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(a) The total mean curvature of a smooth oriented surface $S$ is defined as $$\iint_S H dA$$ where $H$ is the mean curvature (defined w.r.t. a choice of continuous unit normals on $S$.) Is there a standard way to generalize this integral to more general surfaces (rectifiable varifolds, perhaps)?

(b) For the special case of oriented piecewise linear (PL) surfaces without boundary, there is a formula in the literature for the total mean curvature, given by $$\sum_e {\rm length}(e) \,\theta(e),$$ where the sum is over all the edges and $\theta(e)\in (-\pi, \pi)$ is the signed angle between the normals to the adjacent faces at $e$. It is said that this formula is 'exact'.

I cannot trace the origin of this formula (despite seeing it two different references), and cannot figure out in what sense is the formula exact.

Seems like the 'exactness' in (b) hinges on a definite answer to (a).

Thanks.

p.s. A heuristical argument suggests that for any sequence of smooth surfaces $S^k$ converging to a PL surface S, the smooth total mean curvatures of $S^k$ converge to the discrete total mean curvature of S defined by the sum above.

(a) The total mean curvature of a smooth oriented surface $S$ is defined as $$\iint_S H dA$$ where $H$ is the mean curvature (defined w.r.t. a choice of continuous unit normals on $S$.) Is there a standard way to generalize this integral to more general surfaces (rectifiable varifolds, perhaps)?

(b) For the special case of oriented piecewise linear (PL) surfaces without boundary, there is a formula in the literature for the total mean curvature, given by $$\sum_e {\rm length}(e) \,\theta(e),$$ where the sum is over all the edges and $\theta(e)\in (-\pi, \pi)$ is the signed angle between the normals to the adjacent faces at $e$. It is said that this formula is 'exact'.

I cannot trace the origin of this formula (despite seeing it in two references), and cannot figure out in what sense is the formula exact.

Seems like the 'exactness' in (b) hinges on a definite answer to (a).

Thanks.

p.s. A heuristical argument suggests that for any sequence of smooth surfaces $S^k$ converging to a PL surface S, the smooth total mean curvatures of $S^k$ converge to the discrete total mean curvature of S defined by the sum above.

(a) The total mean curvature of a smooth oriented surface $S$ is defined as $$\iint_S H dA$$ where $H$ is the mean curvature (defined w.r.t. a choice of continuous unit normals on $S$.) Is there a standard way to generalize this integral to more general surfaces (rectifiable varifolds, perhaps)?

(b) For the special case of oriented piecewise linear (PL) surfaces without boundary, there is a formula in the literature for the total mean curvature, given by $$\sum_e {\rm length}(e) \,\theta(e),$$ where the sum is over all the edges and $\theta(e)\in (-\pi, \pi)$ is the signed angle between the normals to the adjacent faces at $e$. It is said that this formula is 'exact'.

I cannot trace the origin of this formula (despite seeing it two different references), and cannot figure out in what sense is the formula exact.

Seems like the 'exactness' in (b) hinges on a definite answer to (a).

Thanks.

(a) The total mean curvature of a smooth oriented surface $S$ is defined as $$\iint_S H dA$$ where $H$ is the mean curvature (defined w.r.t. a choice of continuous unit normals on $S$.) Is there a standard way to generalize this integral to more general surfaces (rectifiable varifolds, perhaps)?

(b) For the special case of oriented piecewise linear (PL) surfaces without boundary, there is a formula in the literature for the total mean curvature, given by $$\sum_e {\rm length}(e) \,\theta(e),$$ where the sum is over all the edges and $\theta(e)\in (-\pi, \pi)$ is the signed angle between the normals to the adjacent faces at $e$. It is said that this formula is 'exact'.

I cannot trace the origin of this formula (despite seeing it two different references), and cannot figure out in what sense is the formula exact.

Seems like the 'exactness' in (b) hinges on a definite answer to (a).

Thanks.

p.s. A heuristical argument suggests that for any sequence of smooth surfaces $S^k$ converging to a PL surface S, the smooth total mean curvatures of $S^k$ converge to the discrete total mean curvature of S defined by the sum above.