Strong Pliability: Control, Geometry, Mechanics

• Strong pliability is a flexible property defined by non-rigidity in various settings, such as Carnot groups (via endpoint maps), birational geometry (via Mori fibre spaces), and mechanics (via large strain tolerance).
• In Carnot groups, strong pliability is equivalent to the openness of the multiexponential map and to regularity of the endpoint control, establishing its link with local controllability.
• In mechanics, strong pliability characterizes exceptional reversible deformation, as shown in ultrathin nanofilms and phosphorene which exhibit high strain tolerance and progressive stiffening under load.

Strong pliability is a domain-dependent term with distinct technical meanings in contemporary research. In Carnot-group geometry and control, it denotes a local endpoint-flexibility property of horizontal directions under small perturbations, and recent work proves that strong pliability, pliability, and the openness of a multiexponential map are equivalent (Jean et al., 17 Jul 2025). In birational geometry, pliability is the set of Mori fibre spaces birational to a given Mori fibre space up to square equivalence, and “strong” usage refers to unusually large finite or infinite pliability (Sarikyan, 2022). In mechanics and materials, the term is used descriptively for systems that undergo very large reversible deflection or strain, or that are initially compliant and later stiffen under bending, as in polyethylene cellular nanofilms, phosphorene, and tensegrity-inspired polymer films (Gao et al., 2020).

1. Terminological range

The literature uses “strong pliability” in several non-equivalent senses. The common element is non-rigidity under a constrained class of deformations, but the object being deformed, the admissible perturbations, and the relevant invariants differ substantially.

Domain	Object	Technical content
Carnot groups	Horizontal vector or curve	Local openness of endpoint behavior under small horizontal perturbations
Sarkisov category	Mori fibre space	Cardinality of the birational Mori fibre space class up to square equivalence
Mechanics and materials	Film, membrane, or crystal	Large recoverable deformation, high strain tolerance, or progressive bending response

In the Carnot-group setting, the relevant perturbations are horizontal $C^1$ curves or controls in the horizontal layer, and the central question is whether nearby perturbations fill a neighborhood of the endpoint data. In birational geometry, the pertinent structure is the set of all Mori fibre spaces in the birational class, modulo square equivalence. In materials science, pliability is established through mechanical metrics such as recoverable central deflection, critical tensile strain, modulus contrast, and curvature-dependent bending stiffness (Jean et al., 17 Jul 2025).

2. Strong pliability in Carnot groups

For a Carnot group $G$ with horizontal layer $\mathfrak g_1$, the note "A note on pliability and the openness of the multiexponential map in Carnot groups" defines the endpoint map

$$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$

where $\gamma$ solves

$$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$

Given $X\in \mathfrak g_1$, the endpoint map based at $X$ is

$E_X(Y):=E(X+Y).$

In this framework, $X$ is $G$0-pliable if the map $G$1 restricted to $G$2 is open at $G$3. The strengthened notion is: $G$4 is strongly $G$5-pliable if for every $G$6 there exists $G$7 such that

$G$8

and $G$9 is a submersion at $\mathfrak g_1$0. For $\mathfrak g_1$1, this is the paper’s strong pliability (Jean et al., 17 Jul 2025).

The same paper introduces the finite-dimensional multiexponential map

$\mathfrak g_1$2

and the $\mathfrak g_1$3-condition: $\mathfrak g_1$4 Its main theorem states that, for every $\mathfrak g_1$5, the following are equivalent: $\mathfrak g_1$6 Moreover, $\mathfrak g_1$7 satisfies the submersive $\mathfrak g_1$8-condition $\mathfrak g_1$9 if and only if $$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$0 is a regular point of the endpoint map $$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$1, and both imply the preceding four conditions.

This equivalence is the decisive clarification in the recent theory: strong pliability is presented as a strengthened local controllability or non-rigidity property, but it is not a genuinely new notion. It is equivalent to ordinary pliability and to a finite-dimensional openness property of a multiexponential map (Jean et al., 17 Jul 2025).

3. Directional, uniform, and Whitney-theoretic variants

The paper "Pliability, or the whitney extension theorem for curves in carnot groups" defines a horizontal curve $$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$2 to be pliable if for every neighborhood $$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$3 of $$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$4 in $$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$5, the set

$$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$6

is a neighborhood of $$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$7 in $$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$8. A horizontal vector $$E:L^\infty(I,\mathfrak g_1)\to G,\qquad E(Y)=\gamma(1),$$9 is pliable if the straight curve $\gamma$0 is pliable. The same paper introduces the stronger-looking notion of local uniform pliability and proves that if $\gamma$1 is pliable, then every horizontal vector is locally uniformly pliable; however, pliability and local uniform pliability are not equivalent in general (Juillet et al., 2016).

The later paper "Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groups" shifts the emphasis from all directions to a subset $\gamma$2. Its main theorem states: if every vector in $\gamma$3 is pliable, then $\gamma$4 has the $\gamma$5 Whitney extension property on $\gamma$6. The directional Whitney condition is expressed by requiring that, for compact $\gamma$7, continuous $\gamma$8, and continuous $\gamma$9,

$$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$0

which then implies the existence of $$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$1 such that

$$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$2

Proposition 3.1 gives the uniform form used in the construction: if $$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$3 is compact and every vector in $$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$4 is pliable, then for every $$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$5 there exists $$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$6 such that the same boundary-value solvability conclusion holds uniformly for all $$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$7 (Speight et al., 20 May 2025).

The Engel group furnishes the sharp model example. With

$$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$8

and brackets

$$\begin{cases} (L_{\gamma(t)})^*\dot\gamma(t)=Y(t),\ \gamma(0)=1_G. \end{cases}$$9

the paper proves: $X\in \mathfrak g_1$0 Thus the only non-pliable nonzero directions are multiples of $X\in \mathfrak g_1$1. This yields the directional Whitney extension property on

$X\in \mathfrak g_1$2

a partial Lusin approximation theorem, and the further conclusion that every horizontal curve in the Engel group intersects some $X\in \mathfrak g_1$3 horizontal curve on a set of positive measure (Speight et al., 20 May 2025).

4. Pliability in birational geometry

In birational geometry, pliability is not a local deformation property but a Sarkisov-theoretic birational invariant. For a Mori fibre space $X\in \mathfrak g_1$4,

$X\in \mathfrak g_1$5

where $X\in \mathfrak g_1$6 is square birational equivalence. A birational map

$X\in \mathfrak g_1$7

between Mori fibre spaces $X\in \mathfrak g_1$8 and $X\in \mathfrak g_1$9 is a square equivalence if there exists a birational map $X$0 such that the induced birational map on the generic fibres is an isomorphism (Abban, 2013).

Several papers exhibit unusually large pliability. "On pliability of del Pezzo fibrations and Cox rings" constructs explicit Sarkisov links for a smooth complete intersection

$X$1

with

$X$2

and proves that $X$3 is a Mori fibre space whose generic fibre is a del Pezzo surface of degree $X$4, that there exist at least two non-trivial Sarkisov links from $X$5 to other Mori fibre spaces, and that $X$6 is not rational. Since $X$7 itself is one Mori fibre space, this gives

$X$8

The same paper computes

$X$9

and invokes Alexeev’s theorem to conclude nonrationality (Abban, 2013).

"On the Rationality of Fano-Enriques Threefolds" gives a more striking finite example. For a general Fano-Enriques threefold whose canonical covering $E_X(Y):=E(X+Y).$0 is the double covering of a quadric branched in a divisor of degree $E_X(Y):=E(X+Y).$1, the paper constructs eight explicit Sarkisov links of type I, each producing a Mori fibre space $E_X(Y):=E(X+Y).$2 whose general fibre is a del Pezzo surface of degree $E_X(Y):=E(X+Y).$3, and proves that every birational map from $E_X(Y):=E(X+Y).$4 to a Mori fibre space factors through one of these models. In particular, for a general such $E_X(Y):=E(X+Y).$5,

$E_X(Y):=E(X+Y).$6

In this literature, “strong pliability” is used informally for such large finite values (Sarikyan, 2022).

"High-pliability Fano hypersurfaces" proves that five of Reid’s Fano 3-fold hypersurfaces containing at least one compound Du Val singularity of type $E_X(Y):=E(X+Y).$7 have pliability at least two. The two elements of the pliability set are the singular hypersurface itself and another non-isomorphic Fano hypersurface of the same degree, embedded in the same weighted projective space, but with different compound Du Val singularities. The endpoints are also proved factorial (Campo, 2022).

The strongest phenomenon appears in "Birational Geometry of sextic del Pezzo surfaces". That paper does not explicitly define a separate notion called “strong pliability”; instead, the relevant phenomenon is infinite pliability. Its central theorem states that if $E_X(Y):=E(X+Y).$8 is a birationally solid del Pezzo surface over a perfect field and $E_X(Y):=E(X+Y).$9, then $X$0 is a sextic del Pezzo surface with $X$1 or $X$2. It further proves the existence of a solid sextic del Pezzo surface $X$3 with $X$4, and shows that degree $X$5 del Pezzo surfaces are the only solid surfaces that admit infinite pliability (Kurz et al., 29 Jul 2025).

5. Strong pliability as exceptional mechanical deformability

In the mechanics and materials literature represented here, pliability is established by quantitative response under load rather than by a single formal definition. The paper "Ultrastrong, Ultraflexible, and Ultratransparent Polyethylene Cellular Nanofilms" gives the clearest example of extreme recoverable flexibility. Its key result is a spherical indentation test on a $X$6 nm-thick freestanding film that could be deflected reversibly up to $X$7 mm and sustained this behavior for $X$8 cycles. The corresponding deflection-to-thickness ratio is about $X$9, the maximum rupture force is $G$00 N, and this force is described as about $G$01 million times the film’s weight. The same material has in-plane specific tensile strength $G$02, Young’s modulus $G$03 GPa for the multilayer film, work of fracture $G$04, linear viscoelasticity up to about $G$05 strain for the multilayer film, and an annealed monolayer 2D film modulus $G$06 GPa. The authors attribute this behavior to an ultrathin geometry, a stretch-dominated 2D cellular topology with Delaunay triangulations, highly crystalline molecularly anisotropic cell edges made of extended-chain PE fibrils, and a sequential biaxial planar extension route. For stretch-dominated Delaunay cells, the porosity-scaling relation uses $G$07, whereas for bending-dominated Voronoi tessellations $G$08, and the 2D Maxwell stability metric is

$G$09

The same paper demonstrates a freestanding ultratransparent respiratory face covering with area density $G$10, effective freestanding area $G$11, air flow rate $G$12, pressure drop $G$13 Pa, and NaCl aerosol filtration efficiency $G$14 (Gao et al., 2020).

"Superior mechanical flexibility of phosphorene and few-layer black phosphorus" uses first-principles calculations to characterize strong pliability as large in-plane tensile strain tolerance. A monolayer phosphorene can sustain tensile strain up to $G$15 in the zigzag direction and $G$16 in the armchair direction; for few-layer black phosphorus, the corresponding critical strains are $G$17 in the zigzag direction and $G$18 in the armchair direction. The ideal tensile strengths for monolayer phosphorene are approximately $G$19 GPa along zigzag and $G$20 GPa along armchair; for few-layer phosphorene they are approximately $G$21 GPa and $G$22 GPa, respectively. The Young’s modulus of monolayer phosphorene varies from $G$23 GPa along armchair to $G$24 GPa along zigzag, with an average value of about $G$25 GPa. The paper attributes this to the puckered honeycomb crystal structure: under $G$26 armchair strain, bond lengths change only slightly, bond angle $G$27 changes very little, the puckered-layer distance $G$28 drops from $G$29 Å to $G$30 Å, and the dihedral angles decrease by about $G$31. This means the strain is taken up mainly by flattening the puckers rather than by severely stretching the P–P bonds (Wei et al., 2014).

"Tensegrity-Inspired Polymer Films: Progressive Bending Stiffness through Multipolymeric Patterning" addresses a different mechanical regime: a material that is highly pliable at small deformation but becomes progressively harder to bend as curvature increases. The film combines a soft membrane material and rigid rod domains with a large modulus contrast: rod modulus $G$32 MPa and membrane modulus $G$33 MPa. Sample B, with vertically oriented rows and alternating rows half-phase shifted, is the configuration that best realizes the tensegrity-like bending response. Under compression of tunnel-shaped films by $G$34 mm, Sample B reached a maximum load of $G$35 N $G$36, whereas Sample A reached $G$37 N $G$38. In bending, Sample B showed an early load jump around $G$39 mm displacement, and between about $G$40 and $G$41 mm displacement, with curvature around $G$42 to $G$43, the load increased gradually as rods began to protrude and stretch the membrane. The paper analyzes this with

$G$44

and attributes progressive stiffening to membrane tension generated by rod protrusions together with an increase in second moment of area in the maximum-curvature region. It also states that the present response is not yet a fully true J-shaped bending law (Kuwada et al., 2024).

6. Comparative interpretation and recurrent misunderstandings

A first recurrent misunderstanding is terminological: the phrase does not denote a single invariant across fields. In Carnot groups, strong pliability concerns endpoint openness and submersion properties of horizontal controls. In birational geometry, pliability is a set of Mori fibre space models up to square equivalence. In mechanics, pliability is evidenced by recoverable deflection, large admissible strain, or curvature-dependent bending response (Jean et al., 17 Jul 2025).

A second misunderstanding concerns the relation between pliability and strong pliability in Carnot groups. The 2025 note proves that pliability, strong pliability, and the $G$45-condition are equivalent, while the submersive $G$46-condition is stronger and equivalent to regularity of the endpoint map. By contrast, the 2016 paper does not use the exact phrase “strong pliability”; its strengthened variant is local uniform pliability (Juillet et al., 2016).

A third misunderstanding is to treat directional pliability as equivalent to full pliability. The Engel-group analysis shows that this is false: nonzero multiples of $G$47 are non-pliable, yet

$G$48

consists entirely of pliable directions, and this suffices for partial Whitney and Lusin theorems on the corresponding subset of directions (Speight et al., 20 May 2025).

A fourth misunderstanding is to identify high pliability with rationality in birational geometry. The cited examples show the opposite pattern can occur. The degree-$G$49 del Pezzo fibration with $G$50 is not rational; the Fano-Enriques threefold with $G$51 is presented precisely as a non-rational variety with many Mori fibre space models; and infinite pliability for sextic del Pezzo surfaces is analyzed within the class of solid surfaces (Sarikyan, 2022).

A fifth misunderstanding is to equate mechanical pliability with weakness. The polyethylene cellular nanofilm combines ultrahigh flexibility with ultrahigh in-plane specific tensile strength, work of fracture, and self-standing integrity; phosphorene combines large critical strains with anisotropic ideal strengths; and the tensegrity-inspired polymer film is designed to be initially compliant and then progressively stiffer under bending (Gao et al., 2020).

This suggests that the unifying motif behind the different uses of strong pliability is not a common formal definition but a recurring contrast with rigidity: local endpoint rigidity in Carnot groups, birational rigidity in the Sarkisov category, and brittle or bending-dominated response in thin materials.