On the Evolution of Regularized Dirac-Harmonic Maps from Closed Surfaces

We study the evolution equations for a regularized version of Dirac-harmonic maps from closed Riemannian surfaces. We establish the existence of a global weak solution for the regularized problem, which is smooth away from finitely many singularities. Moreover, we discuss the convergence of the evolution equations and address the question if we can remove the regularization in the end.


Introduction and Results
Harmonic maps from Riemannian surfaces to Riemannian manifolds are a variational problem with rich structure. Due to their conformal invariance the latter share a lot of special properties. Among these are for example their regularity and the removal of isolated singularities. The existence of harmonic maps from surfaces has been established by several methods. The approach by Sacks and Uhlenbeck [32] uses a perturbation of the energy functional such that it satisfies the Palais-Smale condition. The heat flow method was successfully applied in this case by Struwe [33].
An extension of harmonic maps motivated from supersymmetric field theories in physics are Dirac-harmonic maps introduced in [17]. These also arise as critical points of an action functional and couple the equation for harmonic maps with spinor fields. A Dirac-harmonic map is given by a pair , which is smooth away from at most finitely many singular points (x k , t k ), 1 ≤ k ≤ K with K = K(ε, φ 0 , ψ 0 ). The weak solution constructed here is unique and the energy functional (1.1) of the weak solution is decreasing with respect to time. There exists a sequence t k → ∞ such that (φ(·, t k ), ψ(·, t k )) converges weakly in H 1 (M, N ) × H 1 (M, ΣM ⊗ φ −1 t T N) to a regularized Dirac-harmonic map (φ ∞ , ψ ∞ ) as k → ∞ suitably and strongly away from finitely many points (x k , t k = ∞). The pair (φ ∞ , ψ ∞ ) is smooth on M \{x 1 , . . . , x K }. Remark 1.3. (1) It seems that we have to impose the condition (1.4) in order to be able to prove Theorem 1.2. (2) Unfortunately, taking the limit ε → 0 after t → ∞ to obtain a Diracharmonic map does not seem to be possible. We will see later, that both the number of singularities and the regularity of (φ ∞ , ψ ∞ ) crucially depend on ε.
A similar approach in the one-dimensional case was performed in [9], see also [24]. Recently, a new heat-flow approach for Dirac-harmonic maps has been studied in which the Dirac equation is considered as a constraint while 57 Page 4 of 30 V. Branding Results Math the map is deformed by a heat-type equation. Several existence results using this approach could be obtained in the case of a one-dimensional domain [18] and for the domain being a compact surface with boundary [26]. The short time existence for this flow in the case of a closed manifold was recently established in [39].
The results presented in this article are part of the author's PhD thesis [4].
We would also like to point out that several existence results for Diracwave maps could be established [12,13,23] which are Dirac-harmonic maps from a domain with a Lorentzian metric.
This article is organized as follows. After introducing the framework for Dirac-harmonic maps, we present a regularized version of Dirac-harmonic maps. Afterwards, we study the L 2 -gradient flow of the regularized functional in Sect. 2. In Sect. 3 we establish the existence of a long-time solution and Sect. 4 then discusses the convergence of the evolution equations. In the last section we analyze the limit ε → 0.
Let us now describe the setup in more detail. We suppose that M is a closed Riemannian spin surface and N a compact Riemannian manifold. Every orientable Riemannian surface admits a spin structure, the number of different spin structures can be counted by the genus of the surface. For more details on spin geometry, see the book [29]. Coordinates on M will be denoted by x, whereas coordinates on N will be denoted by y. Indices on M are labeled by Greek letters, whereas indices on N are labeled by Latin letters. We use the Einstein summation convention, which means that we will sum over repeated indices.
For a given map φ : M → N , we consider the pull-back bundle φ −1 T N of T N and twist it with the spinor bundle ΣM . On this twisted bundle ΣM ⊗ φ −1 T N there is a metric induced from the metrics on ΣM and φ −1 T N. The induced connection on ΣM ⊗φ −1 T N will be denoted by∇. We will always assume that all connections are metric and free of torsion. Locally, sections of ΣM ⊗ φ −1 T N, called vector spinors, can be expressed as On the spinor bundle ΣM we have the Clifford multiplication of spinors with tangent vectors, which is skew-symmetric, namely for ψ, χ ∈ Γ(ΣM ) and X ∈ T M. We denote the Dirac operator on ΣM by / ∂ and the Dirac operator on the twisted bundle by / D, which is given by where e α is a local basis of T M. In terms of local coordinates / Dψ can be expressed as where Γ i jk are the Christoffel symbols on N . Since the connection on φ −1 T N is metric the operator / D is self-adjoint with respect to the L 2 norm. We may now state the energy functional for Dirac-harmonic maps where τ (φ) is the tension field of the map φ and the right hand side R(φ, ψ) is explicitly given by with R N being the Riemann curvature tensor on N .
In terms of local coordinates, the Euler-Lagrange equations acquire the form where R m lij are the components of the curvature tensor on N . Solutions of the system (1.6), (1.7) are called Dirac-harmonic maps from M → N .
In the analysis of the energy functional E(φ, ψ) one faces the problem that it is unbounded from below, since the operator / D is unbounded. To overcome these analytical difficulties, we "improve" the energy functional E(φ, ψ) by adding a regularizing term, see (1.1). Note that we formally have Of course, we would like to keep the parameter ε as small as possible. Unfortunately, in order to derive energy estimates, we have to drop this assumption.
As a next step we present the Euler-Lagrange equations for E ε (φ, ψ).
Written in local coordinates, the new terms arising from the variation of E ε (φ, ψ) acquire the following form:  A proof can for example be found in [17], p. 416, Lemma 3.1. In particular, this means that the functional E(φ, ψ) is conformally invariant in dimension two. We will see later that the L 4 -norm of ψ plays an important role in the context of a removable singularity theorem. On the other hand, we note that through the regularization the conformal invariance is broken.

Evolution Equations and Energy Estimates
We now turn to the L 2 -gradient flow of the regularized functional E ε (φ, ψ): with initial data (φ 0 , ψ 0 ). As ε > 0 the system (2.1), (2.2) is clearly parabolic and the existence of a smooth short-time solution up to a time T max can be obtained by standard methods, see Theorem 3.24 in [4].
Before turning to the derivation of energy estimates let us make the following remarks.
Proof. The proof is by contradiction. Assume that (φ, ψ) is a Dirac-harmonic map from T 2 → S 2 with deg(φ) = ±1. By the classification theorem for Diracharmonic maps between surfaces obtained in [40] the map φ has to be harmonic in this case. On the other hand, Eells and Wood proved in [22] that there does not exist a harmonic map from T 2 → S 2 of degree ±1 independently of the metrics chosen on the surfaces M and N .

Remark 2.2.
Since the degree of a map is homotopy-invariant, we cannot find a Dirac-harmonic map from T 2 → S 2 in the homotopy class of φ with deg φ = ±1. This example motivates the occurrence of singularities in the heat flow for (regularized) Dirac-harmonic maps.

Remark 2.3.
We cannot hope to find a global smooth solution of (2.1) and (2.2), as already the harmonic map heat flow develops singularities in finite time [14]. In addition, we cannot expect to find a unique solution in general since in [3,37], solutions that are different from Struwe's solution [33], were constructed.
In the following we will often need the following combination of quantities Moreover, for the further analysis it turns out to be useful to introduce the following function space with Q = M × [0, T ) and dQ = dM dt: Let Ω ∈ R 2 be a bounded domain. Then Ladyzhenskaya's inequality holds, that is . Then the following inequality holds: In addition, we need a local version of Ladyzhenskaya's inequality from above. By B R (x) we denote the geodesic ball of radius R around x ∈ M and i M denotes the injectivity radius of M . In terms of these quantities we can formulate the following: Then there exists a constant C such that for any R ∈ (0, i M ) the following inequality holds: Proof. A proof can for example be found in [35], p. 225, Lemma 6.7.
As a first step, we want to obtain a pointwise bound for the norm of the spinor ψ t . Using (2.2) we calculate ∂ ∂t Remark 2.6. If we apply the maximum principle to (2.5), we obtain the estimate In particular, if the initial spinor ψ 0 vanishes, then our system (2.1) and (2.2) reduces to the harmonic map heat flow studied by Struwe in [33]. Moreover, if ψ t = 0 for some time t then ψ t = 0 for all T ≥ t.
be a solution of (2.2) and assume that (1.4) holds. For ε large enough we get a uniform bound on ψ t The constant C depends on M, N, c 1 and the L 2 -norm of ψ 0 .
Proof. We already know that ψ t solves the pointwise equation (2.5). If we can also bound the L 2 -norm of the spinor ψ t we get a uniform pointwise bound by Lemma A.1 (see the "Appendix" for its precise formulation). Thus, we calculate ∂ ∂t where we applied (1.4) in the last step. Hence for ε > 0 big enough the right hand side of the above equation will be negative which gives the desired bound on the L 2 -norm of ψ t .
In the following C denotes a universal constant that may change from line to line. Since our evolution equations are originating from a variational problem, we get bounds in terms of the initial data (φ 0 , ψ 0 ).
Proof. The inequality follows from the fact that the system (2.1), The next Lemma is the analogue of Lemma 3.6 from [33]. We want to get local bounds of the L 2 -norms of dφ t and∇ψ t .
where the constant C only depends on M .
Proof. First of all, we choose a smooth cut-off function η with the following properties where again B R (x 0 ) denotes the geodesic ball of radius R around x 0 ∈ M . In addition, we choose an orthonormal basis {e α , α = 1, 2} on M such that ∇ eα e β = ∇ ∂t e α = 0 at the considered point. By a direct calculation we find ∂ ∂t Multiplying each of the terms with the cut-off function η 2 , adding up the three terms and using the evolution Eqs. (2.1) and (2.2), we find ∂ ∂t Using integration by parts we derive Applying Young's inequality and by the properties of the cut-off function η, Integration with respect to t yields the result.
We can use the previous Lemma to formulate monotonicity formulas for F (φ t , ψ t , B R ). By Young's inequality and the "monotonicity formula" for the local energy E ε (φ t , ψ t , B R ), we get Roughly speaking, we want to make the left hand side of this inequality as small as we have to. This can be achieved by choosing the initial data (φ 0 , ψ 0 ), the radius R of the ball B R and the time T appropriately. More precisely, we get the following (2.9) Proof. From Lemma 2.9 and the bound on the norm of ψ t , it follows that for any δ 1 and (φ 0 , ψ 0 ) suitably, there exists a number R > 0 for which (2.10) Vol. 75 (2020) On the Evolution of Regularized Dirac-Harmonic Page 11 of 30 57 For T 1 = δ1R 2 2C we then get sup such that the desired estimate holds.
In order to turn the Laplace type terms into full second derivatives, we will make use of the following Bochner type formulas: Lemma 2.11. (Bochner type formulas) For a map φ : M → N and a vector spinor ψ ∈ Γ(ΣM ⊗ φ −1 T N) the following Bochner type formulas hold: Proof. This follows from a direct calculation.
Proof. Using the evolution equations (2.1) and (2.2) we compute ∂ ∂t Applying Young's inequality and estimating the terms on the right hand side, we get ∂ ∂t As a next step we transform the Laplace type terms into second derivatives, therefore we apply the Bochner type formulas (2.12), (2.13) and find ∂ ∂t where we estimated all curvature contributions. Finally, we apply the local Sobolev inequality (2.4) to M |dφ t | 4 dM and M |∇ψ t | 4 dM , which leads to ∂ ∂t Choosing δ 1 small enough, the terms containing the second derivatives on the right hand side can be absorbed into the left hand side. Integrating with respect to t yields the result.
Using the bounds on the second derivatives, we can apply the Sobolev embedding theorem to bound Q |dφ t | 4 dQ and Q |∇ψ t | 4 dQ.
Proof. The bounds follow from the Sobolev embedding in two dimensions and the previous estimates, namely The estimate on Q |∇ψ t | 4 dQ can be derived by the same method. So far, we have derived integral estimates on Q = M × [0, T 1 ) of the second derivatives. In order to turn these into estimates on M , we have to gain control over the derivatives with respect to t of the pair (φ t , ψ t ).
Lemma 2.15. Let (φ t , ψ t ) ∈ V be a solution of (2.1) and (2.2). Then we have ∂ ∂t Proof. This follows by a direct calculation.
Proof. First of all, we choose an orthonormal basis {e α , α = 1, 2} on M such that ∇ ∂t e α = 0 at a considered point. Combining both equations from Lemma 2.15 we get ∂ ∂t We have to estimate all terms on the right hand side, starting with the A 1 term Calculating directly using the fact that ψ t is bounded uniformly we find for the Performing the same manipulations with the A 3 term, we get As a next step, we want to control the terms arising from interchanging covariant spinorial derivatives, namely A 4 , A 5 and A 6 .
Regarding A 6 , we use the pointwise bound on ψ t , interchange covariant derivatives, estimate the curvature terms and find Note that ∇ ∂φ ∂t = ∇ ∂t dφ, which is due to the torsion freeness of the connection. We sum up the different contributions and find the following inequality ∂ ∂t We used part of the second order terms on the left hand side to absorb the second order terms from the right hand side.
Integrating with respect to t over the domain τ ≤ s < t ≤ T we get t s dt ∂ ∂t The last term can be bounded in terms of the initial data and the L 2 -norm of ψ t by Lemma 2.8. We use another type of Sobolev inequality (similar to (2.4) for |t − s| ≤ 1) to bound the mixed terms like and similarly for both of the other two terms. Choosing t − s < δ 2 sufficiently small, applying the Sobolev inequality and the estimates from Corollary 2.14, we can absorb part of the right hand side in the left and obtain Finally, we estimate the infimum by the mean value, more precisely Hence, we get the desired bound.
Proof. With the help of the previous estimates we can now bound the full second derivatives of (φ t , ψ t ) in L 2 . By the evolution equations (2.1), (2.2) and Young's inequality, we find The assertion then follows from applying the local Sobolev inequality (2.4) and the Bochner formulas (2.12), (2.13) with δ 1 small enough. Corollary 2.18. (Higher regularity) Suppose the pair (φ t , ψ t ) is a weak solution of (2.1) and (2.2). The pair (φ t , ψ t ) is smooth as long as δ 1 , δ 2 are small enough.
Proof. Since we have a bound on the L 2 -norm of the second derivatives of φ t and ψ t by (2.17), we can apply the Sobolev embedding theorem and get that both |dφ t | ∈ L p and |∇ψ t | ∈ L p for p < ∞. From the evolution equations (2.1) and (2.2) we may conclude that ∂φt ∂t , |∇ 2 φ t | ∈ L p and also ∇ ψt ∂t , |∇ 2 ψ t | ∈ L p . By the regularity theory for parabolic partial differential equations we obtain that |dφ t | and |∇ψ t | are Hölder continuous, see [28], Theorem IV.9.1 and Lemma II.3.3. At this point the smoothness of the pair (φ, ψ) follows from a standard bootstrap argument using Schauder theory, for more details see Theorem 3.24 in [4].

Long-Time Existence and Singularities
In this section we establish the existence of a long-time solution to the evolution equations. Thus, we first of all derive a uniqueness and stability result. To avoid the problem of identifying sections in different vector bundles, we will make use of the Nash embedding theorem. Hence, assume N ⊂ R q isometrically and denote the isometric embedding by ι. Then, u = ι•φ : M → R q can be thought of as a vector-valued function. The vector spinor ψ turns into a vector of usual spinors ψ = (ψ 1 , . . . , ψ q ) with ψ i ∈ Γ(ΣM ), i = 1, . . . , q. The condition that ψ is along the map φ is encoded by Now, the function u satisfies the following equation: with the initial condition u 0 = ι(φ 0 ) and with the initial condition ψ 0 = dι(ψ 0 ), where ψ 0 ∈ Γ(ΣM ⊗ φ −1 0 T N). For a derivation of (3.1) and (3.2) see [4], Section 3.4. Here, II is the second fundamental form of the embedding and P denotes the shape operator. By projecting to a tubular neighborhoodÑ of ι(N ) ⊂ R q we can think of II as a vector-valued function in R q . For more details, see [31], p. 132. Assuming ) < δ 1 we obtain by Corollary 2.17 and the Sobolev embedding theorem that such that we can prove the following  Proof. The first singular time T 0 is characterized by the condition is a weak solution of (2.1) and (2.2). By iteration, we obtain a weak solution (φ t , ψ t ) on a maximal time interval T 0 + δ for some δ > 0. If T 0 + δ < ∞ then by the above argument the solution (φ t , ψ t ) may be extended to infinity, hence T 0 + δ = ∞. The uniqueness follows from Proposition (3.1). Proposition 3.3. Assume (φ t , ψ t ) is a solution of (2.1) and (2.2) satisfying (1.4) and ε sufficiently large. There are only finitely many singular points (x k , t k ), 1 ≤ k ≤ K. The number K depends on M, ε, ψ 0 , dφ 0 and∇ψ 0 .
Proof. We follow the presentation in [30], p. 138, for the harmonic map heat flow. We assume that T 0 > 0 is the first singular time and define the singular set as Recall that by assumption (1.4) we have, where we have renamed the positive constant c 1 to δ 5 . From this we obtain the global estimate (with ε suitably large) . We choose R > 0 such that all the B 2R (x j ), 1 ≤ j ≤ K are mutually disjoint and small enough to have Then, we have by (3.7) for any τ ∈ [T 0 − δ1R 2 4δ3 , T 0 ]. We conclude that which implies the finiteness of the singular set S(φ, ψ, T 0 ). Our next aim is to show that there are only finitely many singular spatial points. Therefore we set Now suppose T 0 < · · · < T j are j singular times and by K 0 , . . . , K j we denote the number of singular points at each singular time. Set By iterating (3.8) we get , which can be rearranged as We conclude that there are only finitely many singularities. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/ licenses/by/4.0/.

Appendix
The following Lemma combines the pointwise maximum principle with an integral norm. It can be thought of as a simple version of Moser's parabolic Harnack inequality. Proof. A proof can for example be found in [36], p. 284.