Failure of semiconvexity of the squared sub-Riemannian distance at the cut locus in the presence of abnormal minimizing geodesics

On a Riemannian manifold, the squared distance function $d^2(x,\cdot)$ fails to be semiconvex precisely at the cut locus, and this second-order fact underlies the differentiability theory of optimal transport maps with quadratic cost. Figalli and Rifford (Geom. Funct. Anal. 2010, Section 5.8) asked for the sub-Riemannian analogue: assuming the sub-Riemannian distance $d_{SR}$ is locally semiconcave off the diagonal, does the existence at $y$ of a function $\varphi$ twice differentiable at $y$ with $\varphi(y)=d_{SR}^2(x,y)$ and $d_{SR}^2(x,z)\geq\varphi(z)$ for all $z$ force $y$ outside the cut locus of $x$? Barilari and Rizzi (Invent. Math. 2019) answered this affirmatively for ideal structures (those with no non-trivial abnormal minimizing geodesics) and showed that the resulting characterization of the cut locus fails in the presence of abnormal minimizers: on the product $\mathbb{H}\times\mathbb{R}$ of the three-dimensional Heisenberg group with the line, the squared distance is not smooth at the endpoints of abnormal minimizers while its symmetric second-difference quotients remain bounded, the loss of smoothness being caused by failure of semiconcavity. They reformulated the surviving general question as: for a complete sub-Riemannian structure, does the optimal cut locus coincide with the set where the squared distance fails to be semiconvex, and does the set of endpoints of abnormal minimizing geodesics coincide with the set where it fails to be semiconcave? The literature status is as follows: the ideal case is settled affirmatively, while the general case — structures admitting abnormal minimizers, such as step-three or rank-varying structures — remains open, with positive confirmations so far only in special classes: the Martinet structure, the Engel group, and step-2 Carnot groups of Métivier, free, GM, or SA type (the GM and SA types are the classes of step-two Carnot groups characterized in Li–Zhang 2021; see Reference 13, arXiv:2102.09860). The problem asks for a proof or refutation in the general case.

Background

A sub-Riemannian structure on a smooth connected manifold $M$ is a distribution $\Delta\subset TM$ (spanned by a family of globally defined smooth vector fields) equipped with a smoothly varying inner product $g$ on $\Delta$. A curve is horizontal if its velocity stays in $\Delta$, and the sub-Riemannian or Carnot–Carathéodory distance $d_{SR}(x,y)$ is the infimum of the lengths, computed with $g$, of horizontal curves joining $x$ to $y$; under the Chow–Rashevskii bracket-generating condition it is finite and induces the topology of $M$. Minimizing curves come in two analytic types: normal geodesics, projections of Hamiltonian trajectories, and abnormal (or singular) geodesics — locally minimizing horizontal curves that are the projection of a covector curve annihilating the distribution and that need not satisfy the normal Hamiltonian equations. A structure with no non-trivial abnormal minimizing geodesics is called ideal.

Two classical notions of second-order regularity: a function $f$ is semiconvex (respectively semiconcave) on an open set $U$ of local coordinates if there is a constant $C\geq 0$ such that the symmetric second difference quotients $[f(z+v)+f(z-v)-2f(z)]/|v|^2$ are bounded below by $-C$ (respectively above by $C$) for every $z\in U$ and all sufficiently small $v$; and $f$ fails to be semiconvex (respectively semiconcave) at a point $y$ if it is not semiconvex (respectively semiconcave) on any neighborhood of $y$. This is the classical local definition, and it is the one this problem uses; Barilari–Rizzi's original text worked with the weaker pointwise reading in which only the quotients at $y$ itself are required to be bounded, and a published addendum to their open questions (recorded by Li–Zhang) specifies that the local definition is the intended one. In Riemannian geometry, $d^2(x,\cdot)$ is smooth off the cut locus and fails to be semiconvex at every cut point (Cordero-Erausquin–McCann–Schmuckenschläger, Invent. Math. 2001, Proposition 2.5) — the fact on which the a.e. differentiability of optimal transport maps with quadratic Riemannian cost rests.

The cut locus itself appears in two conventions. The singular cut locus $\operatorname{Cut}(x)$ is the set of points $y\in M$ at which $d_x^2:z\mapsto d_{SR}^2(x,z)$ fails to be smooth; for genuinely sub-Riemannian structures it contains the base point $x$ itself (by the Ball–Box theorem, a small-scale estimate saying that sub-Riemannian balls are sandwiched between anisotropic boxes, which forces a corner-type singularity of the squared distance at its base point). Figalli–Rifford's notation $\operatorname{Cut}_{SR}(x)$ denotes the corresponding singular set off the diagonal — the points $y\neq x$ at which $d_{SR}(x,\cdot)$ is not continuously differentiable — and under the semiconcavity hypothesis of their regularity theory the two descriptions coincide off the diagonal. The optimal cut locus $\operatorname{CutOpt}(x)$ is defined through loss of optimality: a geodesic $\gamma$ (a horizontal curve locally minimizing the energy between its endpoints) is maximal if it is not the restriction of a geodesic defined on a larger interval, its cut time is $t_{\mathrm{cut}}(\gamma)=\sup\{t>0:\ \gamma|_{[0,t]}\text{ is minimizing}\}$, and $\operatorname{CutOpt}(x)$ is the set of endpoints $\gamma(t_{\mathrm{cut}}(\gamma))$ over maximal geodesics starting at $x$. In the ideal case, which includes the Riemannian case, $\operatorname{CutOpt}(x)=\operatorname{Cut}(x)\setminus\{x\}$; in general the two notions differ.

After the Heisenberg-group transport theorem of Ambrosio–Rigot (J. Funct. Anal. 2004) and the extension by Agrachev–Lee (Trans. AMS 2009), Figalli–Rifford (Geom. Funct. Anal. 2010) proved existence, uniqueness, and a.e. approximate differentiability of the optimal map for the quadratic sub-Riemannian cost $c=\frac{1}{2}d_{SR}^2$ under hypotheses including local semiconcavity of $d_{SR}$ off the diagonal, total geodesic convexity, and a multiplicity condition at cut points. The one Riemannian input they could not replace was the failure of semiconvexity at the cut locus; in Section 5.8 they posed the open problem: assuming $d_{SR}$ locally semiconcave on $M\times M\setminus D$ ($D$ the diagonal), if $d_{SR}^2(x,\cdot)$ admits at $y$ a twice-differentiable lower support — a function $\varphi$ twice differentiable at $y$ with $\varphi(y)=d_{SR}^2(x,y)$ and $d_{SR}^2(x,z)\geq\varphi(z)$ for all $z\in M$ — must $y\notin\operatorname{Cut}_{SR}(x)$?

Barilari–Rizzi (Invent. Math. 2019) resolved this affirmatively for ideal structures (Theorem 6: for ideal structures, $x\in\operatorname{Cut}(y)$ if and only if $d_{SR}^2(\cdot,y)$ fails to be semiconvex at $x$; Corollary 30 gives the local version whenever the minimizing geodesics reaching a neighborhood of the contact point contain no abnormal segment, and its proof proceeds exactly in the support-function form of the Figalli–Rifford question). However, the statement is false in this form without idealness: on the product $\mathbb{H}\times\mathbb{R}$ ($\mathbb{H}$ the three-dimensional Heisenberg group), at the endpoints of abnormal minimizers the squared distance is not smooth — such points belong to the cut locus — yet the semiconvexity quotients remain finite, the loss of smoothness being caused by failure of semiconcavity. Barilari–Rizzi therefore formalized the general question: writing $SC^+(x)$ and $SC^-(x)$ for the sets of points where $d_{SR}^2(x,\cdot)$ fails to be semiconvex and semiconcave, and $\operatorname{Abn}(x)$ for the set of endpoints of abnormal minimizing geodesics from $x$ (the trivial constant geodesic counts as abnormal, so $x\in\operatorname{Abn}(x)$, and a structure is ideal exactly when $\operatorname{Abn}(x)=\{x\}$ for every $x$), is $\operatorname{CutOpt}(x)=SC^+(x)$ and $\operatorname{Abn}(x)=SC^-(x)$ (whence $\operatorname{Cut}(x)=SC^+(x)\cup SC^-(x)$ and $\operatorname{Cut}(x)=\operatorname{CutOpt}(x)\cup\operatorname{Abn}(x)$)? The inclusion $\operatorname{Abn}(x)\supseteq SC^-(x)$ is known in general (Cannarsa–Rifford, Ann. Inst. H. Poincaré Anal. Non Linéaire 2008); the reverse inclusion and the first identity are the open content, and they contain the residual of the Figalli–Rifford question: whether local semiconcavity of $d_{SR}$ off the diagonal (their hypothesis) already forces the absence of abnormal minimizers. The transport program is surveyed in Rifford's SpringerBriefs volume (2014); Badreddine (ANIHPC 2019) extended the Monge problem to rank-two structures in dimension four with many singular minimizers via localization techniques that bypass the cut-locus regularity question rather than resolve it.

Problem Statement

Let $(M,\Delta,g)$ be a complete sub-Riemannian structure, bracket-generating, with no idealness assumption, and let $d_{SR}$ be its Carnot–Carathéodory distance. For a point $x\in M$, writing $d_x^2(z)=d_{SR}^2(x,z)$, define:

• $\operatorname{Cut}(x)$: the singular cut locus of $x$, the set of points $y\in M$ at which $d_x^2$ fails to be smooth (Figalli–Rifford's $\operatorname{Cut}_{SR}(x)$ is the same singular set off the diagonal, described as the points $y\neq x$ where $d_{SR}(x,\cdot)$ is not continuously differentiable);
• $\operatorname{CutOpt}(x)$: the optimal cut locus of $x$, the set of endpoints $\gamma(t_{\mathrm{cut}}(\gamma))$ of maximal geodesics $\gamma$ starting at $x$ at their cut time $t_{\mathrm{cut}}(\gamma)=\sup\{t>0:\ \gamma|_{[0,t]}\text{ is minimizing}\}$;
• $SC^+(x)$ (resp. $SC^-(x)$): the set of points $y$ at which $d_x^2$ fails to be semiconvex (resp. semiconcave) in the classical local sense — i.e. $d_x^2$ is semiconvex (resp. semiconcave) on no neighborhood of $y$, semiconvexity on an open set $U$ meaning a uniform lower bound $[d_x^2(z+v)+d_x^2(z-v)-2d_x^2(z)]/|v|^2\geq -C$ for all $z\in U$ and all sufficiently small $v$ (resp. a uniform upper bound with $\leq C$);
• $\operatorname{Abn}(x)$: the set of endpoints of abnormal minimizing geodesics starting at $x$, the trivial constant geodesic counting as abnormal, so $x\in\operatorname{Abn}(x)$ always.

The problem is to determine, for general complete sub-Riemannian structures, whether for every $x$:

(1) $\operatorname{CutOpt}(x)=SC^+(x)$ — failure of semiconvexity of the squared distance exactly characterizes loss of geodesic optimality;

(2) $\operatorname{Abn}(x)=SC^-(x)$ — failure of semiconcavity exactly characterizes the abnormal endpoints;

and whether $\operatorname{Cut}(x)=SC^+(x)\cup SC^-(x)$ and $\operatorname{Cut}(x)=\operatorname{CutOpt}(x)\cup\operatorname{Abn}(x)$ (Barilari–Rizzi's open questions (28), (29), (30)). The inclusion $\operatorname{Abn}(x)\supseteq SC^-(x)$ is known in general, so a counterexample to (2) can only witness the failure of the reverse inclusion.

This contains the residual of Figalli–Rifford's original Section 5.8 question: assuming $d_{SR}$ locally semiconcave on $M\times M\setminus D$ — a hypothesis known to hold for ideal structures and equivalent to the absence of abnormal minimizers in every class where it has been verified, such as step-2 Carnot groups (Montanari–Morbidelli 2016) — does the existence of a function $\varphi$ twice differentiable at $y$ with $\varphi(y)=d_{SR}^2(x,y)$ and $d_{SR}^2(x,z)\geq\varphi(z)$ for all $z\in M$ imply $y\notin\operatorname{Cut}_{SR}(x)$? Under that hypothesis $d_x^2$ is locally semiconcave near every $y\neq x$, so $SC^-(x)\subseteq\{x\}$; hence a positive answer to (2) forces the absence of non-trivial abnormal minimizers under the hypothesis, placing the question in the ideal regime where Barilari–Rizzi's Corollary 30 gives the affirmative conclusion. A negative answer, to the Figalli-Rifford question or to (1)/(2), is witnessed by an explicit structure, base point, and contact point; for the support-function form, the witness includes the explicit support $\varphi$.

The problem is to prove or refute (1), (2), and the Cut identities (30), for complete sub-Riemannian structures, including rank-varying and step-three-or-higher structures. A refutation of (1) or (2) is witnessed by a complete sub-Riemannian structure and point $x$ together with a point $y\in\operatorname{CutOpt}(x)$ at which $d_x^2$ does not fail to be semiconvex, or a point $y\notin\operatorname{CutOpt}(x)$ at which $d_x^2$ fails to be semiconvex, or a point $y\in\operatorname{Abn}(x)$, $y\neq x$, at which $d_x^2$ does not fail to be semiconcave; a refutation of a Cut identity is witnessed by a complete sub-Riemannian structure and point $x$ together with a point lying in exactly one of the two sides of that identity.

References

1. Luigi Ambrosio, Séverine Rigot, "Optimal mass transportation in the Heisenberg group", Journal of Functional Analysis 208 (2004), no. 2, 261-301. DOI: 10.1016/s0022-1236(03)00019-3, https://doi.org/10.1016/s0022-1236(03)00019-3
2. Dario Cordero-Erausquin, Robert J. McCann, Michael Schmuckenschläger, "A Riemannian interpolation inequality à la Borell, Brascamp and Lieb", Inventiones Mathematicae 146 (2001), no. 2, 219-257. DOI: 10.1007/s002220100160, https://doi.org/10.1007/s002220100160
3. Andrei Agrachev, Paul Lee, "Optimal transportation under nonholonomic constraints", Transactions of the American Mathematical Society 361 (2009), no. 11, 6019-6047. DOI: 10.1090/s0002-9947-09-04813-2, https://doi.org/10.1090/s0002-9947-09-04813-2
4. Alessio Figalli, Ludovic Rifford, "Mass Transportation on Sub-Riemannian Manifolds", Geometric and Functional Analysis 20 (2010), no. 1, 124-159. DOI: 10.1007/s00039-010-0053-z, https://doi.org/10.1007/s00039-010-0053-z (also arXiv:0803.2917, https://arxiv.org/abs/0803.2917)
5. Piermarco Cannarsa, Ludovic Rifford, "Semiconcavity results for optimal control problems admitting no singular minimizing controls", Annales de l'Institut Henri Poincaré C, Analyse non linéaire 25 (2008), no. 4, 773-802. DOI: 10.1016/j.anihpc.2007.07.005, https://doi.org/10.1016/j.anihpc.2007.07.005
6. P. Cannarsa, H. Frankowska, "From pointwise to local regularity for solutions of Hamilton–Jacobi equations", Calculus of Variations and Partial Differential Equations 49 (2014), no. 3-4, 1061-1074. DOI: 10.1007/s00526-013-0611-y, https://doi.org/10.1007/s00526-013-0611-y
7. Ludovic Rifford, "Sub-Riemannian Geometry and Optimal Transport", SpringerBriefs in Mathematics, Springer, Cham, 2014. DOI: 10.1007/978-3-319-04804-8, https://doi.org/10.1007/978-3-319-04804-8
8. Annamaria Montanari, Daniele Morbidelli, "On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups", Journal of Mathematical Analysis and Applications 444 (2016), no. 2, 1652-1674. DOI: 10.1016/j.jmaa.2016.07.032, https://doi.org/10.1016/j.jmaa.2016.07.032 (also arXiv:1508.00997, https://arxiv.org/abs/1508.00997)
9. Annamaria Montanari, Daniele Morbidelli, "On the subRiemannian cut locus in a model of free two-step Carnot group", Calculus of Variations and Partial Differential Equations 56 (2017), no. 2, Paper No. 36. DOI: 10.1007/s00526-017-1149-1, https://doi.org/10.1007/s00526-017-1149-1 (also arXiv:1608.06431, https://arxiv.org/abs/1608.06431)
10. Davide Barilari, Luca Rizzi, "Sub-Riemannian interpolation inequalities", Inventiones Mathematicae 215 (2019), no. 3, 977-1038. DOI: 10.1007/s00222-018-0840-y, https://doi.org/10.1007/s00222-018-0840-y (also arXiv:1705.05380, https://arxiv.org/abs/1705.05380)
11. Zeinab Badreddine, "Mass transportation on sub-Riemannian structures of rank two in dimension four", Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire 36 (2019), no. 3, 837-860. DOI: 10.1016/j.anihpc.2018.10.003, https://doi.org/10.1016/j.anihpc.2018.10.003 (also arXiv:1706.07308, https://arxiv.org/abs/1706.07308)
12. Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska, "Semiconcavity results and sensitivity relations for the sub-Riemannian distance", Nonlinear Analysis 184 (2019), 298-320. DOI: 10.1016/j.na.2019.02.008, https://doi.org/10.1016/j.na.2019.02.008
13. Hong-Quan Li, Ye Zhang, "Sub-Riemannian geometry on some step-two Carnot groups", preprint (2021), arXiv:2102.09860, https://arxiv.org/abs/2102.09860
14. Ludovic Rifford, "Subdifferentials and minimizing Sard conjecture in sub-Riemannian geometry", Journal de l'École Polytechnique — Mathématiques 10 (2023), 1195-1244. DOI: 10.5802/jep.242, https://doi.org/10.5802/jep.242 (also arXiv:2302.02660, https://arxiv.org/abs/2302.02660)
15. Federica Dragoni, Qing Liu, Ye Zhang, "Horizontal semiconcavity for the square of Carnot–Carathéodory distance on step 2 Carnot groups and applications to Hamilton–Jacobi equations", Nonlinearity 38 (2025), no. 4, 045009. DOI: 10.1088/1361-6544/adb930, https://doi.org/10.1088/1361-6544/adb930 (also arXiv:2402.19164, https://arxiv.org/abs/2402.19164)
16. Haowei Lin, "Sub-Finslerian Interpolation Inequalities", preprint (2026), arXiv:2607.16817, https://arxiv.org/abs/2607.16817
17. Annamaria Montanari, Daniele Morbidelli, "New properties of length-extremals in free step-2 rank-4 Carnot groups", Ricerche di Matematica 75 (2026), no. 3, 1433-1467. DOI: 10.1007/s11587-026-01054-3, https://doi.org/10.1007/s11587-026-01054-3 (also arXiv:2503.00021, https://arxiv.org/abs/2503.00021)