Laplace operator

Laplacian

The differential operator $ \Delta $ in $ \mathbf R ^ {n} $ defined by the formula

$$ \tag{1 } \Delta = \ \frac{\partial ^ {2} }{\partial x _ {1} ^ {2} } + \dots + \frac{\partial ^ {2} }{\partial x _ {n} ^ {2} } $$

(here $ x _ {1}, \dots, x _ {n} $ are coordinates in $ \mathbf R ^ {n} $), as well as some generalizations of it. The Laplace operator (1) is the simplest elliptic differential operator of the second order. The Laplace operator plays an important role in mathematical analysis, mathematical physics and geometry (see, for example, Laplace equation; Laplace–Beltrami equation; Harmonic function; Harmonic form). Let $ M $ be an $ n $-dimensional Riemannian manifold with metric

$$ \tag{2 } d s ^ {2} = g _ {ij} d x ^ {i} d x ^ {j} ,\ \ g _ {ij} = g _ {ji} , $$

let $ \| g ^ {ij} \| $ be the matrix inverse to the matrix $ \| g _ {ij} \| $ and let $ g = \mathop{\rm det} \| g _ {ij} \| $. Then the Laplace operator (or Laplace–Beltrami operator) on $ M $ with the Riemannian metric (2) has the form

$$ \tag{3 } \Delta u = - \frac{1}{\sqrt g} \frac \partial {\partial x ^ {i} } \left ( \sqrt g g ^ {ij} \frac{\partial u }{\partial x ^ {j} } \right ) , $$

where $ ( x ^ {1}, \dots, x ^ {n} ) $ are local coordinates on $ M $. (The operator (1) differs in sign from the Laplace operator on $ \mathbf R ^ {n} $ with the standard Euclidean metric $ d s ^ {2} = ( d x ^ {1} ) ^ {2} + \dots + ( d x ^ {n} ) ^ {2} $.)

A generalization of the operator (3) is the Laplace operator on differential forms (cf. also Differential form). Namely, in the space of exterior differential forms on $ M $ the Laplace operator has the form

$$ \tag{4 } \Delta = ( d + d ^ {*} ) ^ {2} = d d ^ {*} + d ^ {*} d , $$

where $ d $ is the operator of exterior differentiation of a form and $ d ^ {*} $ is the operator formally adjoint to $ d $, defined by means of the following inner product on smooth forms with compact support:

$$ \tag{5 } ( \alpha , \beta ) = \int\limits \alpha \wedge \star \beta , $$

where $ \star $ is the Hodge star operator induced by the metric (2) taking a $ p $-form into an $ ( n - p ) $-form. In (5) the forms $ \alpha $ and $ \beta $ are assumed to be real; on complex forms one must use the Hermitian extension of the inner product (5). The restriction of the operator (4) to 0-forms (that is, functions) is specified by (3). On $ p $-forms with an arbitrary integer $ p \geq 0 $ the Laplace operator in local coordinates can be written in the form

$$ \Delta ( a _ {i _ {1} \dots i _ {p} } d x ^ {i _ {1} } \wedge \dots \wedge d x ^ {i _ {p} } ) = $$

$$ = \ \left \{ - \nabla ^ {i} \nabla _ {i} a _ {i _ {1} \dots i _ {p} } + \sum _ {\nu = 1 } ^ { p } (- 1) ^ \nu R _ {i _ \nu } ^ {n} a _ {n i _ {1} \dots \widehat{i} _ \nu \dots i _ {p} } \right . + $$

$$ \left . + 2 \sum _ {\mu < \nu } (- 1) ^ {\mu + \nu } R _ {\cdot i _ \nu \cdot i _ \mu } ^ {n \cdot k \cdot } a _ {k n i _ {1} \dots \widehat{i} _ \mu \dots \widehat{i} _ \nu \dots i _ {p} } \right \} \times $$

$$ \times d x ^ {i _ {1} } \wedge \dots \wedge d x ^ {i _ {p} } . $$

Here $ \nabla ^ {i} $ and $ \nabla _ {i} $ are the covariant derivatives with respect to $ x ^ {i} $( cf. Covariant derivative), $ R _ {\cdot j \cdot l } ^ {i \cdot k \cdot } $ is the curvature tensor and $ R _ {k} ^ {n} = R _ {\cdot i \cdot k } ^ {n \cdot i \cdot } $ is the Ricci tensor.

Suppose one is given an arbitrary elliptic complex

$$ \tag{6 } \dots \rightarrow \Gamma ( E _ {p- 1} ) \rightarrow ^ { d } \ \Gamma ( E _ {p} ) \rightarrow ^ { d } \Gamma ( E _ {p+} 1 ) \rightarrow \dots , $$

where the $ E _ {p} $ are real or complex vector bundles on $ M $ and the $ \Gamma ( E _ {p} ) $ are their spaces of smooth sections. Introducing a Hermitian metric in each vector bundle $ E _ {p} $ and also specifying the volume element on $ M $ in an arbitrary way, one can define a Hermitian inner product in the space of smooth sections of $ E _ {p} $ with compact support. Then operators $ d ^ {*} $ formally adjoint to the operator $ d $ are defined. The Laplace operator (4) is then constructed on each space $ \Gamma ( E _ {p} ) $ by formula (3). If for the complex (6) one takes the de Rham complex, then for a natural choice of the metric on the $ p $- forms and the volume element induced by the metric (2), one obtains for the Laplace operator of the de Rham complex the Laplace operator on forms, described above.

On a complex manifold $ M $, together with the de Rham complex there are also the elliptic complexes

$$ \tag{7 } \dots \rightarrow \Lambda ^ {p- 1,q} \mathop \rightarrow \limits ^ \partial \ \Lambda ^ {p,q} \mathop \rightarrow \limits ^ \partial \Lambda ^ {p+ 1,q} \rightarrow \dots , $$

$$ \tag{8 } \dots \rightarrow \Lambda ^ {p,q- 1} \mathop \rightarrow \limits ^ { {\overline \partial }} \Lambda ^ {p,q} \mathop \rightarrow \limits ^ { {\overline \partial }} \Lambda ^ {p,q + 1 } \rightarrow \dots , $$

where $ \Lambda ^ {p,q} $ is the space of smooth forms of type $ ( p , q ) $ on $ M $. Introducing a Hermitian structure in the tangent bundle on $ M $, one can construct the Laplace operator (4) of the de Rham complex and the Laplace operators of the complexes (7) and (8):

$$ \square = \partial \partial ^ {*} + \partial ^ {*} \partial , $$

$$ \overline \square = {\overline \partial }\; {\overline \partial ^ {*} } + \overline \partial ^ {*} \overline \partial . $$

Each of these operators takes the space $ \Lambda ^ {p,q} $ into itself. If $ M $ is a Kähler manifold and the Hermitian structure on $ M $ is induced by the Kähler metric, then

$$ \Delta = 2 \square = 2 \overline \square . $$

An important fact, which determines the role of the Laplace operator of an elliptic complex, is the existence in the case of a compact manifold $ M $ of the orthogonal Weyl decomposition:

$$ \tag{9 } \Gamma ( E _ {p} ) = d ( \Gamma ( E _ {p- 1} ) ) \oplus {\mathcal H} ^ {p} ( E) \oplus d ^ {*} ( \Gamma ( E _ {p+ 1} ) ) . $$

In this decomposition $ {\mathcal H} ^ {p} ( E) = \mathop{\rm ker} \Delta \mid _ {\Gamma ( E _ {p} ) } $, where $ \Delta $ is the Laplace operator of the complex (6), so that $ {\mathcal H} ^ {p} ( E) $ is the space of "harmonic" sections of $ E _ {p} $ (in the case of the de Rham complex, this is the space of all harmonic forms of degree $ p $). The direct sum of the first two terms on the right-hand side of (9) is equal to $ \mathop{\rm Ker} d \mid _ {\Gamma ( E _ {p} ) } $, and the direct sum of the last two terms coincides with $ \mathop{\rm Ker} d ^ {*} \mid _ {\Gamma ( E _ {p} ) } $. In particular, the decomposition (9) gives an isomorphism between the cohomology space of the complex (6) in the term $ \Gamma ( E _ {p} ) $ and the space of harmonic sections $ {\mathcal H} ^ {p} ( E) $.

References

[1]	G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French)
[2]	S.S. Chern, "Complex manifolds" , Univ. Recife (1959)
[3]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)

Comments

Let $ V $ be a finite-dimensional vector space with an inner product $ \langle , \rangle $ and suppose an orientation is given on $ V $. Choose an orthonormal basis $ ( e _ {1}, \dots, e _ {n} ) $ of $ V $ in the given orientation class. The Hodge $ \star $-operator (Hodge star operator, star operator)

$$ \star : \wedge ^ {p} V \rightarrow \wedge ^ {n - p } V $$

is defined by

$$ \star ( e _ {i _ {1} } \wedge \dots \wedge e _ {i _ {p} } ) = \ \pm e _ {j _ {1} } \wedge \dots \wedge e _ {j _ {n - p } } , $$

where $ \{ i _ {1}, \dots, i _ {p} ; j _ {1}, \dots, j _ {n - p } \} = \{ 1, \dots, n \} $ and the plus (respectively, minus) sign is taken depending on whether the permutation $ \{ i _ {1}, \dots, i _ {p} , j _ {1}, \dots, j _ {n - p } \} $ of $ \{ 1, \dots, n \} $ is even or odd.

Declaring that the $ e _ {i _ {1} } \wedge \dots \wedge e _ {i _ {p} } $, $ i _ {1} < \dots < i _ {p} \leq n $, form an orthonormal basis for $ \wedge ^ {p} V $ defines an inner product on $ \wedge ^ {p} V $, said to be induced from that of $ V $. Let $ \mathop{\rm vol} $ be the volume form determined by the chosen orientation. Then (extending $ \star $ by linearity to all of $ \wedge ^ {p} V $),

$$ \alpha \wedge \star \beta = \langle \alpha , \beta \rangle fnme vol $$

for all $ \alpha , \beta \in \wedge ^ {p} V $.

The Hodge star operator on an oriented Riemannian manifold $ M $ is defined pointwise:

$$ ( \star \phi ) ( x) = \star ( \phi ( x)) $$

for a $ p $-form $ \phi $ on $ M $.

Let $ E $ be a complex vector space of (complex) dimension $ n $ and let $ E ^ \prime $ be the underlying $ 2n $-dimensional real vector space. Let $ h $ be a Hermitian inner product on $ E $. Then the fundamental 2-form associated to $ h $, $ \Omega = - ( \mathop{\rm Im} h)/2 $, provides an inner product on $ E ^ \prime $ and $ \Omega ^ {n} $ provides an orientation. In this case the Hodge $ \star $-operator is defined relative to this inner product and this orientation. It is again extended pointwise to forms on complex manifolds with a Hermitian metric.

The Laplace operator of a Riemannian metric $ g $ can also be defined as the real symmetric second-order linear partial differential operator which annihilates the constant functions and for which the principal symbol (cf. Symbol of an operator) is equal to the quadratic form on the cotangent bundle which is dual to $ g $.

References

[a1]	W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1952)