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Suppose that I have a smooth curved surface, and I choose an arbitrary point $Q$ on that surface. Say the Gaussian curvature at that point is $K$. What I am wondering is, is there an expression for the minimal perimeter $P_{min}$ of a disk of area $A$ which contains point $Q$, in the limit of a very small disk with $AK \ll 1$? What I imagine is something like

$$ P_{min} = 2 \pi \sqrt{\frac{A}{\pi}}+\text{powers of $K$ or derivatives of $K$ at $Q$} $$

The zeroth-order term comes from the planar case. My question is:

  • Is such an expansion possible at all, and if it is, are any lowest order terms known?

I have searched but I could not find any result of the sort. For geodesic disks, I have used the Bertrand–Diguet–Puiseux theorem to obtain

$$ P_{geo} = 2 \pi \sqrt{\frac{A}{\pi}} \left( 1 - \frac{AK}{8\pi} \right) + O(\sqrt{A}A^2K^2) $$ or equivalently $$ P_{geo}^2 = 4 \pi A - A^2 K + O(A^3K^2) $$ which implies that a small geodesic disk of a given area will have larger perimeter in a negatively curved region than in a positively curved region. A second, weaker question is therefore

  • Does a similar result hold for the small perimeter-minimizing disk? I.e. will $P_{min}$ always be larger for $K<0$ than for $K>0$ in the limit of small area?
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    $\begingroup$ Your terminology is not clear. For example, what do you mean by a "geodesic disk"? Is that a disk of constant Gaussian curvature? $\endgroup$
    – Mikhail Katz
    Commented Sep 10, 2023 at 11:45
  • $\begingroup$ No, it's the region enclosed by a geodesic circle. The geodesic circle of radius r centered at a point p is the set of all points whose geodesic distance from p is equal to r. But for an arbitrary surface, the geodesic disk thus defined is not necessarily the perimeter-minimizing disk, in the sense of the isoperimetric inequality. I am trying to find out if there is something that can be said about the mininal perimeter of infinitesimally small disks at a a surface, based on the local curvature of the surface. $\endgroup$
    – m3tro
    Commented Sep 10, 2023 at 13:42
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    $\begingroup$ That would be "distance sphere" (or "circle"), not "geodesic circle" but perhaps the authors you cited use this term. Are you looking for a precise expression, or something like the formula that you reproduced from Bertrand et al.? $\endgroup$
    – Mikhail Katz
    Commented Sep 10, 2023 at 13:45
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    $\begingroup$ Have you thought of using the exponential map to reduce the question to the constant curvature case? $\endgroup$
    – Mikhail Katz
    Commented Sep 11, 2023 at 9:28
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    $\begingroup$ The exponential map is the map from the tangent space to the Riemannian manifold which is an isometry in the radial directions, and satisfies Gauss's lemma. What is relevant here is a variation of the exponential map, which would be the composition of the inverse of the exponential map of the sphere of curvature $K$ (its value at the point $Q$), with the exponential map of the surface. This would be an almost isometry in small neighborhoods, and may be enough to answer your question. $\endgroup$
    – Mikhail Katz
    Commented Sep 11, 2023 at 11:36

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