Laplace–Beltrami operators and self-adjointness

In local coordinates, the metric determines both the gradient coefficients and the volume density. The divergence-form expression includes the derivative of that density; omitting it generally changes the operator. Integration by parts on a closed manifold gives an energy identity equating the Laplacian pairing with the squared gradient norm. This establishes nonnegativity and symmetry on smooth functions, but symmetry on an initial domain is only a starting point for a self-adjoint realization.

Heat evolution consequently uses $e^{-t\Delta}$, and an eigenfunction of eigenvalue $\lambda$ decays by $e^{-t\lambda}$. Sources using the negative Laplacian write the heat equation with the opposite operator sign. Translating conventions is necessary before comparing formulas. Rescaling the metric by the square of a positive length factor multiplies scalar eigenvalues by the reciprocal square factor: frequencies scale inversely with length, while eigenvalues are squared frequencies.

On a closed manifold, the positive energy form has form domain $H^1(M)$. Its associated self-adjoint operator has domain $H^2(M)$, and smooth functions form a core. On a bounded smooth Euclidean domain, the Dirichlet realization has operator domain $H^2\cap H_0^1$, while the Neumann realization has domain consisting of $H^2$ functions with zero outward normal derivative. These regularity descriptions use smooth boundary; rough boundaries require the weaker form-domain formulation instead.

The Dirichlet form domain is $H_0^1$, whereas the Neumann form domain is $H^1$. This distinction will control variational comparison. Boundary conditions remove the integration-by-parts boundary term in different ways and can give different spectra for the same interior expression. In particular, prescribing a zero normal derivative is not equivalent to prescribing zero boundary values. Self-adjointness also involves equality with the adjoint domain, not merely a formal integration-by-parts cancellation on a convenient set of test functions.

Compact Sobolev embedding makes the resolvent compact in the closed or bounded-domain settings above. The spectrum is therefore discrete with finite multiplicities, and the eigenfunctions form an orthonormal basis. Elliptic regularity makes closed-manifold eigenfunctions smooth. None of this should be transplanted unchanged to a noncompact manifold, where continuous spectrum can appear and an ordinary sum of eigenvalues may fail to describe the operator. Compactness is an analytic hypothesis, not merely a visual convenience.

For scalar functions on a closed connected manifold, the energy identity forces a zero eigenfunction to be constant. On several connected components, each component contributes one independent constant. Neumann conditions likewise retain componentwise constants, while a bounded connected Dirichlet domain has strictly positive first eigenvalue. Zero modes must be retained in heat traces and counting functions unless explicitly removed; later, spectral zeta functions will deliberately remove them because taking a negative power of zero is undefined.