Short-time existence of the α-Dirac-harmonic map flow and applications

Abstract In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow for α-Dirac-harmonic maps and blow-up analysis. More precisely, given any initial map along which the Dirac operator has nontrivial minimal kernel, we first prove the short time existence of the heat flow for α-Dirac-harmonic maps. The obstacle to the global existence is the singular time when the kernel of the Dirac operator no longer stays minimal along the flow. In this case, the kernel may not be continuous even if the map is smooth with respect to time. To overcome this issue, we use the analyticity of the target manifold to obtain the density of the maps along which the Dirac operator has minimal kernel in the homotopy class of the given initial map. Then, when we arrive at the singular time, this density allows us to pick another map which has lower energy to restart the flow. Thus, we get a flow which may not be continuous at a set of isolated points. Furthermore, with the help of small energy regularity and blow-up analysis, we finally get the existence of nontrivial α-Dirac-harmonic maps ( ) from closed surfaces. Moreover, if the target manifold does not admit any nontrivial harmonic sphere, then the map part stays in the same homotopy class as the given initial map.


Introduction
Motivated by the supersymmetric nonlinear sigma model from quantum field theory, see [1], Dirac-harmonic maps from spin Riemann surfaces into Riemannian manifolds were introduced in [2]. They are generalizations of the classical harmonic maps and harmonic spinors. From the variational point of view, they are critical points of a conformal invariant action functional whose Euler-Lagrange equations are a coupled elliptic system consisting of a second order equation and a Dirac equation.
It turns out that the existence of Dirac-harmonic maps from closed surfaces is a very difficult problem. Different from the Dirichlet problem, even if there is no bubble, the nontriviality of the limit is also an issue. Here, a solution is considered trivial if the spinor part w vanishes identically. So far, there are only a few results about Dirac-harmonic maps from closed surfaces, see [3][4][5] for uncoupled Dirac-harmonic maps (here uncoupled means that the map part is harmonic; by an observation of Bernd Ammann and Johannes Wittmann, this is the typical case) based on index theory and the Riemann-Roch theorem, respectively. In an important contribution [6], Wittmann investigated the heat flow introduced in [7] and showed the short-time existence of this flow; for reasons that will become apparent below this is not as easy as for other parabolic systems. The problem has also been approached by linking and Morse-Floer theory. See [8,9] for one dimension and [10] for the two dimensional case.
In critical point theory, the Palais-Smale condition is a very strong and useful tool. It fails, however, for many of the basic problems in geometric analysis, and in particular for the energy functional of harmonic maps from spheres [11]. Therefore, it is not expected to be true for Dirac-harmonic maps. To overcome this problem for harmonic maps, Sacks-Uhlenbeck [12] introduced the notion of a-harmonic maps where the integrand in the energy functional is raised to a power a > 1: These a-harmonic maps then satisfy the Palais-Smale condition. However, when we analogously introduce a-Dirac-harmonic maps, the Palais-Smale condition fails due to the following existence result for uncoupled a-Dirac-harmonic maps, which directly follows from the proof of Theorem 4.1.
Theorem 1.1. For a closed spin surface M and a closed manifold N, consider a homotopy class ½/ of maps / : M m ! N n for which ½dim H ðker = D / Þ Z 2 is non-trivial. Assume that / 0 2 ½/ is an a-harmonic map. Then there is a real vector space V of real dimension 4 such that all ð/ 0 , wÞ, w 2 V, are a-Dirac-harmonic maps.
To overcome this issue, in [8,9], the authors add an extra nonlinear term to the action functional of Dirac-geodesics. As for the two dimensional case [10], we even cannot directly prove the Palais-Smale condition for the action functional of perturbed Dirac-harmonic maps into non-flat target manifolds. Instead, we are only able to prove it for perturbed a-Dirac-harmonic maps, and then approximate the a-Dirac-harmonic map by a sequence of perturbed a-Dirac-harmonic maps. However, in this approach, it is not easy to control the energies of the perturbed a-Dirac-harmonic maps, which are constructed by a Min-Max method over increasingly large domains in the configuration space.
Due to these two problems, in this paper, we would like to use the heat flow method to get the existence of Dirac-harmonic maps from closed surfaces to general manifolds where the harmonic map type equation is parabolized and the first order Dirac equation is carried along as an elliptic side constraint [7]. As already mentioned, the short-time existence of the heat flow for Dirac-harmonic map was proved by Wittmann [6]. He constructed the solution to the constraint Dirac equation by the projector of the Dirac operator along maps. By assuming that the Dirac operator along the initial map has nontrivial minimal kernel, he showed that the kernel would stay minimal for small time in the homotopy class of the initial map. This minimality implies a uniform bound for the resolvents and the Lipschitz continuity of the normalized Dirac kernel along the flow. This Lipschitz continuity makes the Banach fixed point theorem available. If one follows this approach, the first issue is how to deal with the kernel jumping problem. Observe that if the Dirac operators converge at the jumping time, the symmetry of the spectrum of Dirac operator guarantees that the limiting Dirac operator has odd dimensional kernel. Therefore, it is natural to try to extend Wittmann's short time existence to the odd dimensional case. However, the eigenvalues in this case may split at time t ¼ 0. Then the projector may not be continuous even if the Dirac operator is smooth with respect to time along the flow (see [13]), which means that the Lipschitz continuity of the kernel is not available in general. To overcome this issue, we need the density mentioned in the abstract, which gives us a piecewise smooth flow.
As for the convergence, it is sufficient to control the energy of the spinor field because the energy of the map decreases along the flow. To do so, one can impose a restriction on the energy of the initial map as in [14] and get the existence of Dirac-harmonic maps when the initial map has small energy. Alternatively, we use another type flow, that is, the heat flow for a-Dirac-harmonic maps (also called a-Dirac-harmonic map flow in the literatures). Our motivation comes from the successful application of this flow to the Dirichlet problem [15]. Different from there, we cannot uniquely solve the constraint equation. Moreover, our equations of the flow are different. We never write the constraint equation in the Euclidean space R q : Instead, we just solve it in the target manifold N. Last, our flow is not unique due to the absent of a boundary. Instead, only a weak uniqueness is available. Consequently, we need prove the fact that the flow takes value in the target manifold N in a different way. Eventually, we shall obtain the following results on the general existence of Dirac-harmonic maps.
Theorem 1.2. Let M be a closed spin surface and (N, h) a real analytic closed manifold. Suppose there exists a map u 0 2 C 2þl ðM, NÞ for some l 2 ð0, 1Þ such that dim H ker = D u 0 ¼ 1. Then there exists a nontrivial smooth Dirac-harmonic map ðU, WÞ satisfying EðUÞ Eðu 0 Þ and jjWjj L 2 ¼ 1: Furthermore, if (N, h) does not admit any nontrivial harmonic sphere, then the map part U is in the same homotopy class as u 0 and ðU, WÞ is coupled if the energy of the map is strictly bigger than the energy minimizer in the homotopy class ½u 0 : Remark 1.3. Our result can at least keep the possibility of the existence of coupled solutions while other solutions produced in the literatures are all uncoupled. The analyticity of the target manifold is a sufficient condition which is used to get the density mentioned in the abstract. In fact, it is easy to see from the proof that we only need the density of the following set Y :¼ fe 2 ðm a i 0 , þ 1Þj there exists at least one map u such that dim H ker = D u ¼ 1 and E a i ðuÞ ¼ eg at the a i -energy minimizer m a i 0 in the homotopy class ½u 0 for a sequence a i & 1 as i ! 1: In [16], Wittmann discussed the density of those maps along which all the Dirac operators have minimal kernel. In particular, we have the following corollary. We also assume that 1. M is connected, oriented and of positive genus; 2. N is connected. If N is even-dimensional, then we assume that it is non-orientable.
Then there exists a nontrivial smooth Dirac-harmonic map.
The rest of paper is organized as follows: In Section 2, we recall some definitions, notations and lemmas about Dirac-harmonic maps and the kernel of Dirac operator. In Section 3, under the minimality assumption on the kernel of the Dirac operator along the initial map, we prove the short time existence, weak uniqueness and regularity of the heat flow for a-Dirac-harmonic maps. In Section 4, we prove the existence of a-Dirac-harmonic maps and Theorem 1.2. In the Appendix, we solve the constraint equation and prove Lipschitz continuity of the solution with respect to the map.

Preliminaries
Let (M, g) be a compact surface with a fixed spin structure. On the spinor bundle RM, we denote the Hermitian inner product by hÁ, Ái RM : For any X 2 CðTMÞ and n 2 CðRMÞ, the Clifford multiplication satisfies the following skew-adjointness: Let r be the Levi-Civita connection on (M, g). There is a connection (also denoted by r) on RM compatible with hÁ, Ái RM : Choosing a local orthonormal basis fe b g b¼1, 2 on M, the usual Dirac operator is defined as = Here and in the sequel, we use the Einstein summation convention. One can find more about spin geometry in [17]. Let / be a smooth map from M to another compact Riemannian manifold (N, h) of dimension n ! 2: Let / Ã TN be the pull-back bundle of TN by / and consider the twisted bundle RM / Ã TN: On this bundle there is a metric hÁ, Ái RM/ Ã TN induced from the metric on RM and / Ã TN: Also, we have a connectionr on this twisted bundle naturally induced from those on RM and / Ã TN: In local coordinates fy i g i¼1, :::, n , the section w of RM / Ã TN is written as where each w i is a usual spinor on M. We also have the following local expression ofr where C i jk are the Christoffel symbols of the Levi-Civita connection of N. The Dirac operator along the map / is defined as which is self-adjoint [11]. Sometimes, we use = D / to distinguish the Dirac operators defined on different maps. In [2], the authors introduced the functional Lð/, wÞ :¼ They computed the Euler-Lagrange equations of L: where s m ð/Þ is the m-th component of the tension field [11] of the map / with respect to the coordinates on N, r/ l Á w j denotes the Clifford multiplication of the vector field r/ l with the spinor w j , and R m lij stands for the component of the Riemann curvature tensor of the target manifold N. Denote Rð/, wÞ : We can write (2.4) and (2.5) in the following global form: and call the solutions ð/, wÞ Dirac-harmonic maps from M to N.
With the aim to get a general existence scheme for Dirac-harmonic maps, the following heat flow for Dirac-harmonic maps was introduced in [7]: When M has boundary, the short time existence and uniqueness of (2.8)-(2.9) was also shown in [7]. Furthermore, the existence of a global weak solution to this flow in dimension two under some boundary-initial constraint was obtained in [14]. In [15], to remove the restriction on the initial maps, the authors refined an estimate about the spinor in [7] as follows: Motivated by this lemma, they considered the a-Dirac-harmonic flow and got the existence of Dirac-harmonic maps. For a closed manifold M, the situation is much more complicated because the kernel of the Dirac operator is a linear space. If the Dirac operator along the initial map has one dimensional kernel, Wittmann proved the short time existence on M whose dimension is m 0, 1, 2, 4ðmod 8Þ: By [18], we can isometrically embed N into R q : Then (2.6)-(2.7) is equivalent to following system: where II is the second fundamental form of N in R q , and

10)
ReðPðSðduðe b Þ, e b Á wÞ; wÞÞ :¼ PðSð@ z C , @ z B Þ; @ z A ÞReðhw A , du C Á w B iÞ: Here Pðn; ÁÞ denotes the shape operator, defined by hPðn; XÞ, Yi ¼ hAðX, YÞ, ni for X, Y 2 CðTNÞ and Re(z) denotes the real part of z 2 C: Together with the nearest point projection: on ð0, TÞ Â M, for A ¼ 1, :::, q: Here we denote the A-th component function of u : for the B-th partial derivative of the A-th component function of p : R q ! R q and the global sections w A 2 CðRMÞ are defined by w ¼ w A ð@ A uÞ, where ð@ A Þ A¼1, :::, q is the standard basis of TR q : Moreover, r and hÁ, Ái denote the gradient and the Riemannian metric on M, respectively.
For future reference, we define Note that for u 2 C 1 ðM, NÞ and w 2 CðRM u Ã TNÞ we have IIðdu p ðe a Þ, du p ðe a ÞÞÞ ¼ ÀF A 1 ðuÞj p @ A j uðpÞ , (2.16) for all p 2 M, where fe a g is an orthonormal basis of T pM . Next, for every T > 0, we denote by X T the Banach space of bounded maps: For any map v 2 X T , the closed ball with center v and radius R in X T is defined by We denote by P u t , v s ¼ P u t , v s ðxÞ the parallel transport of N along the unique shortest geodesic from pðuðx, tÞÞ to pðvðx, sÞÞ: We also denote by P u t , v s the inducing mappings and Now, let us define and c t ðxÞ : ½0, 2p ! C as In general, we also denote by c the curve cðxÞ : ½0, 2p ! C as for some constant K to be determined. Then the orthogonal projection onto kerð = D pu t Þ, which is the mapping can be written by the resolvent by Finally, the following density lemma is very useful for us to extend the flow beyond the singular time.

The heat flow for a-Dirac-harmonic maps
In this section, we will prove the short-time existence of the heat flow for a-Dirac-harmonic maps. Since we are working on a closed surface M, we cannot uniquely solve the Dirac equation in the following system: Rðu, wÞÞ, The short time existence and its extension are the obstacles. This system (if it converges) leads to a a-Dirac-harmonic map which is a solution of the system and equivalently a critical point of functional where s is the tension field.

Short time existence
As in Section 2, we now embed N into R q : Let u : M ! N with u ¼ ðu A Þ and denote the spinor along the map u by w ¼ w A ð@ A uÞ, where w A are spinors over M. For any smooth map g 2 C 1 0 ðM, R q Þ and any smooth spinor field n 2 C 1 0 ðRM R q Þ, we consider the variation where p is the nearest point projection as in Section 2. Then we have Lemma 3.1. The Euler-Lagrange equations for L a are Proof. Suppose ðu, wÞ is a critical point of L a , then for the variation (3.5) we have Then the lemma directly follows from the following computations.
Now, let us state the main result of this subsection.
Since q u 2 T ? pu N and ðdpÞ u : we get @ @t À D uðx, tÞ Cu, where C ¼ Cðjjujj C 2, 1 ðMÂ½0, TÞ Þ: Since uðx, tÞ ! 0 and uðx, 0Þ ¼ 0 for any ðx, tÞ 2 M Â ½0, T, we conclude u ¼ 0 on M Â ½0, T: We have shown that uðx, tÞ 2 N for all ðx, tÞ 2 M Â ½0, T: Finally, by using the -regularity (see Lemma 3.7 below), we conclude that and Since the equations for a-Dirac-harmonic maps are invariant under multiplying the spinor by elements of H with unit norm, by uniqueness we always mean uniqueness up to multiplication of the spinor by such elements. This kind of uniqueness for the Diracharmonic map flow was proved by the Banach fixed point theorem in [6]. However, we cannot apply the fixed point theorem to the a-Dirac-harmonic map flow. Therefore, it is interesting to consider the uniqueness of the a-Dirac-harmonic map flow from closed surfaces. By considering the evolution inequality of jju 1 À u 2 jj C 0 ðMÞ , we can prove the following uniqueness which is weaker than that in [6] because when the quaternions h a are different, we can no longer bound the C 0 -norm of the difference of the maps.

Regularity of the flow
In this subsection, we will give some estimates on the regularity of the flow. Let us start with the following estimate of the energy of the map part. Proof. N ote that (3.1) can be written as: ð1 þ jruj 2 Þ aÀ1 @ t u ¼ divðð1 þ jr u j 2 Þ aÀ1 ruÞ À ð1 þ jr g uj 2 Þ aÀ1 Aðdu, duÞ À 1 a ReðPðAðduðe b Þ, e b Á wÞ; wÞÞ: (3.48) Multiplying the inequality above by @ t u and using 0 ¼ which directly gives us the lemma.
w Consequently, we can also control the spinor part along the heat flow of the a-Diracharmonic map. This lemma follows from applying Lemma 2.1 to gw, where g is a cutoff function.
w To get the convergence of the flow, we also need the following -regularity.
Then there exist three constants 2 ¼ 2 ðM, NÞ > 0, 3 and for any 0 < b < 1, Since M is closed, x 0 has to be an interior point of M. Therefore, our Lemma is just a special case of the Lemma 3.4 in [15]. So we omit the proof here.

Existence of a-Dirac-harmonic maps
In this section, we will prove Theorem 1.2 by the following theorem on the existence of a-Dirac-harmonic maps for a > 1: Suppose there exists a map u 0 2 C 2þl ðM, NÞ for some l 2 ð0, 1Þ such that dim H ker = D u 0 ¼ 1. Then for any a 2 ð1, 1 þ 1 Þ, there exists a nontrivial smooth a-Diracharmonic map ðu a , w a Þ such that the map part u a stays in the same homotopy class as u 0 and jjw a jj L 2 ¼ 1: Proof of Theorem 4.1. By Theorem 2.3 in [21], all the following a-Dirac-harmonic maps are smooth. Let us denote the energy minimizer by Therefore, by Theorem 3.2 and Lemma 3.7, we know that the singular time can be characterized as and there exists a sequence ft i g % T such that ðuðÁ, t i Þ, wðÁ, t i ÞÞ ! ðuðÁ, TÞ, wðÁ, TÞÞ in C 2þl ðMÞ Â C 1þl=2 ðMÞ (4.7) and jjwðÁ, TÞjj L 2 ¼ 1: Together with Lemma 3.7, there is a subsequence, still denoted by ft i g, and an a-Diracharmonic map ðu a , w a Þ such that ðuðÁ, t i Þ, wðÁ, t i ÞÞ converges to ðu a , w a Þ in C 2 ðMÞ Â C 1 ðMÞ and jjw a jj L 2 ¼ 1: If Z 6 ¼ ; and T 2 Z, let us assume that E a ðuðÁ, TÞÞ > m a 0 and ðuðÁ, TÞ, wðÁ, TÞÞ is not already an a-Dirac-harmonic map. We extend the flow as follows: By Lemma 2.3, there is a map u 1 2 C 2þl ðM, NÞ such that m a 0 < E a ðu 1 Þ < E a ðuðÁ, TÞÞ (4.10) and dim H ker = D u 1 ¼ 1: Thus, picking any w 1 2 ker = D u 1 with jjw 1 jj L 2 ¼ 1, we can restart the flow from the new initial values ðu 1 , w 1 Þ: If there is no singular time along the flow started from ðu 1 , w 1 Þ, then we get an a-Dirac-harmonic map as in the case of Z ¼ ;: Otherwise, we use again the procedure above to choose ðu 2 , w 2 Þ as initial values and restart the flow. This procedure will stop in finitely or infinitely many steps.
If infinitely many steps are required, then there exist infinitely many flow pieces fu i ðx, tÞg i¼1, :::, 1 and fT i g i¼1, :::, 1 such that where u i ðÁ, 0Þ ¼ u i 2 C 2þl ðM, NÞ: If the T i are bounded away from zero, then there is ft i g such that (4.9) hold for t i 2 ð0, T i Þ: Therefore, we have an a-Dirac-harmonic map as before. If T i ! 0, then we look at the limit of E a ðu i Þ: If the limit is strictly bigger than m a 0 , we again choose another map satisfying (4.10) and (4.11) as a new starting point. If the limit is exactly m a 0 , then we choose ft i g such that t i 2 ð0, T i Þ for each i. By Lemma 3.7, u i ðt i Þ converges in C 2 ðMÞ Â C 1 ðMÞ to a minimizing a-harmonic map u a : If = D u a has minimal kernel, then for any w 2 ker = D u a , ðu a , wÞ is an a-Dirac-harmonic map as we showed in the beginning of the proof. If = D u a has non-minimal kernel, we use the decomposition of the twisted spinor bundle through the Z 2 -grading G id (see [3]). More precisely, for any smooth variation ðu s Þ s2ðÀ, Þ of u 0 , we split which is orthogonal in the complex sense and parallel. Consequently, for any w 0 2 ker = D u 0 , we have TN and w 6 2 R 6 : Therefore, w 6 s :¼ w 6 u Ã s TN are smooth variations of w 6 0 , respectively, such that d dt t¼0 ð = D u s w 6 s , w 6 s Þ L 2 ¼ 0: (4.14) By taking u 0 ¼ u a and w 0 ¼ w a 2 ker = D u a , the first variation formula of L a implies that ðu a , w 6 a Þ are a-Dirac-harmonic maps (see Corollary 5.2 in [3]). In particular, we can choose w a such that jjw þ a jj L 2 ¼ 1 or jjw À a jj ¼ 1: If it stops in finitely many steps, there exists a sequence ft i g and some 0 < T k þ1 such that lim t i %T ðuðÁ, t i Þ, wðÁ, t i ÞÞ ! ðu a , w a Þ in C 2 ðMÞ Â C 1 ðMÞ, (4.15) where ðu a , w a Þ either is an a-Dirac-harmonic map or satisfies E a ðu a Þ ¼ m a 0 : And in the latter case, u a is a minimizing a-harmonic map. Then we can again get a nontrivial a-Dirac-harmonic map as above. for any 1 < p < 2: Then it is natural to consider the limit behavior when a decreases to 1. Since the blow-up analysis was already well studied in [15], we can directly prove Theorem 1.2.

Appendix
In Section 3, we used some convenient properties of the elements in B T R ð u 0 Þ: Those properties were already discussed in [6]. However, the function space used there is where p is the heat kernel of M, F 1 and F 2 are defined as in (2.17) and (2.18), respectively. Our proof for the short-time existence is different from there, and the space B T R ð u 0 Þ is more natural and convenient in our situation. Therefore, we cannot directly use the statement in [6]. Although the space is changed, the proofs of those nice properties are parallel. In fact, one can see from the following that to make the elements in B T R ð u 0 Þ satisfy nice properties (5.11) and (5.12), it is sufficient to choose R small, namely, T is independent of R. This is the biggest advantage. In the following, we will give the precise statement of the properties we need in Section 3 and proofs for the most important lemmas.
For every T > 0, we consider the space B T R ð u 0 Þ :¼ fu 2 X T jjju À u 0 jj X T Rg \ fuj t¼0 ¼ u 0 g where u 0 ðx, tÞ ¼ u 0 ðxÞ for any t 2 ½0, T: To get the necessary estimate for the solution of the constraint equation, we will use the parallel transport along the unique shortest geodesic between u 0 ðxÞ and p u t ðxÞ in N. To do this, we need the following lemma which tells us that the distances in N can be locally controlled by the distances in R q : Lemma 5.1. [6] Let N & R q be a closed embedded submanifold of R q with the induced Riemannian metric. Denote by A its Weingarten map. Choose C > 0 such that jjAjj C, where Then there exists 0 < d 0 < 1 C such that for all 0 < d d 0 and for all p, q 2 N with jjp À qjj 2 < d, it holds that where we denote the Euclidean norm by jj Á jj 2 in this section.
Lemma 5.2. Choose , d and R as in (5.10). If > 0 is small enough, then there exists C ¼ CðRÞ > 0 such that where exp denotes the exponential map of the Riemannian manifold N. Note that FðÁ, 0Þ ¼ f 0 , FðÁ, 1Þ ¼ f 1 and t 7 ! Fðx, tÞ is the unique shortest geodesic from f 0 ðxÞ to f 1 ðxÞ: We denote by the parallel transport in F Ã TN with respect to r F Ã TN (pullback of the Levi-Civita connection on N) along the curve c x ðtÞ :¼ ðx, tÞ from c x ðt 1 Þ to c x ðt 2 Þ, x 2 M, t 1 , t 2 2 ½0, 1: In particular, P 0, 1 ¼ P v s , u t : Let w 2 C C 1 ðRM ðf 0 Þ Ã TNÞ: We have where w ¼ w i ðb i f 0 Þ, fb i g is an orthonormal frame of TN, w i are local C 1 sections of RM, and fe a g is an orthonormal frame of TM.
We So far, we only know that the T ij are continuous. In the following, we will perform some formal calculations and justify them afterwards. By a straightforward computation, we have for all u, v 2 B T R ð u 0 Þ, x 2 M and t, s 2 ½0, T: The proofs of these two lemmas only depend on the existence of the unique shortest geodesic between any two maps in B T R ð u 0 Þ, which was already shown in (5.12). Therefore, we omit the detailed proof here. Besides, by Lemma 5.2, one can immediately prove the following Lemma by the Min-Max principle as in [6].
Lemma 5.5. Assume that dim K kerð= D u 0 Þ ¼ 2l À 1, where l 2 N and Choose , d and R as in Lemma 5.2. If R is small enough, then for any u 2 B T R ð u 0 Þ and t 2 ½0, T, where Kðu 0 Þ is a constant such that specð = D u 0 Þ n f0g & R n ðÀKðu 0 Þ, Kðu 0 ÞÞ: Once we have the minimality of the kernel in Lemma 5.5, we can prove the following uniform bounds for the resolvents, which are important for the Lipschitz continuity of the solution to the Dirac equation.
Lemma 5.6. Assume we are in the situation of Lemma 5.5. We consider the resolvent Rðk, = D pu t Þ : By the L p estimate (see Lemma 2.1 in [6]), we know the restriction is well-defined and bounded for any 2 p < 1. If R > 0 is small enough, then there exists C ¼ Cðp, RÞ > 0 such that sup jkj¼ K 2 jjRðk, @@D pu t Þjj LðL p , W 1, p Þ < C (5.33) for any u 2 B T R ð u 0 Þ, t 2 ½0, T: Now, by the projector of the Dirac operator, we can construct a solution to the constraint equation whose nontrivialness follows from the following lemma.
Lemma 5.7. In the situation of Lemma 5.5, for any fixed u 2 B T R ð u 0 Þ and any w 2 kerð = D u 0 Þ with jjwjj L 2 ¼ 1, we have ffiffi ffi 1 2 r jjw u t 1 jj L 2 1, (5.34) In Section 3, to show the short-time existence of the heat for a-Dirac-harmonic maps, we need the following Lipschitz estimate. for any u 2 B T R ð u 0 Þ, where c is defined in the Section 2 with K ¼ 1 2 Kðu 0 Þ. In particular, wðu t Þ 2 kerð = D pu t Þ & C C 0 ðRM ðp u t Þ Ã TNÞ. We write wðu t Þ :¼ wðuðÁ, tÞÞ ¼ wðu t Þ jj wðu t Þjj L 2 : (5.36) Let w A ðu t Þ be the sections of RM such that wðu t Þ ¼ w A ðu t Þ ð@ A p u t Þ (5.37) for A ¼ 1, :::, q: Then there exists C ¼ CðR, , w 0 Þ > 0 such that jjP u t , v s wðu t ÞðxÞ À wðu t ÞðxÞjj Cjju t À v s jj C 0 ðM, R q Þ (5.38) and jjw A ðu t ÞðxÞ À w A ðv s ÞðxÞjj Cjju t À v s jj C 0 ðM, R q Þ (5.39) for all u, v 2 B T R ð u 0 Þ, A ¼ 1, :::, q, x 2 M and s, t 2 ½0, T: Proof. Using the following resolvent identity for two operators D 1 , D 2 Rðk, D 1 Þ À Rðk, D 2 Þ ¼ Rðk, D 1 Þ ðD 1 À D 2 Þ Rðk, D 2 Þ, (5.40) we have Rðk, P u t , v s = D pu t ðP u t , v s Þ À1 ÞðP u t , v s P u 0 , u t w 0 À P u 0 , v s w 0 Þ À 1 2pi ð c ðRðk, P u t , v s = D pu t ðP u t , v s Þ À1 Þ À Rðk, = D pv s ÞÞP u 0 , v s w 0 ¼ À 1 2pi ð c Rðk, P u t , v s = D pu t ðP u t , v s Þ À1 ÞðP u t , v s P u 0 , u t w 0 À P u 0 , v s w 0 Þ À 1 2pi ð c ðRðk, P u t , v s = D pu t ðP u t , v s Þ À1 Þ P u t , v s = D pu t ðP u t , v s Þ À1 À = D pv s À Á Rðk, = D pv s ÞÞP u 0 , v s w 0 , (5.41) where c is defined in (2.29) with K ¼ 1 2 Kðu 0 Þ: Therefore, for p large enough, we get jjP u t , v s wðu t ÞðxÞ À wðu t ÞðxÞjj C 1 jjP u t , v s w u t À w v s jj W 1, p ðMÞ C 2 jj ð c Rðk, P u t , v s = D pu t ðP u t , v s Þ À1 ÞðP u t , v s P u 0 , u t w 0 À P u 0 , v s w 0 Þjj W 1, p ðMÞ þC 2 k ð c ðRðk, P u t , v s = D pu t ðP u t , v s Þ À1 Þ ðP u t , v s = D pu t ðP u t , v s Þ À1 À = D pv s Þ Rðk, = D pv s ÞÞP u 0 , v s w 0 k W 1, p ðMÞ C 2 ð c jjRðk, P u t , v s = D pu t ðP u t , v s Þ À1 ÞðP u t , v s P u 0 , u t w 0 À P u 0 , v s w 0 Þjj W 1, p ðMÞ þC 2 ð c jjðRðk, P u t , v s D pu t ðP u t , v s Þ À1 Þ ðP u t , v s = D pu t ðP u t , v s Þ À1 À = D pv s Þ Rðk, D pv s ÞÞP u 0 , v s w 0 jj W 1, p ðMÞ C 3 sup ImðcÞ jjRðk, P u t , v s = D pu t ðP u t , v s Þ À1 Þjj LðL p , W 1, p Þ jjP u t , v s P u 0 , u t w 0 À P u 0 , v s w 0 jj L p þ C 3 sup ImðcÞ jjRðk, P u t , v s = D pu t ðP u t , v s Þ À1 Þjj LðL p , W 1, p Þ sup ImðcÞ jjRðk, = D pv s Þjj LðL p , W 1, p Þ jjP u t , v s = D pu t ðP u t , v s Þ À1 À = D pv s jj LðW 1, p , L p Þ jjP u 0 , v s w 0 jj L p : Now, we estimate all the terms in the right-hand side of the inequality above. First, by Lemmas 5.6 and 5.4, we know that all the resolvents above are uniformly bounded. Next, by Lemma 5.2, we have jjP u t , v s = D pu t ðP u t , v s Þ À1 À = D pv s jj LðW 1, p , L p Þ CðRÞjju t À v s jj C 0 ðM, R q Þ : (5.43) Finally, by Lemma 5.3, we obtain jjP u t , v s P u 0 , u t w 0 À P u 0 , v s w 0 jj L p Cð, w 0 Þjju t À v s jj C 0 ðM, R q Þ : (5.44) Putting these together, we get (5.38). Next, we want to show the following estimate which is very close to (5.39).
jj w A ðu t ÞðxÞ À w A ðv s ÞðxÞjj CðR, , w 0 Þjju t À v s jj C 0 ðM, R q Þ : (5.45) In fact, we have jj w A ðu t ÞðxÞ À w A ðv s ÞðxÞjj jj wðu t ÞðxÞ À wðv s ÞðxÞjj R x MR q jjP u t , v s wðu t ÞðxÞ À wðv s ÞðxÞjj R x MR q þ jjP u t , v s wðu t ÞðxÞ À wðu t ÞðxÞjj R x MR q ¼ jjP u t , v s wðu t ÞðxÞ À wðv s ÞðxÞjj R x MT ðpvsðxÞÞ N þ jjP u t , v s wðu t ÞðxÞ À wðu t ÞðxÞjj R x MR q CðR, , w 0 Þjju t À v s jj C 0 ðM, R q Þ þ jjP u t , v s wðu t ÞðxÞ À wðu t ÞðxÞjj R x MR q : It remains to estimate the last term in the inequality above. To that end, let cðrÞ :¼ exp ðpu t ÞðxÞ ðr exp À1 ðpu t ÞðxÞ ðp u t ðxÞÞÞ, r 2 ½0, 1, be the unique shortest geodesic of N from ðp u t ÞðxÞ to ðp v s ÞðxÞ: Let X 2 T cð0Þ N be given and denote by X(r) the unique parallel vector field along c with Xð0Þ ¼ X: Then we have P u t , v s X À X ¼ Xð1Þ À Xð0Þ ¼ Therefore, where II is the second fundamental form of N in R q and C 1 only depends on N. Using (5.9) and the Lipschitz continuity of p we get jjc 0 ð0Þjj N d N ððp u t ÞðxÞ, ðp v s ÞðxÞÞ C 2 jju t ðxÞ À v s ðxÞjjj R q (5.48) and jjP u t , v s X À Xjj R q C 3 jju t ðxÞ À v s ðxÞjjj R q jjXjj N : (5.49)