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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 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.

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  • $\begingroup$ What is an example of constant TMC? $\endgroup$ – Narasimham May 17 '18 at 4:39
  • $\begingroup$ What are the reference(s) that says (b) is "exact"? $\endgroup$ – j.c. May 17 '18 at 21:37
  • $\begingroup$ The lastest manual of Brakke's surface evolver. facstaff.susqu.edu/brakke/evolver/downloads/manual270.pdf $\endgroup$ – Thomas Yu May 17 '18 at 23:44
  • $\begingroup$ Thanks, for anyone else who's curious, the comment is under "mean_curvature_integral" in 15.6.4. There's a brief discussion along the lines of Ivan Izmestiev's answer in section 4.4 of this paper of John Sullivan's arxiv.org/abs/0710.4497 $\endgroup$ – j.c. May 18 '18 at 2:15
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The analogy between these two quantities goes back to Steiner (Über parallele Flächen, 1840). Both of them appear as coefficients of linear terms in the expansion of the area of the surface at distance $\epsilon$ from the given smooth/polyhedral surface.

Minkowski proved the convergence of the discrete to smooth for convex surfaces with respect to the Hausdorff metric on the space of convex bodies. Keywords: intrinsic volumes, quermassintegrals. This also implies that the total mean curvature is defined for arbitrary convex surfaces.

Besides, it can be defined for the boundaries of finite unions of convex bodies by sort of inclusion-exclusion formula.

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