Limiting Behaviour of the Teichmüller Harmonic Map Flow
In this paper we study the Teichmüller harmonic map flow as introduced by Rupflin and Topping . It evolves pairs of maps and metrics into branched minimal immersions, or equivalently into weakly conformal harmonic maps, where maps from a fixed closed surface with metric to a general target manifold . It arises naturally as a gradient flow for the Dirichlet energy functional viewed as acting on equivalence classes of such pairs, obtained from the invariance under diffeomorphisms and conformal changes of the domain metric.
In the construction of a suitable inner product for the gradient flow a choice of relative weight of the map tangent directions and metric tangent directions is made, which manifests itself in the appearance of a coupling constant in the flow equations. We study limits of the flow as approaches 0, corresponding to slowing down the evolution of the metric.
We first show that given a smooth harmonic map flow on a fixed time interval, the Teichmüller harmonic map flows starting at the same initial data converge uniformly to the underlying harmonic map flow when .
Next we consider a rescaling of time, which increases the speed of the map evolution while evolving the metric at a constant rate. We show that under appropriate topological assumptions, in the limit the rescaled flows converge to a unique flow through harmonic maps with the metric evolving in the direction of the real part of the Hopf differential.
Given a smooth closed orientable surface , a metric on and a smooth closed Riemannian manifold , the energy of a map is defined as
We call a harmonic map if it is a critical point of (viewed as a functional on maps). The harmonic map flow is then given by
where is the tension field of the map . One can view this flow as a gradient flow (with respect to the map) for the energy functional . The harmonic map flow has been studied extensively, see e.g. [2, 5, 7, 13, 21]. As expected by its gradient flow nature, it aims to transform an initial map into a harmonic map. In general, this is not possible without singularities in the map forming, as some homotopy classes do not contain any harmonic maps (). However, assuming nonpositive sectional curvature on , Eells and Sampson were able to show long-time existence and uniform derivative bounds for this flow in  and as a result could show that any homotopy class of maps contains a harmonic representative.
If one also allows the domain metric to vary, a different gradient flow for the functional can be found. The energy is invariant under conformal changes of the domain metric in two dimensions, so depending on the genus of one can restrict to flowing through metrics of Gauss curvature , corresponding to hyperbolic surfaces, tori and the sphere respectively. This gradient flow was introduced in , and is called the Teichmüller harmonic map flow. It is given by
where again denotes the tension field of , the coupling constant a choice of scaling in defining a metric on pairs of maps and metrics, the -orthogonal projection of quadratic differentials onto holomorphic quadratic differentials and is the Hopf differential of , a quadratic differential that measures how ‘close’ is to being conformal. In particular vanishing of implies that is a weakly conformal map. This flow thus tries to transform given initial data into a weakly conformal harmonic map.
A result of Gulliver, Osserman and Royden () allows us to give a more geometric characterization of such maps as branched minimal immersions.
In this sense one can view the study of the flow (3) as an alternative approach to the classical variational method for studying branched minimal immersions (as done in ), and indeed in  some of the results in  are recovered using the flow method.
A priori singularities in this flow might appear both in the metric (i.e. parts of the domain may become arbitrarily ‘thin’) and the map. The latter type is well-understood for the harmonic map flow, the general idea is that so-called bubbles can be extracted at points and times where energy concentrates (). These bubbles are non-constant harmonic maps . Away from such concentration points, the harmonic map flow enjoys higher regularity. This principle carries over to the Teichmüller harmonic map flow (), and allows one to employ a lot of the techniques familiar from the study of the harmonic map flow in the development of the theory for the flow (3).
Possible degeneration of the metric on the other hand requires new ideas. When this degeneration only happens at infinite time and is a hyperbolic surface, it was shown in  that the initial map in some sense (sub-)converges to a collection of branched minimal immersions (or constant maps) in the limit. In a joint work with Rupflin, Topping and the author () the properties of this convergence where studied more closely.
When one constructs the flow (3) as a formal gradient flow on a space of maps and hyperbolic metrics (see  for the details of this construction), a choice of inner product is made. This manifests itself in the coupling constant in the definition of the flow (3). In this paper we study the behaviour of the flow when one lets .
We establish two results in this direction. Initially, we fix some initial data and a time interval . Assuming that the corresponding classical harmonic map flow starting from the same initial data does not develop singularities, we prove the following theorem, establishing uniform convergence of the Teichmüller harmonic map flow to the harmonic map flow:
Let be a smooth closed oriented surface of genus and , the space of constant Gauss curvature hyperbolic metrics on . Further take to be a smooth closed Riemannian manifold and a smooth map. Consider a fixed time interval such that the harmonic map flow satisfying (2) (with respect to ), starting at the initial condition , is smooth. Then the flows satisfying (3) with initial condition and some coupling constant converge smoothly to the harmonic map flow , in the following sense as :
The metrics converge to the initial metric in uniformly in for each .
The maps converge to smoothly on .
The proof will be carried out in Section 2.
We can also adopt a different viewpoint on the relative behaviour of the map evolution and the metric evolution for small . By rescaling time appropriately, we can think of the map evolution happening increasingly quickly, while we fix the ‘speed’ of the metric evolution. In particular the rescaling yields
where . Studying now corresponds to . In the limit, we heuristically expect the map to become instantaneously harmonic at all times, while the metric evolves in the direction of the real part of the Hopf differential (as for harmonic).
Consider a target with strictly non-positive curvature and fix a homotopy class of maps which does not contain any constant maps or maps to closed geodesics. Work by Hartman () then guarantees the existence of a unique harmonic map in that homotopy class for any choice of metric on the domain . Thus given a curve of metrics , we can find a corresponding curve of harmonic maps . In this setting we prove that the flows (4) do indeed converge to a flow through harmonic maps.
Let be a smooth closed oriented surface of genus and be a smooth closed Riemannian manifold. Given smooth initial data for (4), take with from Lemma 3.1, and consider the sequence of solutions to (4) with rescaled coupling constant on the fixed time interval , which we further assume to be smooth up to . Then the following is true:
There exists a limit curve of hyperbolic metrics (i.e. each has Gauss curvature ) on , continuous in time and smooth in space in the sense that for all , is an element of . After possibly selecting a subsequence in , The curves converge to in (i.e. uniformly in time in ), again for all .
Further assume that has strictly negative sectional curvature and that the homotopy class of does not contain maps to closed geodesics in the target or constant maps. Let be the unique curve of harmonic maps homotopic to corresponding to , then the limit curve of metrics is differentiable in time at each point away from , with derivative given by , where as usual denotes the Hopf differential. Finally, the maps also converge to uniformly in in away from 0 for all , and the convergence of to does not require a choice of subsequence in .
We give the proof and the definitions of the involved spaces in Section 3.
A consequence of Theorem 1.2 is that the flow through harmonic maps
enjoys short-time existence when one works in the above setting. On the way to proving Theorem 1.2 we in fact also show a uniqueness statement. It would be interesting to analyse this flow further, in particular investigating whether the resulting curves in Teichmüller space are geodesics (with respect to the Weil-Petersson metric) and if finite-time singularities in the metric can occur (see e.g. ).
Acknowledgements: Large parts of the work in this paper were originally carried out during my thesis, and I would like to thank my advisor Peter Topping for his guidance and encouragement. I also enjoyed the support of the Pacific Institute for the Mathematical Sciences (PIMS) during the later stages of this project.
2 Small Coupling Constant Limit
In this section we prove Theorem 1.1. To this end, we first establish the metric convergence using estimates from . We then prove that the map part converges, which is accomplished through showing that the energy concentration along the flow (3) stays controlled if the corresponding harmonic map flow is smooth.
2.1 Convergence of the Metric
We collect some results on the behaviour of the metric under the flow (3) here.
Let be a smooth one-parameter family of metrics for . Define the -length of on by
Let be evolving under . Recall the energy identity
from . As a consequence we have an estimate for the -length.
Let , and assume that is a solution to with coupling constant on for some initial data . Then we have the estimate
Integrating the energy identity (6) in time from to yields
for denoting the initial energy. Hence
and Hölder’s inequality yields the claim. ∎
Note that after projecting a curve of metrics down to a path in Teichmüller space the -length of defined in the above lemma corresponds to the length of , computed with respect to the classical Weil-Petersson metric (up to a constant) .
We call curves of metrics evolving in the direction of the real part of a holomorphic quadratic differential horizontal curves:
Definition 2.4 (See also ).
In our setting a horizontal curve is a smooth one-parameter family of hyperbolic metrics on for , such that for all such we have , where is some holomorphic quadratic differential on .
Proposition 2.5 (Proposition 2.2 in ).
For every and every there exists a number such that the following holds true. Let , i.e. a hyperbolic metric of class , for which the length of the shortest closed geodesic in is no less than . Then there is a number such that for any horizontal curve with and we have
The -norm here and in the following is to be understood with respect to some fixed set of local coordinate charts on , thus in particular refers to the set of hyperbolic metrics with coefficients in .
We also have the following result from  controlling the projection .
Lemma 2.6 (Lemma 2.9 in ).
For any and any there exists a neighbourhood of in and a constant such that for all and we have
Here depends on , , and and we used the fact that embeds continuously into for sufficiently large . We first collect here a number of useful consequences of the above results.
By the previous estimates we can find such that for and the following properties hold:
For any vector field we have .
For any smooth map we have .
For any -tensor we have .
The injectivity radius satisfies . In particular we necessarily have existence of a weak solution in the sense of  up to time .
For all and metric balls satisfy .
The difference of the inverse metric tensors is bounded by . Similarly, the difference of the Christoffel symbols of is bounded by .
The metrics lie in the neighbourhood from Lemma 2.6, and in particular with depending only on , and .
An immediate consequence is the desired metric convergence on fixed time intervals.
Under the assumptions of Theorem 1.1, we have uniform convergence in of to on in for any as .
The convergence can be seen as in (10). Note that the above remark guarantees that the flows considered actually exist up to time (as degeneration of the metric is ruled out for sufficiently small ). ∎
2.2 -Convergence of the Map for small times
We proceed by proving -closeness of the flow (3) for small to the underlying harmonic map flow under a boundedness assumption we will justify later.
Take to be a smooth closed oriented surface of genus and to be a smooth closed Riemannian manifold. As before denote by , the solutions (with respect to initial data ) to the usual harmonic map flow (2), respectively Teichmüller harmonic map flow (3) with coupling constant . Let be such that the flow is smooth on . Further assume there exist constants (allowed to depend on and ) such that the flow admits a smooth solution for all which satisfies
Then given any , we can find , depending on , , , , , and , such that for all we have
where denotes the second fundamental form of the target and we write .
Multiplying this equation by and integrating over together with partial integration (with respect to ) we obtain
We now estimate the terms on the right hand side. All norms in the following, as well as the volume form , are taken to be defined with respect to . Also assume from now on that so Remark 2.7 applies, then for the first term we find
where we used integration by parts (with respect to both and , as the flow (3) leaves the volume form invariant, see ), and Remark 2.7 to estimate the difference of the inverse metric tensors and finally .
Here and in the following we let denote positive constants, which may depend on , and the initial energy .
We proceed to estimate the second fundamental form terms. Note that
which we estimate pointwise
with . The first inequality follows by an application of the mean value theorem to (which is a smooth function on ) (see also  for this estimate). The second inequality again follows by estimating the difference of the inverse metric tensors as in Remark 2.7. Multiplying (17) and (18) by , integrating over and applying Hölder’s inequality gives
The strategy is now to derive an estimate for that allows us to apply Gronwall’s lemma to deduce our desired smallness. In particular we need to control all the terms in (20) through quantities integrable in time (e.g. by the assumption on and the smoothness of ), and .
To this end, recall the following consequence of Sobolev’s inequality :
Using Young’s inequality we further find
Together with (21) and Young’s inequality this implies
which is of the desired form.
Similarly we can estimate the first term in the right hand side of (20) via Young’s inequality to obtain
We now additionally assume , so . Writing (cf. ) we arrive at
We can apply Gronwall’s lemma to this inequality for the function and finally get
This estimate will be valid as long as . Hence from (10) we see that we can indeed choose small such that
for all and . ∎
The basic ingredient for proving the bound required in the above proposition is following consequence of a standard small energy regularity estimate for almost-harmonic maps:
Take to be a smooth closed manifold and a smooth closed oriented surface of genus and . Consider a smooth map with energy bounded by . Then there exists a constant such that the following holds: Given such that
for all , there exists (depending on , , and ) such that
For we have the following local regularity estimate (e.g. ) for with :
Now let , be as in the lemma and further assume for now. We can then cover by finitely many balls , , and apply estimate (29) on each, using . After summing and absorbing additional terms into the constant we obtain the claim. Note that if , we can carry out the same argument replacing with e.g. .
To apply this lemma, we need to find some such that , i.e. control the concentration of energy along the flow. Assuming the existence of such an for a moment, we obtain our desired bound as a consequence of the following lemma.
As before, take to be a smooth closed manifold and a smooth oriented closed surface of genus . Assume that there exists such that for all , with as in Lemma 2.10. Let and consider the solution with coupling constant to (3) defined on with initial data . Then there exists such that
for all and some constant , with and only depending on , , , and . Furthermore is uniformly bounded in for each (with a bound only depending on ).
We apply the bound (28) at each time to the map with respect to the initial metric and radius . We find
with a constant depending on , , and , where we take to denote the connection induced by .
We would like to use the energy identity (6) to integrate this inequality and further estimate the right hand side. Thus, as in , we want to replace with in (31). To do this, we estimate pointwise
This can be seen using Remark 2.7 to in particular estimate the difference of the Christoffel symbols with respect to and , see also the author’s thesis  for more detail. This allows us to estimate
which we can use together with estimate (31) to find
where we used (as by Remark 2.7 the energy densities are comparable). Thus after choosing sufficiently small, now assuming we can absorb the remaining extra terms on the right and obtain
with a constant also only depending on , , , and . We now drop the term (it was only required to control the error introduced by switching the metric of the tension) and integrate (35) over :
Here we used the energy identity (6) to estimate the integral of the tension.
Finally is uniformly bounded in for by our assumption on the metric combined with the monotonicity of the energy . ∎
Hence it remains to show that the energy concentration along the flow is controlled. We first recall that under certain assumptions on the metric (which in particular by the previous section hold in our case) the evolution of energy along the flow is controlled uniformly for short times.
Assume to be a weak (as defined in ) solution to (3) on , with smooth initial data and some coupling constant . Further assume that there exists some such that for all and smooth maps , we have , as well as . Then for all the following estimate holds with a positive constant for all , :
Here the energies and geodesic balls are taken with respect to the initial metric .
Take to be a standard cut-off function, satisfying on and (with some universal constant ). We can then multiply equation (3) for by and integrate over with respect to (exactly as in ) to arrive at
where we view the target as isometrically embedded via . Using integration by parts we find
We then note that
with an error term given by