Smoothness of Function on Manifold

Motivations

Dirichlet Energy

Dirichlet energy is defined as the integral of squared norm of functions’ gradient in a set. So it’s a functional over smooth function on the set $C^\infty (M)\to \R$.

$$ D[f]=\int_M\|\nabla f\|^2 d\omega\\ $$

An important property of Dirichlet Energy is that the stable points of $D[f]$ gives harmonic functions.

Sketch of Proof: Given a perturbation $u$ then the change of Dirichlet energy w.r.t. the perturbation amplitude $\epsilon$ will be the following. The 2nd equation follows from Gauss theorem.

$$ \frac{dD[f+\epsilon u]}{d\epsilon}=2\int_M \nabla u \cdot\nabla f d\omega=-2\int_Mu \Delta fd\omega $$

Thus if this quantity vanish for any given $u$, then $D[f]$ is stable around $f$. which is equivalent to $\Delta f$ vanishes everywhere on $M$.

Dirichlet Energy on Manifold

Given a manifold and a function on it, we would like to define the Dirichlet energy on it.

Integration on Manifold

First recall, the definition of an integration on manifold, esp. integration of a scalar field, is the integration of the volume (n-)form.

Here we assume we work with some local coordinates $\phi:D\to M,u\mapsto p$, then the volume form could be written as this, i.e. the scalar functions are weighted by the determinant of metric tensor $\sqrt{\det g}$.

$$ \mathcal I=\int_M f(p)dvol_M=\int_D f(\phi(u))\sqrt{\det g}\; du_1\wedge...\wedge du_n $$

Moreover, if the manifold $M$ is embedded in a ambient Euclidean space, then we could explicitly write out the coordinate map $\phi$ and the Jacobian $J=d\phi$ . Then $g=J^TJ$, which entails $\sqrt{\det g}=\epsilon \det J$ . This simply means the integration will be weighted by the volume expansion factor $\det J$, while the sign of it depends on the orientation of the coordinates.

Norm of Gradient

Differential of function is a 1-form on the manifold $df$, while gradient is the counterpart of it in the tangent vector space.

Smoothing and Interpolation on Sphere

Interpolation on point clouds

Two extend this idea further, we want to define the smoothness of tuning with any data point space.