [5, Prop. 1.33]). By Lemma 4.1, the flow (1.1) on \(\overline{M}\) starting at \((\tilde{u}_0, q^*g_0)\) covers the original flow on *M* starting at \((u_0,g_0)\). We can analyse this lifted flow using Theorem 1.11 for a subsequence of the times \(t_i\) at which we analysed the flow on *M*.

It suffices to show that the images of the branched minimal immersions we can construct from the lifted flow consist of *both* of the lifts of *p*, not just one. These distinct points can then only be connected by nontrivial necks.

To see this, note that from the analysis of the original flow on *M* with Theorem 1.1 from [13] (i.e. with Proposition 1.2 and Theorem 1.3), we can find some \(\delta > 0\) sufficiently small such that for sufficiently large *i*, the \(\delta \)-thin part of \((M,g(t_i))\) will consist of a (disjoint) union of (sub)collars that eventually degenerate. For large enough *i*, the image of the \(\delta \)-thick part of \((M,g(t_i))\) will be contained in a small neighbourhood of *p*. For each such large *i*, we pick a point \(y_i\) in the \(\delta \)-thick part, and deform \(\alpha \) to pass through \(y_i\). We view \(\alpha \) then as a path that starts and ends at \(y_i\), and by assumption, the composition \(u_0\circ \alpha \) takes us exactly once round the target \(S^1\). In particular, as we pass once round the lift of \(\alpha \), we move from one lift of \(y_i\) to the other, and the flow map moves from being close to one lift of *p* to being close to the other lift.

In particular, the branched minimal immersions in the lifted picture map to both lifts of *p* as required.

## Notes

### Acknowledgments

The third author was supported by EPSRC Grant Number EP/K00865X/1. We thank the referee for detailed commentary and suggestions.

## References

- 1.Chen, J., Tian, G.: Compactification of moduli space of harmonic mappings. Comment. Math. Helv.
**74**, 201–237 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Ding, W.Y., Li, J., Liu, Q.: Evolution of minimal torus in Riemannian manifolds. Invent. Math.
**165**, 225–242 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Ding, W.Y., Tian, G.: Energy identity for a class of approximate harmonic maps from surfaces. Commun. Anal. Geom.
**3**, 543–554 (1995)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math.
**86**, 109–160 (1964)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
- 6.Hummel, C.: Gromov’s compactness theorem for pseudo-holomorphic curves. Progress in Mathematics, vol. 151, pp. viii+131. Birkhäuser Verlag, Basel (1997)Google Scholar
- 7.Lin, F.-H., Wang, C.-Y.: Energy identity of harmonic map flows from surfaces at finite singular time. Calc. Var. Partial Differ. Equ.
**6**, 369–380 (1998)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Parker, T.H.: Bubble tree convergence for harmonic maps. J. Differ. Geom.
**44**, 595–633 (1996)MathSciNetzbMATHGoogle Scholar - 9.Qing, J., Tian, G.: Bubbling of the heat flows for harmonic maps from surfaces. Commun. Pure Appl. Math.
**50**, 295–310 (1997)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Randol, B.: Cylinders in Riemann surfaces. Comment. Math. Helvetici
**54**, 1–5 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Rupflin, M., Topping, P.M.: Flowing maps to minimal surfaces. Amer. J. Math.
**138**(4). arXiv:1205.6298 (To appear) - 12.Rupflin, M.: Flowing maps to minimal surfaces: Existence and uniqueness of solutions. Ann. I. H. Poincaré-AN
**31**, 349–368 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Rupflin, M., Topping, P.M., Zhu, M.: Asymptotics of the Teichmüller harmonic map flow. Adv. Math.
**244**, 874–893 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Rupflin, M., Topping, P.M.: A uniform Poincaré estimate for quadratic differentials on closed surfaces. Calc. Var. Partial Differ. Equ.
**53**, 587–604 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Rupflin, M., Topping, P.M.: Teichmüller harmonic map flow into nonpositively curved targets. J. Differ. Geom. arXiv:1403.3195 (to appear)
- 16.Rupflin, M., Topping, P.M.: Global weak solutions of the Teichmüller harmonic map flow into general targets. (In preparation)Google Scholar
- 17.Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math.
**113**, 1–24 (1981)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv.
**60**, 558–581 (1985)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Topping, P.M.: Winding behaviour of finite-time singularities of the harmonic map heat flow. Math. Z.
**247**, 279–302 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Topping, P.M.: Repulsion and quantization in almost-harmonic maps, and asymptotics of the harmonic map flow. Ann. Math.
**159**, 465–534 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 21.Tromba, A.: Teichmüller theory in Riemannian geometry. Lecture notes prepared by Jochen Denzler. Lectures in Mathematics ETH-Zürich. Birkhäuser (1992)Google Scholar
- 22.Topping, P.M.: Lectures on the Ricci flow. L.M.S. Lecture note series, vol. 325 C.U.P. (2006). http://www.homepages.warwick.ac.ukmaseqRFnotes.html
- 23.Zhu, M.: Harmonic maps from degenerating Riemann surfaces. Math. Z.
**264**, 63–85 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.