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In Montgomery's A Tour of Subriemannian Geometries, Their Geodesics and Applications, problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?".

Can someone explain what are the major difficulties faced in proving the same ? Suggestions for suitable references are also welcome. Thanks.

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The reason the problem is hard is that we do not have a good handle on what abnormal (=singular) geodesics can look like. See the chapter of my book that describes abnormal geodesics. Progress is being made in the Carnot case, but it is slow.

I would look at a fairly recent article (EXTREMAL CURVES IN NILPOTENT LIE GROUPS ENRICO LE DONNE, GIAN PAOLO LEONARDI, ROBERTO MONTI, AND DAVIDE VITTONE; I think in GAFA) on the Carnot case for state-of-the art stuff, and the first paper by myself and that of Liu-Sussmann just to get a feel of abnormal geodesics.

It is POSSIBLE the problem can be solved without understanding abnormals - by some direct analysis. No one has done anything like this: the abnormals always seem to be there, hiding.

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According to the MathSciNet, the first general contribution to the solution of the problem is given in the following paper:

E. Hakavuori, E. Le Donne, Non-minimality of corners in subriemannian geometry. Invent. Math. 206 (2016), no. 3, 693–704. (MathSciNet review).

The paper mentioned by Richard Montgomery is:

E. Le Donne, G. P. Leonardi, R. Monti, D. Vittone, Extremal curves in nilpotent Lie groups. Geom. Funct. Anal. 23 (2013), no. 4, 1371–1401. (MathSciNet review).

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    $\begingroup$ The Hakavuori-LeDonne 2016 paper is beautifully written and is indeed a major contribution. $\endgroup$
    – Richard Montgomery
    Commented Mar 3, 2020 at 3:24

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