On Existences Of Special Metrics And Connections In Complex Vector Bundle

The content of this paper is divided into three parts. In the first part, we investigate the coupled vortex equation on holomorphic vector bundles, we have solved the Dirichlet problem for coupled vortex equation, proved the existence of coupled vortices (i.e. the solutions of coupled vortex equations) on a class of complete non-compact Ka|¨hler manifolds. In the second part, we study the coupled Yang-Mills-Higgs equation on vector bundle. The coupled Yang-Mills-Higgs equation can be viewed as a generalization of the Yang-Mills equation and Yang-Mills-Higgs equation. We obtain some properties of the coupled Yang-Mills-Higgs heat flow, including the energy inequality, Bochner-type inequality and monotonicity of certain quantities;we also discuss the asymptotic behavior of a regular coupled Yang-Mills-Higgs flow. In the third part, the space-like hypersurfaces with constant scalar curvature in a de Sitter space are studied and a pinching theorem on the square of the length of the second fundamental form is obtained.0.1 Coupled vortex equations on holomorphic vector bundlesLet us recall some standard geometric notations and definitions. Let (M, η) be a compact Ka|¨hler manifold, 77 be the Kahler form, and E be a holomorphic vector bundle over M.The stability of vector bundles (introduced by Mumford in his study of moduli for bundles [Muj) is a well established concept in algebraic geometry. A holomorphic vector bundle E is called stable if for every coherent sub-sheaf E′→ E of lower rank (or, weakly holomorphic sub-bundle) it holds that μ(E′) < μ(E), where the η-slope of E′ is defined bythe quotientand the 77-degree of E' is defined byde9r,(E') = I d^Ar,""1. (0.1.2)Here C\{E') is the first chern class of E' (similarly forHermitian-Einstein metric was introduced by Kobayashi in 1980 in any arbitrary holomorphic vector bundle over a complex manifold, which can be viewed as a generalization of a Kahler-Einstein metric in the tangent bundle.A Hermitian metric H in E is called a Hermitian-Einstein metric if the curvature Fjj of the chern connection Ah in (E,H) (i.e., the unique iiT-unitary integrable connection Ah in E inducing the holomorphic structure Be) satisfies the Einstein condition:V^lAr,FH = -yIdE, (0.1.3)where A^ denotes the contraction of differential forms with Kahler form 77, and the real constant 7 is given by 7 = {n^iA H-unitary connection A is called Hermitian-Einstein if it's curvature 2-form Fa is of type (1,1) and satisfies the above Einstein condition.The classical Hitchin-Kobayashi correspondence states that a holomorphic structure is stable if and only if it is simple (i.e. it admits no non-trivial trace free infinitesimal automorphisms ) and admits a Hermitian-Einstein metric. General solutions of the Hermitian-Einstein equation correspond to polystable holomorphic structure, i.e. to bundles which are the direct sum of stable bundles of the same slope.A short history of the proof of the Hitchin-Kobayashi correspondence is as follows.Narasimhan and Seshadri ([NS], 1965) proved this theorem for vector bundles over a compact Riemannian surface. For higher dimensional cases, the correspondence between the stability of bundles and the existence of Hermitian-Einstein metrics was suggested independently by Hitchin and Kobayashi. Kobayashi [Ko] and Lubke [Lu] found that any holomorphic bundle over a Kahler manifold which admits an irreducible Hermitian-Einstein metric (connection) is stable. Donaldson ([Dol], 1983,) obtained the first existence proof of the Hermitian-Einstein metric for stable bundles over algebraic surfaces by introducing a Yang-Mills gradient flow. Uhlenbeck and Yau ([UY], 1986) used a more direct continuity method to prove the higher dimensional's case. Li and Yau ([LY], 1987) proved the correspondence over Hermitian manifolds with Gauduchon metric (i.e. dd{rfl~1) = 0). P.D.Bartolomeis and G.Tian ([BT], 1996) generalized it to the Almost-Hermitian case with other condition.Some generalizations of the classical Hitchin-Kobayashi correspondence. Theclassical Hitchin-Kobayashi correspondence has several interesting and important generalizations. For example, Higgs bundle was initiated by Hitchin [Hi]. He studied the Higgs bundles on a compact Riemann surface and the moduli space. His work has influenced various fields of mathematics. C.Simpson [Si] studied the Higgs bundles over higher dimensional Kahler manifolds.Holomorphic pair. Different from the Higgs bundle, Bradlow [Brl], [Br2] considered holomorphic vector bundles on which additional data in the form of a prescribed holomorphic global section ). Bradlow investigated the follow vortex equation2V^TAFh + ?* - rIdE = 0, (0.1.4)where * is the adjoint of with respect to metric H, and r is a real number. This equation generalizes the Hermitian-Einstein equation and is the analog of the classical vortex equation over R2.Holomorphic triple. Holomorphic triple consists of two holomorphic vector bundles E\,E<2 over complex manifold M and a holomorphic rnorphism : E2 —? E\ (i.e. dEi?E% — 0). The Coupled Vortex equation is a generalization of the vortex equations, were introduced by Garcia-Prada [GP1]. The equations we shall consider are+ \4> o * - nIdEl = 0,(0.1.5) -iPocf,- T2IdE2 = 0,where Hi and H2 are Hermitian metrics on bundles E\, E2 respectively.Bradlow and Garcia-Prada [BG] proved there is a Hitchin-Kobayashi type correspondence between stable triples, and showed a solution of the above Coupled Vortex equations.Holomorphic pair can be viewed as a special case of holomorphic triple, i.e. let Ei be a line bundle L on M, (E,L,) = (E,cj>). In [GP], Garcia-prada show that the above coupled vortex equation can also be obtained via dimensional reduction of classical Hermitian-Einstein equations under an SU(2) action on certain associated bundles on the manifold M x CP1 (77 auCpi). Recently, Garcia-prada's idea has been used by Tian and Yang to study the compactification of the moduli space of vortices and coupled vortices.In this paper, we want to investigate the Coupled Vortex equation over compact Hermitian manifolds with non-empty smooth boundary and some complete non-compact Kahler manifolds. Firstly, we consider the evolution equation of the Coupled Vortex equations, i.e. the following equations,- T2IdE2).We give the long-time existence of the above heat flow over any compact Hermitian manifolds (with or without boundary), i.e. we obtainTheorem 1 (long-time existence) Let M be a compact Hermitian manifold without boundary (with non-empty smooth boundary). Given any initial Hermitian metrics (givenany data on the boundary), the evolution equations (0.1.6) have a unique solution which exists for 0 < t < oo.When M is closed, the long-time solution of equations (0.1.6) usually will not converge. But when M is compact with non-empty boundary, we can prove that the long time solution of heat flow (0.1.6) must converge to (as t —? oo) a solution of the coupled vortex equations (0.1.5) satisfying the boundary condition. So, we have solved the Dirichlet boundary value problem for the Coupled Vortex equations, i.e.,Theorem 2 (Dirichlet problem) Let E = E\ ? Ei be a holomorphic vector bundle over the compact Hermitian manifold M with non-empty smooth boundary dM, where E\,E : Ei —> E\. For any Hermitian metric tp = ( o 4>*H = nIdEl,- \PH o = r2IdE2, (0.1.7).Hi\dM=. We obtainTheorem 3 Let M be an m-dimensional complete noncompact Kahler manifold without boundary, let E = E\ ? Ei be a holomorphic vector bundle over At with initial Hermitianmetric Hq — (Hq,Hq), here Hq, Hq are Hermitian metrics on E\, ￡2 respectively, and holomorphic morphism (j> : E2 —> E\. Let6^ = |V-lAFHi + -^of10 - rxIdEl\U + W-lkFH2 - -cf>*H° o 0 - T2ldE2\ir2Assume i/iai Ai(M) > 0, where Xi(M) denotes the lower bound of the spectrum of the Lapla-cian operator, and that \\&\\lp(M) < °° for some p > 1 and real numbers t\, T2- Then there exists an Hermitian metric H = (#1,1/2) on E such that H satisfies the Coupled Vortex equations< (0.1.8)I y/=lAFHa - WH o <$> - r2IdE,.Remark: The examples satisfying the assumption of theorem 3 include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric. According to Theorem 1.4-A of [Gro] the universal cover of any Kdhler hyperbolic manifolds (in the sense of Gromov) satisfies the assumption of Theorem 3 too. Therefore Theorem 3 can be applied to a broad class of Kdhler manifolds.On the other hand, we give the long time existence of the coupled vortex flow on any-complete Hermitian manifold under an assumption on the initial metrics.Theorem 4 Let M be a complete noncompact Hermitian manifold without boundary, let E = E\ ? E2 be a holomorphic vector bundle over M with initial Hermitian metric Hq — (Hq,Hq), here Hq, Hq are Hermitian metrics on E\, E2 respectively, and holomorphicmorphism cf> : Ei —* E\. Let2TSuppose that there exists a positive number Co such that 02 < Co everywhere. Then the Coupled Vortex heat flow (0.1.6) has a long-time solution on M x [0, oo).0.2 The Coupled Yang-Mills-Higgs flowGiven a vector bundle E over a closed (i.e. compact and without boundary) Riemannian manifold M, suppose that the bundle E has a Riemannian structure. The Yang-Mills functional is defined on the space of connections of E, where all connections are required to be compatible with the Riemannian structure of E. It can be written asYM(A)= [ \FA\2dVg, (0.2.1)JMwhere A is a connection and FA denotes its curvature and dVg is the volume form of g. We call A a Yang-Mills connection of E if A is a critical point of the Yang-Mills functional i.e. A satisfies the Yang-Mills equationDAFA = 0, (0.2.2)where D*A is the adjoint operator of the covariant differentiation associated with the connection A.The Yang-Mills-Higgs functional is defined through a connection A and a section u of the bundle EYMH(A,u)= I \\FA\2 + \DAu\2 + hl-\u\2)2)dVg. (0.2.3)JM 4Yang-Mills-Higgs fields (A,u) are the critical points for the above Yang-Mills-Higgs functional. Equivalently, the pairs (A, u) satisfy the following Yang-Mills-Higgs equationswhere u* denote the dual of u respect to the given metric. The Yang-Mills-Higgs theory can be viewed as a generalization of the Yang-Mills theory. For further discussions on its physical meaning, we refer the readers to [JT].In this paper, we are interested in a more general case. Let (E\,H\) and (.Eg,#2) be two Riemannian vector bundles on the manifold (M,g), and let A\ denote the set of all connections on (Ei, Hi). Consider the following Yang-Mills-Higgs type functional, which will be called Coupled Yang-Mills-Higgs functional (CYMH), on A\ x A2 x Q,°(Ei E%),YMH(Al,A2,4>) = fM \FAl\2 + \FA2\2 + \DAl9A*\2(0.2.5)+ 110 o 0* - TlIdEl I2 + \\F o - T2IdE2\2dVg,where r\ and t2 are real parameters. We denote the integrand above by e(A\, A2,4>) and call it the CYMH action density for the triple (Ai,A2,The equations we shall consider are> = 0,l = 0, (0.2.7). AFA2 + ^r o cj> + ^r2IdE2 = 0, where the operator A is the contraction with w. The above equations are called the Coupled Vortex equations which were introduce by Garcia-Prada in [GP], and solutions (Ai,A2,) = 2||^||2 + 4(||F°f ||2 + ||F°;2||2)