the Laplace-Beltrami operator
/ lah-PLAHSS bel-TRAH-mee /
The ordinary Laplacian, u_xx + u_yy + u_zz, lives on flat space where coordinates are uniform and right angles are everywhere. But many problems live on a curved surface — the temperature on a sphere, vibrations of a curved membrane, diffusion over the surface of the Earth — where flat coordinates do not exist. The Laplace-Beltrami operator is the Laplacian rebuilt to work on any curved space (a Riemannian manifold), respecting its geometry. It is the geometric summit of elliptic theory: the natural operator of equilibrium and diffusion on a curved world.
On a manifold with metric g (which tells you how to measure lengths and angles at each point), the Laplace-Beltrami operator of a function u is Delta_g u = (1 over the square root of det g) times the sum over i,j of the partial in x_i of (the square root of det g times g^(ij) times the partial of u in x_j), where g^(ij) is the inverse metric and det g its determinant. The clutter is exactly the bookkeeping that makes the expression independent of the coordinate chart you happened to choose — change coordinates and every piece transforms so the operator stays the same geometric object. The cleanest way to say what it is: Delta_g u = div(grad u), where the gradient and divergence are both taken in the metric g. On flat space with the identity metric the square roots and inverses all become trivial and it collapses back to the familiar u_xx + u_yy + u_zz. It is uniformly elliptic (the metric being positive definite is exactly the ellipticity), self-adjoint with respect to the metric's volume integral, and so inherits the entire elliptic toolbox.
Because it is self-adjoint and elliptic on a compact manifold, the Laplace-Beltrami operator has a discrete spectrum and a complete orthonormal basis of eigenfunctions, generalising Fourier series to curved spaces: on the sphere these eigenfunctions are precisely the spherical harmonics. Its spectrum is a deep geometric invariant — it encodes the manifold's volume, dimension, and total curvature through the heat-kernel trace, the curved-space version of Weyl's law and the modern incarnation of 'hearing the shape'. It is the central operator of the heat and wave equations on manifolds, of Hodge theory (where it acts on differential forms and its kernel computes the topology), and of geometric analysis throughout — quantum mechanics on curved spaces, image processing on surfaces, and general relativity all speak its language.
On the unit sphere, write a function in spherical coordinates (theta, phi). The Laplace-Beltrami operator becomes Delta_g u = (1 over sin theta) the partial in theta of (sin theta times u_theta) + (1 over sin^2 theta) u_(phi phi). Its eigenfunctions are the spherical harmonics Y_(l,m), with eigenvalues -l(l+1) for l = 0, 1, 2, ... — the same operator and the same modes that appear in the angular part of the hydrogen atom and in the multipole expansion of a planet's gravity field.
On the sphere the geometric Laplacian's eigenfunctions are the spherical harmonics — Fourier analysis on a curved world.
Sign conventions clash in the literature: many geometers define the Laplace-Beltrami operator with the OPPOSITE sign (as a positive operator, with nonnegative eigenvalues), so on the sphere they write the eigenvalues as +l(l+1). The geometry is identical; only the sign convention differs, so always check which one an author uses before comparing formulas. Also, the operator depends on the metric — change the metric (stretch the surface) and you change the operator and its entire spectrum.