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The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow

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Pith's one-line read Every immersed 2-sphere with Willmore energy at most $12\pi$ admits an energy-decreasing regular homotopy to either a round sphere or a surface in the model family $J$.

desk verdict Genuine progress on the Willmore landscape below 12π, but the main theorem as stated fails for round spheres and the lower-bound half leans on a same-author companion preprint. read the letter →

arxiv 2506.23359 v1 pith:63CX2TVP submitted 2025-06-29 math.DG

Elena Mäder-Baumdicker , Jona Seidel This is my paper · ORCID
classification math.DG MSC 53C4253E4057R42
keywords Willmoreenergyflowregularhomotopyclassimmersed2-spherescatenoidblow-upunavoidablesingularityLi-Yauinequalitytriple-point-freeinvariant
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that the low-energy part of the Willmore energy landscape for immersed 2-spheres in $\mathbb{R}^3$ is completely described by four regular homotopy classes. More precisely, any smooth immersion $f:S^2\to\mathbb{R}^3$ with Willmore energy at most $12\pi$ can be deformed, without ever raising its energy, to either a round sphere or one of the model surfaces in a family $J$. Combined with an invariant from a companion paper, this implies that the sublevel set $I(\omega)$ has exactly four regular homotopy classes for $8\pi<\omega\le 12\pi$, and that the only initial surfaces whose Willmore flow singularities are unavoidable are precisely the surfaces in $J$. The result matters because it turns a global analytic question about the Willmore flow into a manageable topological classification, and it extends the Li–Yau inequality at the $12\pi$ threshold to a large class of triple-point-free immersed spheres.

What carries the argument

The argument is carried by a gluing construction around the catenoid blow-up predicted by the blow-up classification for the Willmore flow below $16\pi$: when $W(f_0)\le 12\pi$, the only possible blow-up at a singularity is a catenoid. The authors cut the surface along the catenoid neck, replace each side by a flat disk, and use the theorem that Willmore flow below $8\pi$ converges to a round sphere. The two flowing hemispheres are glued back to the catenoid at every time, producing catenoid spheres of two types, $\alpha$ and $\beta$; type $\beta$ can be shrunk to lower the energy below $8\pi$ and then flowed to a round sphere, while type $\alpha$ gives exactly the model family $J$. The companion invariant $F$ for triple-point-free immersed spheres is the topological tool that separates the four classes: it is constant along triple-point-free regular homotopies and takes values $e_{\pm 1}$ on round spheres and $e_{\pm 3}$ on $J$.

What would settle it

Find a triple-point-free immersed sphere $f:S^2\to\mathbb{R}^3$ with $W(f)\le 12\pi$ whose invariant $F(f)$ takes a value outside $\{e_{-3},e_{-1},e_1,e_3\}$; Proposition 1.9 would then be false. Alternatively, construct a regular homotopy below $12\pi$ from some $j\in J$ to a round sphere, which would contradict Theorem D and collapse Theorem 1.4.

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Extended reading notes

Core claim

The central claim, Theorem 1.2, is that energy below $12\pi$ forces a very specific global shape: every smooth immersed sphere with $W(f)\le 12\pi$ admits a regular homotopy $H$ with $H_0=f$, $W(H_t)<W(f)$ for every positive time, and $H_1$ either a round sphere or an element of $J$. Theorem 1.4 then says that for $\omega\in(8\pi,12\pi]$ the sublevel set $I(\omega)$ contains exactly four regular homotopy classes: two containing the round spheres of opposite orientation and two containing the model surfaces $\pm j$, whose infimum energy is $8\pi$ but not attained. Proposition 1.6 characterizes unavoidable singularities: a flow starting at $f_0$ with $W(f_0)\le 12\pi$ develops an unavoidable singularity exactly when $f_0\in J$. Proposition 1.9 extends the Li–Yau inequality at $12\pi$: a triple-point-free immersion with invariant $F(f)$ outside $\{e_{\pm 1},e_{\pm 3}\}$ must have $W(f)>12\pi$.

Load-bearing premise

The entire classification depends on the companion paper's invariant and on Theorem D asserting that every regular homotopy from a model surface $j\in J$ to an embedded sphere must pass through at least one triple point; if that theorem were false, the four-class count and the 'unavoidable singularities exactly $J$' statement would both collapse.

Editorial extensions

If this is right

  • For every $\omega\in(8\pi,12\pi]$, the sublevel set $I(\omega)$ splits into exactly four regular homotopy classes, so the energy landscape below $12\pi$ is known at the level of regular homotopy classes.
  • If a smooth immersion with energy at most $12\pi$ fails to converge to a round sphere under the Willmore flow, then the initial surface can be deformed, without energy increase, to a model surface in $J$; the singularity is unavoidable precisely for initial data in $J$.
  • The Li–Yau inequality is extended at the $12\pi$ threshold: a triple-point-free immersed sphere with invariant outside $\{e_{-3},e_{-1},e_1,e_3\}$ has Willmore energy greater than $12\pi$, so the new invariant $F$ controls the complexity of such surfaces.
  • For any initial surface with $W\le 12\pi$ that admits a triple-point-free regular homotopy to a round sphere, there is also an energy-nonincreasing such homotopy; this is the low-energy version of the statement needed for an optimal sphere eversion.
  • Below $16\pi$, if no Enneper-type blow-up occurs, the same gluing method deforms the surface without energy increase to a round sphere or to an $\alpha$-connected catenoid or trinoid model, giving a partial landscape description up to $16\pi$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the companion invariant were replaced by a quadruple-point-free analogue, the same strategy would likely prove the $16\pi$ conjecture once an explicit sphere eversion through a halfway model is shown to be free of quadruple points; the paper explicitly raises this as Question 1.12.
  • The four-class structure suggests a genuine topological phase change at $12\pi$: numerical experiments could test whether energy-decreasing paths from random low-energy immersions land on round spheres or on the $J$-model family, and whether the infimum $8\pi$ is approached by catenoid-sphere gluings.
  • The cut-and-glue procedure described as a possible Willmore flow with surgery is a testable algorithmic construction: implementing it at a catenoid blow-up and checking whether the resulting flow converges to one or two round spheres would provide a concrete dynamical realization of the classification.
  • For spheres of revolution, Proposition 1.10's lower bound $W(f)>4\pi(|\tau|+1/2)$ is sharp only in the limit; constructing explicit near-equality examples by gluing round spheres with catenoid necks would clarify how close the inequality is to being attained.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies the sublevel sets I(omega) of the Willmore energy on the space of smooth immersions of S^2 into R^3. The main theorem (Theorem 1.2) asserts that every such immersion with energy at most 12 pi admits a regular homotopy of strictly decreasing energy whose terminal surface is either a round sphere or one of the model surfaces in the family J. Combining this with an invariant for triple-point-free immersed spheres imported from the companion preprint [Sei25], the paper derives a four-class description of I(omega) for omega in (8 pi, 12 pi], a classification of unavoidable singularities of the Willmore flow (Proposition 1.6), an extension of the Li-Yau inequality at 12 pi (Proposition 1.9), consequences for sphere eversions, and a partial classification below 16 pi. The constructive upper-bound part is carried out by gluing catenoid pieces to round spheres with explicit energy estimates, using the Lamm-Nguyen blow-up classification [LN15] and the Kuwert-Schaetzle convergence theorem [KS04].

Significance. If the results are correct, the paper gives a complete description of the regular homotopy classes of the Willmore energy landscape below 12 pi and identifies exactly which initial surfaces of energy at most 12 pi lead to unavoidable singularities. This would be a substantial advance in the global analysis of the Willmore functional. The paper's strength is its explicit gluing construction: the energy bookkeeping in Proposition 2.7, the estimates in Lemma 2.4, and the detailed computations in Lemma A.1 give a concrete, checkable pathway from a catenoid blow-up to either a round sphere or a surface in J. The upper-bound direction is internally plausible. However, the lower-bound direction and the classification of unavoidable singularities rest on Theorem D, the invariant values F(+-e)=e_{+-1}, F(j)=e_{+-3}, and a generic-triple-point persistence assertion, all imported from the same-author companion preprint [Sei25] without proof or summary in the present manuscript. In addition, the main theorem as stated is false for round initial data because the required strict energy decrease is impossible when W(f)=4 pi. These points make the paper conditional rather than self-contained.

major comments (3)
  1. [Section 1.3, Theorem 1.2] Theorem 1.2 is false as stated for round initial data. If f is a round sphere, then W(f)=4 pi and every immersed sphere has Willmore energy at least 4 pi, with equality only for round spheres; hence no regular homotopy H with H_0=f can satisfy W(H_t)<W(f) for all t in (0,1]. The proof in Section 2.3 also implicitly assumes omega = W(f0)-W(f_{s0}) > 0, which fails for round f0. The theorem should be restricted to non-round initial data, or the strict inequality should be weakened in a way that is still sufficient for the later applications. Since the round spheres are already the endpoints e and -e in the classification, this is a fixable statement error, but it is load-bearing because Theorem 1.2 is cited as the main theorem.
  2. [Section 1.3, Theorem 1.4; Section 1.4, Proposition 1.6; Section 1.5, Proposition 1.9] The lower-bound half of the paper's central classification is not proved in this manuscript. The assertion that j in J is not regularly homotopic to e or -e inside I(omega), the 'only if' direction of Proposition 1.6, and Proposition 1.9 all depend on Theorem D ([Sei25, Theorem 1.7]), on the invariant values F(+-e)=e_{+-1} and F(j)=e_{+-3} ([Sei25, Example 5.2]), and on the claim that regular homotopies from J to embedded spheres have generic triple points. None of these results is stated in sufficient detail or proved here, and [Sei25] is a same-author companion preprint. If any of these imported results fails, the 'at least four classes' part of Theorem 1.4 collapses, and the characterization of unavoidable singularities is unsupported. The paper should either include proofs or sufficiently detailed statements of the imported results, or the main conclusions should be explicitly conditioned on the companion preprint.
  3. [Appendix A, Corollary 1.3] The proof of Corollary 1.3 uses a 'generic triple-point persistence' assertion that is stronger than the quoted Theorem D. Theorem D only says that every regular homotopy joining some j in J to an embedded sphere has triple points; Corollary 1.3 needs the existence of a time at which the triple point is generic, so that all immersions in a C^infinity neighborhood also have triple points. This genericity statement is asserted to follow from [Sei25], but it is not formulated precisely nor proved. Since the strict inequality W(H_t)>12 pi is needed to separate the J-classes from the round classes in I(12 pi), this missing formulation is a load-bearing gap.
minor comments (5)
  1. [Abstract and Section 1.3] The abstract says a homotopy whose Willmore energy 'does not exceed' that of the initial surface, while Theorem 1.2 states the stronger strict inequality W(H_t)<W(f) for all t in (0,1]. These formulations should be reconciled, especially in light of the round-sphere issue raised above.
  2. [Section 2.2, Definition 2.3] The definition says W_alpha(lambda,R,delta) is 'the Willmore energy of some catenoid sphere' in CatSph_alpha. Since the set CatSph_alpha may contain different parametrizations, it would be clearer to define W_alpha and W_beta as the energy of the specific rotationally symmetric gluing constructed in the preceding paragraph, and to note that this value is independent of the chosen parametrization.
  3. [Section 2.2, proof of Lemma 2.4] In the proof of Lemma 2.4(i), the constant delta_0 is set to 2 pi / C, but the statement requires delta_0 in (0,1/2). The definition should be delta_0 = min(1/2, 2 pi / C) to make the stated range valid.
  4. [Section 2.3, Step 5.2 of Proposition 2.7] The text says that by translation and rotation of F_t one can ensure F_t(p)=0 and T_p F_t = R^2 x {0} for all t, but it does not explicitly justify that these ambient motions can be chosen continuously in t. This is likely true, but a sentence explaining the continuity would remove ambiguity.
  5. [Section 2.4, Remark 2.11] There is a typo: 'The discussion aboe' should read 'The discussion above'.

Circularity Check

3 steps flagged · score 4.0 of 10

The gluing construction behind Theorem 1.2 is self-contained, but the exact four-class classification, the unavoidable-singularity characterization, and Proposition 1.9 import their lower-bound content from the same-author companion preprint [Sei25].

  1. self citation load bearing [Section 1.3, Theorem D and proof of Theorem 1.4; Appendix A, proof of Corollary 1.3.]
    "Theorem D. ([Sei25, Theorem 1.7]). Every regular homotopy joining some j ∈ J to an embedded sphere has triple points. ... By Corollary 1.3, it is also not regularly homotopic to e or −e in I(ω). This proves the statement."

    The lower-bound half of the main classification (I(ω) has at least four classes) is not proved in this manuscript. Corollary 1.3 obtains the needed strict inequality W > 12π by quoting [Sei25]: 'since [Sei25] implies that there must be an immersion Ht with a generic triple point'. Theorem 1.4 then uses Corollary 1.3 to conclude that j is not homotopic to ±e inside I(ω). No proof, statement of assumptions, or independent verification of [Sei25] is provided here. If Theorem D fails, j could be connected to an embedded sphere with energy at most 12π, the four-class count collapses, and Proposition 1.6's 'only if' direction loses its justification.

  2. self citation load bearing [Section 2.3, Lemma 2.8.]
    "Now, for given f ∈ I(ω) ∩ J we apply Proposition 2.7 for δ and λ = R > λ1 to obtain a regular homotopy to some i ∈ Imm(S2,R3) with hemispheres i|Si ∈ CatSphα(λ,λ,δ). Type β cannot occur by Theorem D."

    This step excludes the β-attachment solely by importing Theorem D from the same-author preprint [Sei25]. The exclusion is load-bearing for Lemma 2.8, which is exactly the statement used in Theorem 1.4 to organize all f ∈ I(ω) ∩ J into the two classes J±. The paper's own construction shows that a β-type catenoid sphere can be deformed to a round sphere with energy below 8π; without Theorem D that alternative would merge classes and the four-class theorem would not follow. The paper does not establish Theorem D or its independence from the present results.

1 more flagged steps
  1. self citation load bearing [Section 1.5, proof of Proposition 1.9.]
    "Suppose W(f) ≤ 12π. In [Sei25, Example 5.2] it is computed F(±e) = e±1 and F(j) = e±3 for all j ∈ J. Then, Theorem 1.2 provides a regular homotopy without triple points to round spheres e, −e or some j ∈ J. The invariant is constant along regular homotopies without triple points [Sei25, Theorem 1.3] and thus F(f) ∈ {e−3, e−1, e1, e3}."

    The claimed extension of the Li-Yau inequality at 12π is obtained by combining Theorem 1.2 with the invariant F, whose constancy, computed values, and membership criterion are all quoted from [Sei25]. The present paper neither defines F nor proves the quoted facts. Since Proposition 1.9 is presented as a consequence of the landscape classification, its content reduces to the same companion preprint: if the [Sei25] values or constancy fail, the claimed lower bound W > 12π for all other triple-point-free immersions is unsupported.

full rationale

The constructive core of the paper, Theorem 1.2, is not circular: its proof in Section 2.3 is a genuine gluing argument using Proposition 2.7, Lemma 2.4, the external classification of blow-ups by Lamm-Nguyen [LN15], and Kuwert-Schatzle's convergence theorem [KS04]. No equation in the proof is set equal to its own input by construction, no fitted parameter is relabeled as a prediction, and the energy controls are explicit. The main concern is the lower-bound direction of the classification. The four-class statement, the unavoidable-singularity characterization, Proposition 1.9, and the strict version of Corollary 1.3 all depend essentially on Theorem D and the F-invariant facts imported from [Sei25], a companion preprint by the second author that is cited but neither proved nor summarized here. This is load-bearing self-citation rather than an independent derivation. If [Sei25] is later verified externally, the circularity score should drop to 0-2; absent such verification, the paper's central classification rests in part on a same-author citation.

Assumptions & free parameters 1 free parameters · 9 assumptions · 3 invented entities

The paper's claims rest on seven external theorems plus the companion-preprint invariant. The in-paper contributions (gluing estimates, energy bookkeeping, the four-class topology argument) are genuine; the external inputs are standard results in the field ([KS04], [LN15], [Bry84], [LY82], [MB81], [Lan85], [Kus87]) except for [Sei25], which is a same-author preprint. No data-fitted parameters: all constants (Λ, δ0, λ1, ω, ε, ρ) are existential choices in proofs. No genuinely invented entities: J and the catenoid spheres are explicit constructions with external anchors (Blatt's singular examples for J, explicit parametrizations for catenoid spheres).

free parameters (1)
  • Auxiliary proof constants Λ, δ0, λ1, ρ, ω, ε
    Existential constants chosen inside Lemma 2.4, Prop 2.7, and Thm 1.2 (e.g. ε := 4π − W_β(Λ,Λ,δ) > 0; ω := W(f0) − W(f_{s0}) > 0). They are not fitted to data and the results are uniform for all sufficiently large λ, R, but the energy bookkeeping depends on these choices.
assumptions (9)
  • standard math Theorem A (Kuwert-Schätzle [KS04]): the Willmore flow with W(f0) ≤ 8π exists globally and converges to a round sphere.
    Invoked in Prop 2.7 Step 5.1, the proof of Thm 1.2, and Remark 2.11 to force convergence of the two disk flows after gluing.
  • domain assumption Theorem E (Lamm-Nguyen [LN15, Thm 1.6]): for W(f0) < 16π the blow-up is a catenoid, trinoid, or Enneper surface; for W(f0) ≤ 12π only the catenoid case appears.
    Load-bearing for Thm 1.2 and Prop 2.7: the entire gluing construction starts from a catenoid blow-up. Quoted in Section 2.3, not reproven.
  • domain assumption Theorem D (Seidel [Sei25, Thm 1.7]): every regular homotopy from j ∈ J to an embedded sphere has triple points; invariance and values F(±e) = e_{±1}, F(j) = e_{±3} ([Sei25, Thm 1.3, Ex. 5.2]).
    Companion preprint by the second author; used in Cor 1.3, Thm 1.4, Prop 1.6, and Prop 1.9. Cannot be verified inside this manuscript.
  • standard math Theorem B (Max-Banchoff [MB81]): every sphere eversion has a quadruple point; every regular homotopy from f to −f has a quadruple point.
    Used in the proof of Thm 1.4 to separate j from −j in I(ω) below 16π.
  • standard math Li-Yau inequality [LY82]: an immersion with an n-tuple point has W ≥ 4πn.
    Used throughout: Thm 1.4 lower bound, Prop 1.9, Prop 1.10, and Cor 1.3.
  • standard math Bryant classification [Bry84]: Willmore spheres appear only at levels 4πk; below 16π the only smooth Willmore spheres are the round spheres.
    Used in Cor 1.3 ('no critical points at 12π') and in the discussion around Def 1.5.
  • standard math Langer compactness [Lan85, Thm 2.4] and short-time existence of the Willmore flow [KS12].
    Lemma A.2 is proved from [Lan85]; short-time existence is background for starting flows from glued surfaces.
  • standard math Kusner [Kus87, Thm A] on density at infinity of complete minimal surfaces.
    Used at the end of the proof of Prop 1.10 to upgrade the sphere-of-revolution bound to a strict inequality.
  • standard math Smale [Sma58, Sma59]: immersions of S² in R³ form one regular homotopy class; orientation-preserving diffeomorphisms of S² are homotopic.
    Background for the regular-homotopy-class framework in Section 1.2 and for reparametrization steps in Remark 2.6.
invented entities (3)
  • Family J of model surfaces (Definition 1.1) independent evidence
    purpose: Repository of low-energy surfaces that force unavoidable singularities; the endpoint of Theorem 1.2 and the unavoidable class in Prop 1.6.
    J is explicit via a rotation curve (Figure 1) and membership is checkable via the [Sei25] invariant; [Bla09] shows the rotational model develops singularities, giving external evidence the family is nonempty and relevant.
  • Catenoid spheres CatSph_α and CatSph_β (Definition 2.3) independent evidence
    purpose: Building blocks: a catenoid neck glued to one round sphere; type α generates J, type β can be shrunk to reduce energy below 8π and then flowed to a round sphere.
    Explicitly parametrized (Eqs. 4-5) with computed energy asymptotics (Lemma 2.4, Lemma A.1); no hidden postulates.
  • Triple-point-free invariant F (from [Sei25]) independent evidence
    purpose: Certifies triple-point obstructions and yields the extension of the Li-Yau inequality in Prop 1.9; used to verify J-membership.
    Defined and computed in the companion preprint; this paper reproduces the needed values but not the construction. Independent evidence runs through the companion derivation, flagged as a same-author source.

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Cite this review

Pith. "Pith review of The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow." pith.science (2026). https://pith.science/paper/63CX2TVP

@misc{pith2026250623359,
  author       = {Pith},
  title        = {Pith review of: The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/63CX2TVP}},
  note         = {Machine review of arXiv:2506.23359}
}
abstract

We study the sublevel sets of the Willmore energy on the space of smoothly immersed $ 2 $-spheres in Euclidean $ 3 $-space. We show that the subset of immersions with energy at most $ 12\pi $ consists of four regular homotopy classes. Moreover, we show that in certain regular homotopy classes, all singularities of the Willmore flow are avoidable, that is, the initial surface admits a regular homotopy to a round sphere whose Willmore energy does not exceed that of the initial surface. This yields a classification of initial surfaces with energy at most $ 12\pi $ that lead to unavoidable singularities. As a further consequence, we obtain an extension of the Li-Yau inequality at $ 12\pi $ for a large class of immersed spheres without triple points. To prove these results, we glue together different instances of the Willmore flow and employ an invariant for triple-point-free immersed spheres.

Figures

Figures reproduced from arXiv: 2506.23359 by the authors.

Figure 1
Figure 1. An immersion in the family J and its generating curve. In [MS02, Bla09], surfaces of revolution of this kind serve as initial surfaces that lead to singularities. However, surfaces in J are not necessarily rotationally symmetric and may possess additional self-intersections. In fact, the definition of J is rather implicit, but note that for any given smooth immersion f : S 2 → R 3 , an invariant devised in [Sei25] g… view at source ↗
Figure 2
Figure 2. A sketch of the Willmore energy landscape of immersed 2-spheres in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Some immersions without triple points and their values of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Four ways to attach two round spheres to a catenoid. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The profile curves for the smooth rotationally symmetric catenoid spheres of [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: The gluing region and the functions u1, u2, u˜. Definition 2.3. We define CatSphα(λ, R, δ) (resp. CatSphβ (λ, R, δ)) to be the set of all catenoid spheres fα (resp. fβ) of type α (resp. β) for given λ > 1, R > 1, and δ ∈ (0, 1 − 1/λ) ∩ (0, R − 1) as defined above. We a…
Figure 7
Figure 7. Figure 7: The introduced domains on S 2 and the immersion ft0 . 16 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Simultaneous regular homotopies to a catenoid piece and a round sphere glued [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: The homotopy fλt from the proof of Theorem 1.2 for λt = 8, 5, 2. We conclude the section with some observations about the family J defined in Defini￾tion 1.1. Lemma 2.8. Let ω ∈ (8π, 12π]. There exists j ∈ J with W(j) < ω such that for all f ∈ I(ω) ∩ J, there exists a …
Figure 10
Figure 10. Figure 10: The singularity models of Proposition 2.10. Their images in the lower half￾space are displayed and a reflection at the horizontal plane gives their full images. Proof. We repeat many arguments of the proof of Proposition 2.7 and Theorem 1.2 without going into detail. …
Figure 11
Figure 11. Figure 11: An example for the proof of Proposition 1.10. It remains to show the strict inequality. Assume it is an equality, then f is a crit￾ical point of the Willmore energy. Otherwise, the Willmore flow would yield a surface contradicting the inequality since it preserves sur…

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An Invariant for Triple-Point-Free Immersed Spheres

    math.GT 2025-06 accept novelty 8.0 of 10

    The paper defines an invariant whose image is completely described, showing infinitely many regular homotopy classes of triple-point-free immersed spheres.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.