Question

Does anyone know of a modern proof of the First Minkowski Formula for a compact embedded hypersurface $\psi \colon \mathcal{M}^n \hookrightarrow \mathbb{R}^{n+1}$ ? The integral formula is $$ \int_{\mathcal{M}} H \langle \psi , \nu \rangle \mathrm{d}A +A = 0$$ where $A$ is the area of $\mathcal{M}$, $H$ is the mean curvature of $\mathcal{M}$ and $\nu$ is the outward normal field on $\mathcal{M}$.

Many thanks for any suggestions.

Answer 1

Consider the following differential $(n-1)$-form $\omega$ on $M$: for $p\in M$ and $v_1,\dots,v_{n-1}\in T_pM$, define $$ \omega(v_1,\dots,v_{n-1}) = [ \psi(p), \nu(p), d\psi(v_1),\dots,d\psi(v_{n-1})] $$ where the square brackets denote the standard volume form in $\mathbb R^{n+1}$ (in other words, the determinant).

A simple computation shows that $$ d\omega = n(1+H\langle\psi,\nu\rangle)dA . $$ Now the formula follows from the fact that $\int_M d\omega=0$ by Stokes.

This works for any immersed orientable hypersurface. For a non-orientable one, consider the oriented double cover. Note that in the embedded case one may assume that $M$ is a submanifold and $\psi$ is the inclusion, and then the formula for $\omega$ gets simpler: just remove all $\psi$ and $d\psi$.

Unfortunately I don't remember where I have read this proof, it was too long ago. (And I think that that text was only for $n=2$ anyway.)

Answer 2

I highly recommend:

Curves and Surfaces (Graduate Studies in Mathematics) [Hardcover] Sebastian Montiel and Antonio Ros (Author)

(and various papers by the same authors). They do this in a nice way, and have various generalizations...

Answer 3

In fact, it is a straightforward consequence of the Bochner Lichnerowicz formula on M.