By . is named d-bounded if there exists a differential form on X such that = d and L . (iii) is named d-bounded if is d-bounded on X. (ii) Remark 5. When X is compact, these notions bring nothing at all new. When X is non-compact, it’s PHA-543613 supplier uncomplicated to confirm that d-boundedness implies d-boundedness, whereas there is certainly no direct relationship involving boundedness and d-bounxdedness. The K ler hyperbolic manifold is then defined as Definition 5. A K ler manifold ( X, ) is named K ler hyperbolic if is d-bounded. We list some functionality property from the K ler hyperbolicity here. They’re just about obvious, and one could refer to [13] for much more particulars. Proposition 1. (i) Let X be a K ler hyperbolic manifold. Then, just about every complicated submanifold of X is still K ler hyperbolic. In reality, if Y is usually a complex manifold which admits a finite morphism Y X, then Y is K ler hyperbolic. (ii) Cartesian item of K ler hyperbolic manifolds is K ler hyperbolic. (iii) A full K ler manifold ( X, ) with unfavorable sectional curvature must be K ler hyperbolic. This truth was pointed out in [13], whose proof is usually located in [18]. More precisely, if sec -K, there exists a 1-form on X such that = d and three.two. Notations and Conventions We make a short introduction for the basic notations and conventions in K ler geometry to finish this section. We propose readers to view [15] for a sophisticated comprehension. Let ( X, ) be a K ler manifold of dimension n, and let ( L, ) be a holomorphic line bundle on X endowed using a smooth metric . The typical operators, for instance , also as L, , and so forth., in K ler geometry are defined locally and therefore make sense with or without having the compactness or completeness assumptions. For an m-form , we define e := Let D = be the Chern connection on L associated with . Additionally, for an L-valued k-form , we define the operators D := (-1)nnk1 D , := (-1)nnk1 , : = (-1)nnk1 and e ( ) : = (-1)m(k1) e ( ) . Let A p,q ( X, L) be the space of all the smooth L-valued ( p, q)-forms on X. The pointwise inner product , on A p,q ( X, L) is defined by the equation: , , dV := e-LK- two .Symmetry 2021, 13,5 offor , A p,q ( X, L). The pointwise norm | |, is then induced by , . The L2 -inner product is defined by(, ), :=X , , dVfor , A p,q ( X, L), and the norm , is induced by ( , . p,q Let L(two) ( X, L) be the space of each of the L-valued (not essential to be smooth) ( p, q)-forms with bounded L2 -norm on X, and it equipped with ( , becomes a Hilbert space. The operators D , , and are then the adjoint operators of D, , and with respect to ( , if X is compact. Nevertheless, when X is non-compact, the scenario will be much more complex. We will handle it in the next section. 4. The Hodge Decomposition The Hodge decomposition is definitely the ingredient to study the geometry of a Alvelestat tosylate compact K ler manifold. One particular can consult [14,15] for a full survey. In this section, we’ll talk about the Hodge decomposition on a non-compact manifold. Let ( X, ) be a full K ler manifold of dimension n with damaging sectional curvature, and let ( L, ) be a holomorphic line bundle on X endowed with a smooth metric . 4.1. Elementary Supplies We gather from [13] some simple properties concerning the Hodge decomposition right here. Bear in mind that the adjoint partnership in between and generally fails when X is non-compact. In truth, the compactness becomes vital when a single takes an integral. Nevertheless, because X is complete here, we still have.