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Regular Curves, Singular Graphs: Cantor Parts and the Relaxed Willmore Energy

This paper demonstrates that finite relaxed one-dimensional Willmore energy does not necessarily rule out diffuse singularities in the derivative, by constructing a continuous function with a non-zero Cantor part that still possesses finite energy.

Original authors: Hans-Christoph Grunau, Boris Gulyak

Published 2026-07-15

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Hans-Christoph Grunau, Boris Gulyak

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a cartographer trying to draw a map of a very bumpy, jagged mountain range. In the world of math, these "mountains" are curves, and the "bumpiness" is measured by something called elastic energy (or Willmore energy). Think of this energy as the cost of bending a stiff wire into a shape. If the wire is too crinkled, the energy bill goes to infinity, and the shape is considered "broken" or impossible in the eyes of standard physics.

For a long time, mathematicians believed a specific rule: if a shape has a finite energy bill, it must be "clean." They thought the map could only have smooth slopes and sharp, sudden jumps (like a cliff edge). They believed that any "fuzzy," infinitely detailed, yet non-jumping mess (called a Cantor part) would automatically make the energy bill infinite. It was like saying, "If your map has any invisible, dust-like fuzziness, the cost is too high, so that fuzziness can't exist."

The Big Surprise
In this paper, Hans-Christoph Grunau and Boris Gulyak prove that this rule is wrong. They constructed a specific, continuous curve that has a finite energy bill, yet it is covered in that "fuzzy" Cantor dust.

Here is how they pulled off the magic trick:
They realized that the "fuzziness" (the Cantor part) only becomes expensive if it sits on a flat or gentle slope. So, they built a curve where the fuzziness is concentrated exactly where the slope of the curve goes vertical (blows up to infinity).

Think of it like this: Imagine a staircase where the steps get infinitely steep. Usually, adding a fuzzy, invisible layer to a step would cost extra energy. But if the step is already so steep it's practically a vertical wall, the math allows you to add that fuzzy layer without increasing the total energy cost. The "fuzz" hides perfectly in the "vertical" chaos.

What This Means for the Curve
The authors didn't just find a weird number; they built a physical shape.

The Shape: It is a continuous line that never breaks or jumps.
The Twist: If you look at the curve from the standard "up-and-down" view, its derivative (the slope) has a Cantor part. It looks like a messy, fractal-like mess.
The Secret: However, if you rotate the entire curve slightly, that mess disappears! The curve becomes a perfectly smooth, Lipschitz graph (a nice, well-behaved line) with no fuzziness at all.
The Conclusion: The "fuzziness" isn't a fundamental flaw in the curve itself; it's an illusion caused by the angle you are looking at it from. The curve is actually a smooth, $C^1$ (continuously turning) object that can be described by a smooth path, even though its standard graph looks broken.

What They Ruled Out
The paper explicitly argues against the idea that finite energy automatically forces a shape to be "clean" (specifically, belonging to the class $SBV$, which bans Cantor parts). They show that you cannot simply say, "If the energy is low, there can be no Cantor dust." They also rule out the idea that there is a "safety threshold" of energy below which Cantor parts are forbidden. You can make the energy bill as small as you want (even close to zero) and still have a non-zero Cantor part, provided you stretch the curve vertically enough.

How Sure Are They?
This isn't a guess or a simulation. The authors have proved it mathematically. They constructed the function step-by-step using a generalized Cantor set (a specific type of fractal) and a carefully designed "slope profile." They calculated the energy and showed, with rigorous math, that the total cost is finite while the Cantor part remains non-zero.

The Bigger Picture
This discovery changes how we think about "smoothness" in geometry.

For $p=2$ (The Willmore Energy): They proved the existence of such a curve.
For all $p > 1$ : They showed this trick works for a whole family of similar energy rules, not just the one they started with.
Surfaces of Revolution: They even showed that if you spin this weird curve around an axis, you get a 3D surface (like a vase) that has finite energy, even though the profile line has that Cantor fuzz.

The Takeaway
The paper demonstrates that nature (or at least, the math of elastic curves) is more flexible than we thought. You can have a shape that looks messy and fractal from one angle, yet is perfectly smooth and finite in energy. The "fuzz" isn't a bug; it's a feature that only appears when you look at the curve the wrong way. The authors have proven that finite energy does not rule out these diffuse singularities, shattering the old expectation that "finite energy means clean."

Technical Summary: Regular Curves, Singular Graphs: Cantor Parts and the Relaxed Willmore Energy

Problem Statement
The paper addresses a fundamental question in the calculus of variations concerning the regularity of minimizers (or finite-energy limits) for curvature-dependent functionals, specifically the one-dimensional Willmore energy (elastic energy) and its $L^p$ generalizations. In the context of free-discontinuity problems and the theory of Special Functions of Bounded Variation (SBV), it is a standard expectation that finite relaxed energy excludes "diffuse" singularities in the derivative, leaving only absolutely continuous and jump parts. Jump parts correspond geometrically to vertical segments in graph limits, while Cantor parts represent diffuse, atomless singular measures.

The central problem investigated is whether the finiteness of the relaxed one-dimensional Willmore energy, $W(u)$ , for a function $u \in BV((0,1))$ , necessarily implies that $u \in SBV((0,1))$ (i.e., that the Cantor part of the distributional derivative, $D^c u$ , vanishes). While previous results for certain exponents $p$ close to 1 suggested that Cantor parts might be energetically neutral or excluded, the behavior for $p=2$ (the Willmore case) and general $p>1$ remained an open question, particularly regarding whether the geometric constraints of finite curvature energy could rule out such singularities.

Methodology
The authors employ a constructive approach based on the relaxation theory developed by Dal Maso, Fonseca, Leoni, and Morini (DMFLM09). Their methodology involves the following steps:

Relaxation Framework: They utilize the characterization of the relaxed functional $F_p(u)$ (which includes the Willmore term plus a total variation term) on the space $X^p_\psi$ . A key condition for a function to have finite relaxed energy is that its singular part must be concentrated on the set where the absolutely continuous representative of the derivative, $(u')_a$ , blows up (tends to $\pm \infty$ ).
Superposition Construction: To construct a counterexample, the authors define a function $u = U + f_\delta$ $u = U + f_{δ}$ .
- $f_\delta$ is a generalized Cantor function (monotone, continuous, with a non-trivial Cantor part $D^c f_\delta$ ).
- $U$ is a continuous, monotone function constructed specifically so that its derivative $U'$ (which serves as the absolutely continuous part of $u'$ ) tends to $+\infty$ precisely on the Cantor set $D_\delta$ associated with $f_\delta$ .
Singular Profile Design: The construction of $U$ $U$ relies on a specific singular profile $\Phi$ $Φ$ defined on the intervals removed during the Cantor set construction. This profile is designed such that:
- $U'$ blows up on $D_\delta$ , satisfying the concentration requirement for the singular part.
- The composition $\Psi_p \circ U'$ (where $\Psi_p$ is the primitive of the weight function $\psi^{1/p}$ ) belongs to the Sobolev space $W^{1,p}$ , ensuring the curvature energy remains finite.
- The parameters of the construction (scaling factors and exponents) are carefully tuned to ensure the convergence of the energy integrals despite the unbounded slope.
Geometric Analysis: The authors analyze the geometric properties of the graph of $u$ . They demonstrate that while the graph function $u$ has a singular derivative in the vertical projection, the curve itself admits an arc-length parametrization $\gamma$ that is of class $C^1 \cap W^{2,p}$ . Furthermore, they show that a suitable rotation of the graph eliminates the Cantor part entirely, rendering the derivative of the rotated graph function Lipschitz (and thus in $SBV$).

Key Results
The paper establishes the following main theorems and corollaries:

Theorem 1 (Willmore Case): There exists a continuous function $u \in BV((0,1)) \cap C^0([0,1])$ such that the relaxed Willmore energy $W(u)$ is finite, yet the Cantor part of its derivative is non-zero ( $D^c u \neq 0$ ). Consequently, finite relaxed Willmore energy does not imply $u \in SBV((0,1))$ .
Theorem 2 (General Weighted Case): The result extends to general weighted curvature energies $F_p$ for all $p > 1$ , provided the weight function $\psi$ decays sufficiently fast ( $\psi(t) \leq C t^{-\alpha}$ with $\alpha > 2p-1$ ).
Corollary 3 ( $L^p$ -Curvature Energies): For the specific case of relaxed $L^p$ -curvature energies $W_p$ (where the weight decays as $t^{-(3p-1)}$ ), finite energy does not exclude non-trivial Cantor parts for any $p > 1$ .
Geometric Interpretation (Theorem 9 & Remark 10): The graph of the constructed function admits an arc-length parametrization $\gamma$ which is a regular $C^1$ curve with square-integrable ( $L^2$ ) curvature (or $L^p$ for general $p$ ). In a specific variant, the graph can even be a $C^2$ -submanifold of $\mathbb{R}^2$ . The presence of the Cantor part is shown to be a projection artifact; it vanishes if the graph is rotated.
Corollary 6 (Energy Scaling): The relaxed Willmore energy of such functions can be made arbitrarily small while maintaining a non-trivial Cantor part. This implies there is no positive energy threshold below which Cantor parts are excluded.
Corollary 15 (Surfaces of Revolution): By revolving a $C^2$ -regular profile curve (constructed via the $C^2$ -variant) around an axis, one obtains a surface of revolution with finite Willmore energy, whose generating radius function possesses a non-trivial Cantor part in its distributional derivative.

Significance and Claims
The paper claims that its construction fundamentally alters the understanding of regularity in relaxed curvature functionals:

Failure of SBV Expectation: It disproves the intuition that finite relaxed curvature energy rules out diffuse singularities (Cantor parts). This contrasts with the behavior of first-order functionals like the Mumford-Shah energy, where energy bounds typically enforce SBV regularity.
Projection Dependence: The authors emphasize that the Cantor part is not an intrinsic singularity of the underlying geometric curve. The curve itself is geometrically "tame" (regular $C^1$ with integrable curvature), but the specific choice of projection (the graph representation) introduces the singular part. This highlights a distinction between the intrinsic geometry of a curve and the regularity of its graph representation.
Compatibility with Relaxation Theory: The construction demonstrates that the "blow-up" mechanism required by the relaxation theorem (Dal Maso et al.) is sufficient to accommodate Cantor parts even for $p=2$ and higher, extending previous results that were limited to $p$ close to 1.
Implications for Minimizers: The authors note that their result suggests that regularity of minimizers for relaxed Willmore functionals cannot be deduced solely from small-energy arguments (as is common in parametric Willmore theory). Instead, regularity must rely on minimality conditions or lower-order terms, as the energy itself does not penalize the Cantor part if it is concentrated on the blow-up set of the slope.

The paper concludes by noting that while the construction is one-dimensional, it can be embedded into two-dimensional graph problems via product constructions, suggesting that similar phenomena may occur in higher dimensions, though the existence of Cantor parts in actual minimizers remains an open question.

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