Constrained Isometric Co-extension Theory

• Constrained isometric co-extension is a boundary-value formulation where prescribed isometric data on a hypersurface is extended to a neighborhood while preserving both trace and metric constraints.
• The construction utilizes convex integration techniques and adapted short maps to correct metric defects by decomposing them into primitive metrics and employing localized corrugation profiles.
• The approach differentiates equidimensional rigidity from one-sided flexibility, highlighting the role of curvature, second fundamental forms, and regularity thresholds in isometric embedding.

“Constrained isometric co-extension” (Editor's term) denotes a boundary-value form of the isometric immersion problem in which isometric data prescribed on a hypersurface or submanifold are extended to a neighborhood, or to one side of that submanifold, while preserving both the trace data and the metric constraint. In its basic equidimensional form one fixes a smooth $n$-manifold $(M^n,g)$, a hypersurface $\Sigma\subset M$, and a smooth isometric immersion $f:\Sigma\to\mathbb R^n$, and asks for a map $v$ on a neighborhood $U\subset M$ such that

$v^*g_0=g,\qquad v|_\Sigma=f,$

or, in low regularity, $v^*g_0=g$ almost everywhere. The modern theory is stratified by codimension, sidedness, and regularity: equidimensional extensions are highly rigid above the Lipschitz threshold, one-sided codimension-one problems admit convex-integration flexibility, and smooth submanifold-based formulations are governed by curvature and second-fundamental-form compatibility (Wasem, 2015, Hungerbühler et al., 2014, Yu, 2019).

1. Boundary-value formulation and geometric setup

In the equidimensional setting studied by Wasem, the target has the same dimension as the source, so the problem is codimension zero. The prescribed datum is an isometric immersion

$f:\Sigma\to \mathbb R^n,$

with $\mathbb R^n$ carrying the Euclidean metric $(M^n,g)$0, and the extension problem is local near a point of $(M^n,g)$1. The paper distinguishes full neighborhoods from one-sided neighborhoods, and the constructive theory is one-sided: in local coordinates one works on an $(M^n,g)$2-polytope $(M^n,g)$3, sets

$(M^n,g)$4

and treats $(M^n,g)$5 and $(M^n,g)$6 as the relevant one-sided domains (Wasem, 2015).

A parallel codimension-one formulation appears in the one-sided isometric extension literature. There the prescribed datum is a smooth isometric immersion or embedding $(M^n,g)$7, $(M^n,g)$8, or $(M^n,g)$9, and the extension is sought on a one-sided neighborhood of $\Sigma\subset M$0 constructed from the exponential map. In Cao and Inauen’s global formulation, if $\Sigma\subset M$1 is the unit normal respecting the orientation, then

$\Sigma\subset M$2

defines

$\Sigma\subset M$3

and one asks for an isometric embedding $\Sigma\subset M$4 satisfying $\Sigma\subset M$5 (Cao et al., 2020).

Regularity is intrinsic to the formulation. In codimension zero, the exact equation is often relaxed from $\Sigma\subset M$6 pointwise to

$\Sigma\subset M$7

because the available flexible constructions are Lipschitz and generally non-$\Sigma\subset M$8. In codimension one, by contrast, one-sided $\Sigma\subset M$9 and $f:\Sigma\to\mathbb R^n$0 extensions exist under suitable positivity hypotheses, and in that regime the output can be chosen as an embedding if the seed short map is an embedding (Wasem, 2015, Hungerbühler et al., 2014, Cao et al., 2018).

2. Equidimensional rigidity and obstruction

The equidimensional problem is fundamentally different from the classical Nash–Kuiper regime. Wasem emphasizes that in codimension zero there is no positive-dimensional sphere of normal oscillatory directions, and the classical Liouville theorem implies that images of $f:\Sigma\to\mathbb R^n$1 equidimensional Euclidean isometric immersions are congruent. This is sharpened by a local nonexistence theorem: an $f:\Sigma\to\mathbb R^n$2-dimensional Riemannian manifold $f:\Sigma\to\mathbb R^n$3 can be locally isometrically embedded by a differentiable map into $f:\Sigma\to\mathbb R^n$4 if and only if $f:\Sigma\to\mathbb R^n$5 is flat, and in that case the map is in fact $f:\Sigma\to\mathbb R^n$6 (Wasem, 2015). Thus any differentiable equidimensional constrained co-extension forces flatness.

For hypersurface data there is a more refined obstruction based on second fundamental forms. Let $f:\Sigma\to\mathbb R^n$7 be the second fundamental form of $f:\Sigma\to\mathbb R^n$8, with scalar form

$f:\Sigma\to\mathbb R^n$9

and let $v$0 be the second fundamental form of $v$1, with scalar form

$v$2

If there exists a unit vector $v$3 such that

$v$4

then no isometric Lipschitz extension $v$5 can exist (Wasem, 2015). The proof is a length-comparison argument: along a geodesic $v$6, any Lipschitz isometric extension would imply

$v$7

while the asymptotic expansions of these distances are controlled by the scalar second fundamental forms.

An analogous obstruction survives in higher codimension for two-sided local extension. In Hungerbühler–Wasem’s one-sided theory, if there exists a unit vector $v$8 such that

$v$9

then no isometric extension $U\subset M$0 can exist on a full neighborhood $U\subset M$1 of $U\subset M$2 (Hungerbühler et al., 2014). The point is not that all low-regularity extension is obstructed, but that full local extension is more rigid than one-sided extension.

A higher-regularity codimension-one rigidity phenomenon appears in $U\subset M$3. Cao and Inauen prove that if $U\subset M$4, $U\subset M$5, and $U\subset M$6 is an isometric extension of prescribed smooth codimension-one data, then

$U\subset M$7

for every tangent field $U\subset M$8 (Cao et al., 2020). Equivalently, along $U\subset M$9, the tangential connection induced by the extension must agree with the Levi-Civita connection. This is a first-order compatibility law forced by regularity above the Hölder threshold $v^*g_0=g,\qquad v|_\Sigma=f,$0.

3. Weak convex integration in codimension zero

The positive equidimensional theory begins with an adapted short map. For a one-sided neighborhood $v^*g_0=g,\qquad v|_\Sigma=f,$1 of $v^*g_0=g,\qquad v|_\Sigma=f,$2, a map

$v^*g_0=g,\qquad v|_\Sigma=f,$3

is a short map adapted to $v^*g_0=g,\qquad v|_\Sigma=f,$4 if $v^*g_0=g,\qquad v|_\Sigma=f,$5 and

$v^*g_0=g,\qquad v|_\Sigma=f,$6

as quadratic forms, with equality on $v^*g_0=g,\qquad v|_\Sigma=f,$7 only; equivalently, $v^*g_0=g,\qquad v|_\Sigma=f,$8 is positive definite on $v^*g_0=g,\qquad v|_\Sigma=f,$9 and zero on $v^*g_0=g$0 (Wasem, 2015). Such a seed exists if there are choices of normals $v^*g_0=g$1 for which

$v^*g_0=g$2

is positive definite.

From an adapted short map, Wasem constructs a one-sided Lipschitz a.e.-isometric extension. The main theorem states that for every $v^*g_0=g$3 there exists a Lipschitz map $v^*g_0=g$4 whose singular set has Hausdorff dimension $v^*g_0=g$5, such that

$v^*g_0=g$6

(Wasem, 2015). The metric defect is not eliminated pointwise by classical Nash spirals or codimension-one corrugations. Instead, the defect is decomposed into primitive metrics

$v^*g_0=g$7

and corrected one rank-one term at a time by a degenerate corrugation ansatz adapted to codimension zero.

The crucial geometric obstruction to classical convex integration is that in codimension zero the “sphere” of oscillatory directions degenerates to two points. Accordingly, the exact circle equation is replaced by inequalities and average identities. In the regular case, the corrugation profile $v^*g_0=g$8 is chosen so that

$v^*g_0=g$9

$f:\Sigma\to \mathbb R^n,$0

$f:\Sigma\to \mathbb R^n,$1

thereby producing only an averaged correction to the metric defect (Wasem, 2015). This is why the scheme is described as weak convex integration.

Iteration proceeds by steps and stages. After correcting all primitive metrics in a stage, the map is approximated again by an adapted piecewise affine short map. The key stage estimate is that for any $f:\Sigma\to \mathbb R^n,$2 one obtains $f:\Sigma\to \mathbb R^n,$3 with

$f:\Sigma\to \mathbb R^n,$4

and

$f:\Sigma\to \mathbb R^n,$5

Derivative control is obtained from the $f:\Sigma\to \mathbb R^n,$6 estimate

$f:\Sigma\to \mathbb R^n,$7

which yields $f:\Sigma\to \mathbb R^n,$8-Cauchy convergence of the gradients and finally

$f:\Sigma\to \mathbb R^n,$9

(Wasem, 2015).

The singular set is explicit: $\mathbb R^n$0 so

$\mathbb R^n$1

unless no new simplices are introduced. The paper also proves a density statement: if

$\mathbb R^n$2

is the $\mathbb R^n$3-closure of adapted short maps and

$\mathbb R^n$4

then the zero set of $\mathbb R^n$5 is dense in $\mathbb R^n$6 (Wasem, 2015).

4. One-sided flexibility in codimension one

In codimension one, the one-sided theory of Hungerbühler–Wasem gives a $\mathbb R^n$7 analogue of the adapted-short-map paradigm. A smooth immersion $\mathbb R^n$8 is adapted to $\mathbb R^n$9 if

$(M^n,g)$00

with equality only on $(M^n,g)$01. If there exists a unit normal field $(M^n,g)$02 such that

$(M^n,g)$03

is positive definite on $(M^n,g)$04, then near every $(M^n,g)$05 there exists an adapted short map, even an adapted short embedding (Hungerbühler et al., 2014). The main existence theorem states that every such adapted short map can be approximated arbitrarily well in $(M^n,g)$06 by a $(M^n,g)$07-isometric one-sided immersion $(M^n,g)$08 satisfying

$(M^n,g)$09

and that the construction satisfies a $(M^n,g)$10-dense parametric $(M^n,g)$11-principle (Hungerbühler et al., 2014).

Codimension-one $(M^n,g)$12 extensions sharpen this picture. Given a smooth isometric immersion $(M^n,g)$13, if there exists a unit normal field $(M^n,g)$14 along $(M^n,g)$15 such that

$(M^n,g)$16

as a quadratic form on $(M^n,g)$17, then there is a one-sided neighborhood $(M^n,g)$18 of $(M^n,g)$19 and, for every

$(M^n,g)$20

a $(M^n,g)$21 isometric immersion

$(M^n,g)$22

with

$(M^n,g)$23

(Cao et al., 2018). The initial seed is an adapted short immersion with defect of the form

$(M^n,g)$24

where $(M^n,g)$25 near $(M^n,g)$26, and the iteration is localized away from $(M^n,g)$27 so that every perturbation vanishes near the prescribed hypersurface.

The codimension-one convex integration mechanism is stronger than the equidimensional one because there is still a genuine normal direction. Primitive metrics are corrected by corrugations satisfying the exact identity

$(M^n,g)$28

and the resulting $(M^n,g)$29 regularity threshold

$(M^n,g)$30

comes from balancing defect decay, frequency growth, and the $(M^n,g)$31-cost of each stage (Cao et al., 2018). This is a low-codimension Nash–Kuiper regime with an explicit boundary-preserving localization.

5. Critical regularity at $(M^n,g)$32

Cao and Inauen identify $(M^n,g)$33 as the critical Hölder exponent for constrained codimension-one $(M^n,g)$34 extensions. Their theorem has a rigidity/flexibility dichotomy. If $(M^n,g)$35, $(M^n,g)$36, and $(M^n,g)$37 is a $(M^n,g)$38 isometric extension of $(M^n,g)$39, then

$(M^n,g)$40

If $(M^n,g)$41, $(M^n,g)$42 with

$(M^n,g)$43

then there exists $(M^n,g)$44 and $(M^n,g)$45 such that

$(M^n,g)$46

at all points where $(M^n,g)$47 (Cao et al., 2020).

The rigidity mechanism above $(M^n,g)$48 is analytic. A result of De Lellis–Inauen implies that for $(M^n,g)$49, $(M^n,g)$50, the weak tangential connection is well defined distributionally, so Gauss-type identities survive at the regularity level needed to compare the induced tangential connection with the Levi-Civita connection along $(M^n,g)$51. Since $(M^n,g)$52 on $(M^n,g)$53 and $(M^n,g)$54 is smooth, the distributional identity becomes a pointwise compatibility law there (Cao et al., 2020).

Below $(M^n,g)$55, the iteration preserves not only the trace but also the differential on $(M^n,g)$56. The short initial extension is built in the form

$(M^n,g)$57

and already satisfies

$(M^n,g)$58

Because the convex integration step is arranged so that

$(M^n,g)$59

the final exact isometric extension inherits the same strict inequality (Cao et al., 2020). The same machinery also yields a global existence theorem: if $(M^n,g)$60 is a compact $(M^n,g)$61-manifold with $(M^n,g)$62 metric and $(M^n,g)$63, then there exist infinitely many $(M^n,g)$64 isometric embeddings

$(M^n,g)$65

6. Canonical constructions and applications

The most vivid equidimensional example is Wasem’s “isometric collapse” of the sphere. There exist infinitely many Lipschitz maps

$(M^n,g)$66

such that

$(M^n,g)$67

where $(M^n,g)$68 is the equator and $(M^n,g)$69 is the standard inclusion (Wasem, 2015). The construction extends the equatorial data separately to the upper and lower hemispheres by one-sided adapted short maps and then glues the resulting Lipschitz a.e.-isometric maps because the iteration leaves a neighborhood of the equator unchanged.

This sphere-collapse example sharply illustrates the regularity barrier. Because $(M^n,g)$70 has Gaussian curvature $(M^n,g)$71, the map cannot be $(M^n,g)$72 or even merely differentiable by the differentiable flatness theorem. The resulting map is Lipschitz, its singular set is dense in this curved example, and it cannot be locally $(M^n,g)$73 or locally injective (Wasem, 2015). Extrinsic dimension is collapsed, but metric preservation survives almost everywhere in the weak sense.

The codimension-one literature contains a conceptually similar boundary-value example. Hungerbühler–Wasem exhibit flexible $(M^n,g)$74 extensions of the equatorial inclusion

$(M^n,g)$75

to maps $(M^n,g)$76, in sharp contrast with Borisov-type rigidity at $(M^n,g)$77, $(M^n,g)$78 (Hungerbühler et al., 2014). This contrast between equidimensional collapse and codimension-one wrinkling is central to the subject: the prescribed trace datum is the same type of object, but the available extension mechanisms depend decisively on codimension and regularity.

7. Related smooth and metric formulations

A smooth counterpart is provided by the submanifold-based Cartan–Ambrose–Hicks theory of Mencattini, Mendonça, and Vlachos. In the equal-dimensional case, given isometric immersions $(M^n,g)$79 and $(M^n,g)$80, a metric- and connection-preserving bundle map

$(M^n,g)$81

and curvature transport relation

$(M^n,g)$82

they construct a local isometry $(M^n,g)$83 such that

$(M^n,g)$84

In positive codimension they prescribe, in addition, a Riemannian vector bundle $(M^n,g)$85 and a symmetric tensor

$(M^n,g)$86

together with the compatibility law

$(M^n,g)$87

and prove an extension theorem under the Gauss, Codazzi, and Ricci equations (Yu, 2019). This is a smooth constrained co-extension theory in which prescribed boundary data include normal bundle and second-fundamental-form information.

A variational boundary analogue arises in thin-sheet theory. Guven, Hanna, and Müller show that for isometric bending of an unstretchable sheet with free edge, the bulk metric constraint does not by itself determine admissible free-edge deformations. To make the problem mechanically well posed, one must constrain not only the local boundary arc-length but also the local boundary geodesic curvature; otherwise the naive free-edge theory implies unphysical conditions such as local planarity at the boundary (Guven et al., 2019). This does not produce an extension theorem in the convex-integration sense, but it identifies the extra edge invariants required for consistent boundary-value formulations of isometric bending.

There is also a distinct equidimensional low-regularity theory based on Gromov’s length-preserving notion of isometry rather than the pullback equation. In that setting a Lipschitz map is isometric if it preserves the length of every rectifiable curve, and a Baire-category argument shows that the typical $(M^n,g)$88-Lipschitz extension of fixed short data on a compact set $(M^n,g)$89 is isometric on $(M^n,g)$90, while the typical short map on $(M^n,g)$91 is the restriction of a global isometric map of $(M^n,g)$92 (Kirchheim et al., 2014). This suggests a broader landscape in which constrained isometric co-extension encompasses smooth curvature-driven extension, one-sided convex-integration flexibility, and metric length-preserving extension, each with its own notion of admissible boundary data and its own rigidity threshold.