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I am looking for a reference stating that

If a graph $z=f(x,y)$ over a convex domain $D$ is minimal, then it is area-minimizing.

  • 5.4.18 in Federer's "Geometric measure theory" and Lemma 1.1. in Colding--Minicozzi's "A Course in Minimal Surfaces" state that it is true with respect to $D\times \mathbb R$.

  • 6.1 in Morgan's "Geometric measure theory" states it right, but (at least formally) the proof only shows that it is area-minimizing among oriented surfaces.

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    $\begingroup$ A naive question: What is a "minimal graph"? $\endgroup$
    – Joseph O'Rourke
    Commented Nov 26, 2019 at 23:28
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    $\begingroup$ Probably the "minimal graph" is defined as a solution of the non-linear PDE which is called the "minimal surface equation in non-parametric form"? $\endgroup$
    – Alexandre Eremenko
    Commented Nov 27, 2019 at 1:06
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    $\begingroup$ If you look below Corollary 1.2 in Colding Minicozzi's book they prove what you are asking by observing projecting to the cylinder is distance non-increasing map as the cylinder is a convex region). $\endgroup$
    – RBega2
    Commented Nov 28, 2019 at 1:59
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    $\begingroup$ That being said, the CM argument uses a calibration so doesn't that restrict its applicability to oriented surfaces? $\endgroup$
    – RBega2
    Commented Nov 28, 2019 at 2:04
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    $\begingroup$ @MohammadGhomi The issue is to establish the existence of a minimizer in the class of of all orientable and non-orientable surfaces (otherwise there might be non-orientable competitors that don't satisfy any equation). You can do this by working with mod 2 currents, but its not exactly elementary... $\endgroup$
    – RBega2
    Commented Nov 28, 2019 at 15:22

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I suppose that we have a fixed contour $\Gamma$ which is a graph over $\partial D$, and you want to show that the minimal graph spanned by $\Gamma$ is area minimizing.

First, by the maximum principle (with respect to vertical planes), the interior of any (compact) minimal surface spanned by $\Gamma$ has to lie in the interior of $D\times R$. Then using Alexandrov's reflection principle, with respect to horizontal planes, I think that one can show that the surface must be a graph over $D$. Then the result should follow from the references you cite.

In short, I am not aware of an explicit reference, but I think what you want should be a straight forward application of maximum principal and Alexandrov's method of moving planes.

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  • $\begingroup$ Thank you --- I know how to prove it --- I just wanted a reference. (This should be a classical statement.) $\endgroup$
    – Anton Petrunin
    Commented Nov 27, 2019 at 21:54
  • $\begingroup$ As you may also know, I think that the moving plane argument here also shows that any minimal surface bounded by $\Gamma$ should be embedded, and therefore orientable, as it will divide $D\times R$ into two regions. So I think that one may assume orientability. $\endgroup$
    – Mohammad Ghomi
    Commented Nov 28, 2019 at 14:34
  • $\begingroup$ In this case you use existence of minimal surface (which is too much for the problem). $\endgroup$
    – Anton Petrunin
    Commented Nov 28, 2019 at 17:52

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