Laplacian operators in differential geometry

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In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them.

1 Connection Laplacian
2 Hodge Laplacian
3 Bochner Laplacian
4 Lichnerowicz Laplacian
5 Conformal Laplacian
6 See also

[edit] Connection Laplacian

The connection Laplacian is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemmanian- or pseudo-Riemannian metric. When applied to functions (i.e, tensors of rank 0), the connection Laplacian is often called the Laplace-Beltrami operator. It is defined as the trace of the second covariant derivative:

$\Delta T= \text{tr}\;\nabla^2 T,$

where T is any tensor, $\nabla$ is the Levi-Civita connection associated to the metric, and the trace is taken with respect to the metric. Recall that the second covariant derivative of T is defined as

$\nabla^2_{X,Y} T = \nabla_X \nabla_Y T - \nabla_{\nabla_X Y} T.$

Note that with this definition, the connection Laplacian has negative spectrum. On functions, it agrees with the operator given as the divergence of the gradient.

[edit] Hodge Laplacian

The Hodge Laplacian, also known as the Laplace-de Rham operator, is differential operator on acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemmanian- or pseudo-Riemannian metric.

$\Delta= \mathrm{d}\delta+\delta\mathrm{d} = (\mathrm{d}+\delta)^2,\;$

where d is the exterior derivative or differential and δ is the codifferential. The Hodge Laplacian has positive spectrum.

The connection Laplacian may also be taken to act on differential forms by restricting in to act on skew-symmetric tensors. The connection Laplacian differs from the Hodge Laplacian by means of a Weitzenböck identity.

[edit] Bochner Laplacian

The Bochner Laplacian is defined differently from the connection Laplacian, but the two will turn out to differ only by a sign, whenever the former is defined. Let M be a compact, oriented manifold equipped with a metric. Let E be a vector bundle over M equipped a fiber metric and a compatible connection, $\nabla$ . This connection gives rise to a differential operator

$\nabla:\Gamma(E)\rightarrow \Gamma(T^*M\otimes E)$

where $Γ(E)$ denotes smooth sections of E, and T^*M is the cotangent bundle of M. It is possible to take the $L 2$ -adjoint of $\nabla$ , giving a differential operator

$\nabla^*:\Gamma(T^*M\otimes E)\rightarrow \Gamma(E).$

The Bochner Laplacian is given by

$\Delta=\nabla^*\nabla$

which is a second order operator acting on sections of the vector bundle E. Note that the connection Laplacian and Bochner Laplacian differ only by a sign:

$\nabla^* \nabla = - \text{tr} \nabla^2$

[edit] Lichnerowicz Laplacian

The Lichnerowicz Laplacian^{[citation needed]} is defined only on symmetric tensors of rank 2, and makes sense on manifolds equipped with a metric. It differs from the connection Laplacian by terms involving the Riemann curvature tensor, and has natural applications in the study of Ricci flow and the Prescribed Ricci curvature problem.

[edit] Conformal Laplacian

On a Riemannian manifold, one can define the conformal Laplacian as an operator on smooth functions; it differs from the Laplace-Beltrami operator by a term involving the scalar curvature of the underlying metric. In dimension $n \geq 3$ , the conformal Laplacian, denoted L, acts on a smooth function u by

$Lu = -4\frac{n-1}{n-2} \Delta u + Ru,$

where $Δ$ is the Laplace-Beltrami operator (of negative spectrum), and R is the scalar curvature. This operator often makes an appearance when studying how the scalar curvature behaves under a conformal change of a Riemannian metric. If $n \geq 3$ and g is a metric and u is a smooth, positive function, then the conformal metric $\tilde g = u^\frac{4}{n-2} g$ has scalar curvature given by