Laplace-Beltrami operator (${\Delta }_g$)

The Laplace-Beltrami operator is the intrinsic Laplacian determined by a Riemannian metric. In local coordinates it is

The spectrum of a specified self-adjoint Laplace-Beltrami operator. On a closed manifold of positive dimension, it is a discrete sequence of nonnegative eigenvalues tending to infinity, counted with the dimensions of their Laplacian eigenspaces. On a noncompact manifold or a manifold with boundary, the domain and boundary conditions are part of the operator definition and continuous spectrum can occur.

Writing $D_r=\mathbf{x}\cdot \nabla =r{\partial }_r$, the spherical-coordinate Laplacian gives $r^2\Delta =D_r^2+D_r+{\Delta }_{S^2}$. Thus the positive angular Laplace-Beltrami operator has the displayed Cartesian expression and eigenvalues $l(l+1)$ on spherical harmonics. Omitting $D_r$ is incorrect when the square means composition: on $x_1$, the truncated expression gives $x_1$, whereas the degree-one angular eigenvalue is two.

// Target: fluid-mechanics.bigb

Choose an orthonormal basis $e_i$ of the tangent space at $p$ and the unit-speed geodesics with initial velocities $e_i$. Their covariant accelerations vanish, so $(f\circ {\gamma }_i{)}^{\prime \prime }(0)={\mathrm{Hess}}_{g}f(e_i,e_i)$. Taking the metric trace of the Riemannian Hessian proves the formula for $\Delta =\mathrm{div}\nabla$. The positive Laplace-Beltrami operator has the opposite sign.

The positive convention for the Laplace-Beltrami operator is ${\Delta }_{+}f=\delta df$, where $\delta$ is the codifferential. It is the restriction of the Hodge Laplacian to functions. In coordinates,On a compact boundaryless Riemannian manifold, or for compactly supported functions, $⟨f,{\Delta }_{+}f⟩=\Vert df{\Vert }^2\ge 0$. The opposite $\mathrm{div}\mathrm{grad}$ convention is also standard; formulas involving the sign must specify which is used.

For a function $f$ and one-form $\alpha$, the codifferential obeys $\delta (f\alpha )=f\delta \alpha -⟨df,\alpha {⟩}_g$. This follows from the graded Leibniz rule and the definition of the Hodge star operator. Applying it to $d(fh)=fdh+hdf$ gives the product formula. In particular ${\Delta }_{+}(f^2)=2f{\Delta }_{+}f-2|df{|}_g^2$. The cross-term sign reverses for the opposite Laplace-Beltrami operator convention.

For a geodesic $\gamma$ with initial velocity $v$, $(f\circ \gamma {)}^{\prime \prime }(0)=\mathrm{Hess}f(v,v)$, because the covariant acceleration vanishes. Summing over an orthonormal basis of the tangent space gives $\Delta f(p)=-{\sum }_i(f\circ {\gamma }_i{)}^{\prime \prime }(0)$ for the nonnegative Laplace-Beltrami operator.

For the nonnegative Hodge Laplacian convention $\Delta f=-\mathrm{div}\mathrm{grad}f$, a smooth function on a closed manifold with $\Delta f\ge 0$ has $\int \Delta f=0$ and hence $\Delta f=0$. Integration by parts gives $\int |df{|}^2=0$, so it is constant on each connected component. Global constancy requires connectedness; boundary conditions are needed if a boundary is allowed.

For a smooth real function on the unit two-sphere, integration of the covariant-derivative commutation formula gives the displayed identity. Explicitly, ${\nabla }^a{\nabla }_a{\nabla }_{b}w={\nabla }_b{\Delta }_{S}w+{\mathrm{Ric}}_b{}^c{\nabla }_{c}w$, and the unit sphere has $\mathrm{Ric}=g$. Integrating by parts once and then again proves the formula. It controls all angular second derivatives by the Laplace-Beltrami operator without singular estimates at coordinate poles.

The surface Laplacian is the Laplace-Beltrami operator of a surface with its induced Riemannian metric. It is ${\Delta }_{s}f={\nabla }_s\cdot {\nabla }_{s}f$, using the surface gradient and surface divergence. On a sphere of radius $a$, a degree-$l$ spherical harmonic has eigenvalue $-l(l+1)/a^2$.

If a twice differentiable real function has a local maximum at an interior point of a Riemannian manifold, its Hessian matrix is negative semidefinite there and hence its Laplacian is nonpositive.