Curvature estimate of the Yang-Mills-Higgs flow on Kähler manifolds

Using the decomposition of Donaldson’s functional and the properties of Harder-Nasimhan-Seshadri filtration, we get the C 0 -bound of the rescaled metrics Abstract: The curvature estimate of the Yang-Mills-Higgs flow on Higgs bundles over compact Kähler manifolds is studied. Under the assumptions that the Higgs bundle is non-semistable and the Harder-Narasimhan-Seshadri filtration has no singularities with length one, it is proved that the curvature of the evolved Hermitian metric is uniformly bounded.


Introduction
Let be a Higgs sheaf over a compact Kähler manifold . Higgs bundles and Higgs sheaves, which were studied by Hitchin [1] and Simpson [2,3] , play an important role in many different areas including gauge theory, Kähler and hyperkähler geometry, group representations, and nonabelian Hodge theory. A Higgs sheaf on is a pair , where is a coherent sheaf on M, and the Higgs field is a holomorphic section such that . A torsion-free Higgs sheaf is -stable (resp.semistable) if for every -invariant proper coherent sub-sheaf such that is called the -slope of , and -degree of is defined as follows: is the first Chern class of .
Let Σ ε be a set of singularities, where is not locally free. If is locally free on the whole M, i.e., Σ ε , there is a Higgs bundle on M such that the Higgs sheaf is generated by the local holomorphic sections of . A locally free Higgs sheaf can be considered as a Higgs bundle, i.e., .
(E, θ) θ Given an unstable torsion-free coherent Higgs sheaf , one can associate a filtration by -invariant coherent subsheaves as follows: such that the quotient is torsion-free, semistable, and , which is called the Harder- [ω] Q i Narasimhan filtration of . The associated graded object is uniquely determined by the isomorphism class of and the Kähler class . Moreover, for every quotient , a further filtration by sub-sheaves exists, such that the quotients are torsion-free andstable, and for each . The double filtration is called the Harder-Narasimhan-Seshadri (HNS) filtration of the Higgs sheaf . The associated graded object (E, θ) [ω] is uniquely determined by the isomorphism class of and Kähler class . The number is the length of the HNS filtration.
H (E,∂ E , θ) Given a Hermitian metric on the Higgs bundle , we consider the Hitchin-Simpson connection: where is the Chern connection with respect to the Hermitian metric and is the adjoint of with respect to . The curvature of the Hitchin-Simpson connection is expressed as follows: where is the curvature of the Chern connection, denoted by . A Hermitian metric on the Higgs bundle is said to be -Hermitian-Einstein if it satisfies the following Einstein condition on , i.e., where and denotes the contraction with the Kähler metric . Hitchin [1] and Simpson [2] proved that a Higgs bundle admits a Hermitian-Einstein metric if and only if it is Higgs poly-stable. Many interesting and important generalizations and extensions can also be found in the literature (see Refs. [1, 4 -12] ).
Let be a Hermitian metric on the complex vector bundle , be the space of connections of compatible with the metric , and be the space of the unitary integrable connection of . A pair is a Higgs pair if and . Let denote the space of all Higgs pairs on the Hermitian vector bundle . We consider the following Yang-Mills-Higgs flow on the Hermitian vector bundle with initial data : The Yang-Mills flow was first introduced by Atiyah-Bott in Ref. [13]. Simpson [2] induces the following Hermitian-Yang-Mills-Higgs (HYMH) flow for Hermitian metrics on the Higgs bundle with initial metric : , Simpson [2] proved the long-time existence of the Hermitian-Yang-Mills-Higgs flow and demonstrated convergence under the condition that the Higgs bundle is stable. In Ref. [14], Li and Zhang showed that by choosing complex gauge transformations that satisfy , is the unique long-time solution of the Yang-Mills-Higgs flow Eq. (8).
According to Uhlenbeck's compactness [15,16] , for any sequence along the flow, there is a subsequent that weakly converges to a Yang-Mills connection in the gauge transformation sense outside a closed subset Σ an of Hausdorff complex co-dimension of at least 2. We call Σ an the bubbling set or the analytic singular set. In contrast, we denote Σ alg as the singular set of the associated graded object , i.e., is locally free away from Σ alg , which is a complex analytic sub-variety of complex co-dimension of at least 2 that we call algebraic singular set. According to the results in Refs. [17,18], it is not difficult to obtain that Σ alg Σ an . It is an interesting problem to demonstrate that Σ an Σ alg . This problem was solved by Daskalopoulos and Wentworth in Ref. [19] for Kähler surfaces, and by Sibley and Wentworth in Ref. [20] for Kähler manifolds. In this paper, we derive the curvature estimate in the case where is non-semistable and the Harder-Narasimhan-Seshadri filtration has no singularities with length one, which generalizes Li, Zhang, and Zhang's result [21] in the Higgs bundle case. In fact, we obtain the following theorem: Theorem 1.1. Let be a Hermitian vector bundle on a compact Kähler manifold , and be the solution of the Yang-Mills-Higgs flow (8) with an initial Higgs . If the Higgs bundle is non-semistable and the HNS filtration has no singularities with length one, then there exists a uniform constant such that If the algebraic singular set Σ alg , we hypothesize that the theorem holds away from the algebraic singular set. The proof is complicated, and we mainly adapt some Li, Zhang, and Zhang's techniques to our cases of interest. However, the proof is even more complicated in the Higgs bundle case. We next provide an overview of the proposed proof. Let be the long-time solution of the HYMH flow (9). It is well known that Therefore, we only need to estimate the curvature tensor of the Hitchin-Simpson connection with respect to the evolved Hermitian metric . For simplicity, we denote as as a fixed Higgs field. According to the assumption of Theorem 1.1, there exists an exact sequence ω such that and become torsion-free,stable Higgs bundles, and Let and be the Hermitian metrics on the Higgs subbundle and quotient bundle , respectively, induced by the evolved metric on , as the second fundamental form, and let be the form induced by the Higgs field and .We derive the evolution equations for , , , and (see Eqs. (20), (21), (23), and (24)). In Ref. [2], Simpson generalized the Hermitian-Yang-Mills flow to the Higgs bundle case. Under the assumption of stability, using the results of Uhlenbeck and Yau [22] , Simpson obtained a uniform -estimate on . This implies uniform higher-order estimates including the uniform curvature estimate. When is unstable, the -norm of the evolved metric may be unbounded. In our case, we first obtain a uniform -bound on the rescaled metrics, i.e., and . This is crucial in the proof of the proposed theorem, where we use the stabilities of and , and the property . Then, we prove that the norms of , , , and are uniformly bounded using the uniformestimate. By choosing suitable test functions and using the maximum principle, we obtain a uniform estimate of .
The remainder of this paper is organized as follows. In Section 2, we derive the evolution equations for the induced metrics and , the second fundamental forms , and the induced forms . We also obtain the decomposition of Donaldson's functional, which is used in Section 3, where we derive a uniform -bound for the rescaled metrics. In Section 4, we obtain the uniform -estimate and complete the proof of Theorem 1.1.
where is the inclusion map and is the orthogonal projection into with respect to metric . Then, the pull-back metric, holomorphic structure, and Higgs field on respectively are and . The corresponding Gauss-Codazzi equation for the Hitchin-Simpson connection is Let be the solution of the HYMH flow (9) on the Higgs bundle with initial metric . Now, we split the HYMH flow into the sub-bundle and quotient bundle . Recall from Ref. [21] that and Using the flow (9) and Gauss-Codazzi (17) equations, we have Now, we consider the evolution of the second fundamental and : Let be the solution of the HYMH flow (9) with initial metric and be the second fundamental form, and let be induced by and . Then we have and Proof. For simplicity, we let and denotē and then For the first evolution equation of , taking the derivative of the above equation with respect to , we have that ) . Thus, we obtain the first evolution Eq. (23).
Concerning the second evolution equation of , taking the derivative of with respect to , we have that As a result, we obtain that .
Let be the bundle isomorphism defined in (15). Then, we have that Proof. Similar to the proof in Ref. [ As a result, we obtain Eq. (28).

C 0
In the proof of the -estimate, we need to decompose Donaldson's functional in the Higgs bundle case, which was proved by Donaldson (see Ref. [23]) for the holomorphic vector bundle case.
Let us recall Donaldson's functional in the Higgs bundle case:

∂H(s) ∂s
) ω n n! ds (31) and where is the path connecting the metrics and on . Donaldson proved that the integral above is independent of the path when the base manifold is Kähler.
If we have an exact sequence of Higgs bundles, then where is -invariant and and are the Higgs fields on and , respectively, induced by . Recall that a Hermitian metric on induces Hermitian metrics , on , , respectively. Note also that is the second fundamental form, and is determined by the Higgs field and Hermitian metric . Then we have the following lemma: For any exact sequence of Higgs bundles (33), Donaldson's functional has the following decomposition: and are the Hermitian metrics on and induced by on . is a path-connecting metric between and .
M 0 E t Proof. Taking derivative of with respect to , we have d dt Then, we obtain Donaldson's functional of the pullback metric : ) ω n n! =: Recall that Using Eq. (36) and Stokes formula, we have Given that d dt we obtain Given that We can obtain Thus, we have Given that d dt we obtain Combining Eqs. (38) and (42), we have d dt 0 t Integrating Eq. (43) from to , we obtain Eq. (34).
At the end of this section, we consider parabolic inequalities for and , which will be used in the next section. Through direct calculation, we obtain and

C 0 -estimate of the rescaled metrics
ω Note that the Higgs bundles and arestable. According to the Donaldson-Uhlenbeck-Yau theorem, we can suppose that and are -Hermitian-Einstein metrics on and , i.e., Let us denote , , and set and . Using (20) and (21), we have that and where we have applied the non-negativity of and . We also used the following equalities: and Using inequalities (48) and (49), and the maximum principle, we can obtain a uniform bound on . In fact, we obtain the following lemma: In the following, we will derive uniform upper bounds on and , which imply uniform upper bounds on and using (52). Then, we will obtain uniform -bounds on and . At the beginning of the proof, the following proposition are required: Proposition 3.2. Along the heat flow (9), we have satisfies , and is a uniform constant.
Proof. From Lemma 2.3, for any exact sequence of Higgs bundles, (55) According to the flow equations (20) and (21), we have that ) ω n n! dl (58) Furthermore, from the definition of Donaldson's functional and Gauss-Codazzi Eq. (17), it follows that Given that and are -stable Higgs bundles over , Donaldson's functional and are bounded from below. Together with the above Eq. (59), we obtain +∞ 0ṽ (t)dt ⩽C 1 < +∞ where is a uniform positive constant. By contrast, along with the heat flow Eq. (9), we have ( According to the estimate of the heat kernel by Cheng and Li in Ref. [24] (or see Theorem 3.2 in Ref. [25]), a positive constant exists such that t > 0 s > 0 for any , . Using the Gauss-Codazzi equation (17) and (64), we have Combining formulas (61) and (65), and noting that , we conclude that is the targeted function.
Using Proposition 3.2, we obtain a uniform -bound on the rescaled metrics and .
be the solution of the Hermitian-Yang-Mills-Higgs flow (9) on the Higgs bundle with initial metrics , , and be the induced Hermitian metrics on and . Set and . Then, there exists a uniform constant such that Noting that the metrics and are fixed Hermitian metrics, we can check that for the entire , where is a uniform constant. Then, we obtain a uniform constant such that and ∆ log According to formulas (70), (71) and Moser's iteration, we have the following mean inequalities, including a uniform constant such that and where denotes a uniform constant. According to (20), it follws that ∂ ∂t From (62), the Gauss-Codazzi equation (17), and the maximum principle, we have and sup (x,t)∈M×[0,+∞) and . Direct calculations yield Substituting (97) into (96), choosing the constants and sufficiently large, and using formulas (90) and (92), at the maximum point we have where is a positive constant that depends only on the local uniform bound of and curvature of .
This completes the proof of Theorem 1.1.