Pith. sign in

REVIEW 4 major objections 5 minor 18 references

On the geometry of the asymptotic boundary of translators in $\mathbb H^2\times \mathbb R$

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For complete translators in $\mathbb{H}^2\times\mathbb{R}$, every connected component of the vertical asymptotic boundary is a vertical line or a vertical half-line, and every connected component of the horizontal asymptotic boundary is a…

desk verdict A plausible and useful classification of asymptotic boundary components for translators in H2 x R, but the proof as written has a load-bearing gap in Claim 7 that the stress-test note correctly identifies, plus similar compactness gaps elsewhere. read the letter →

arxiv 2505.21083 v1 pith:IYVKUI4R submitted 2025-05-27 math.DG

Giuseppe Pipoli , Joao Paulo dos Santos , Giuseppe Tinaglia This is my paper
classification math.DG MSC 53C4253C44
keywords translatorsmeancurvatureflowasymptoticboundaryhyperbolicproductspacetangencyprinciplesymmetricgeodesiccompactification
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

Complete properly immersed translators in the product space $\mathbb{H}^2\times\mathbb{R}$—surfaces that translate at constant speed under mean curvature flow—have asymptotic boundaries that are geometrically rigid. The paper proves that, under a boundary regularity assumption, every connected component of the vertical asymptotic boundary is a single vertical line $\{p\}\times\mathbb{R}$ or a vertical half-line $\{p\}\times[T,\infty)$, and every connected component of the horizontal asymptotic boundary is a complete geodesic. This matters because translators model the self-similar profiles of singularities in mean curvature flow, so a rigidity theorem at infinity constrains the possible singular behavior of the flow in negatively curved product geometries. The argument shows that any non-straight boundary component would force a known symmetric translator to touch the surface from one side at an interior point, which the tangency principle forbids.

What carries the argument

The load-bearing tools are two. First, the tangency principle for translators (Theorem 3): two translators that touch at an interior point with the same normal and lie locally on one side of each other must coincide locally, and if both are complete they coincide globally. Second, a supply of explicit symmetric translators—the rotational bowl, the parabolic bowl, and the hyperbolic translating catenoids (also called hyperbolic v-grim reapers)—used as barriers. By translating or hyperbolically moving these barriers, the proof manufactures a first point of interior contact with an assumed nonstraight boundary component; the tangency principle turns that contact into a contradiction. The geodesic compactification of $\mathbb{H}^2\times\mathbb{R}$, which splits the boundary into vertical, horizontal, and corner parts, provides the language in which the argument is organized.

What would settle it

Produce a complete properly immersed translator in $\mathbb{H}^2\times\mathbb{R}$, satisfying the paper's boundary regularity, whose vertical asymptotic boundary has a connected component other than a vertical line or vertical half-line—for instance a compact vertical segment—or whose horizontal asymptotic boundary has a non-geodesic component. The theorems assert no such surface exists, so any explicit example or numerical construction would falsify the classification.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1: if $M$ is a complete properly immersed translator in $\mathbb{H}^2\times\mathbb{R}$ whose asymptotic boundary $\partial_\infty M$ is a proper family of disjoint immersed continuous curves and $M\cup\partial_\infty M$ is a continuous surface with boundary, then a vertical boundary component $\gamma$ is either $\{p\}\times[T,\infty)$ with $p\in\partial_\infty\mathbb{H}^2$ and $T\in\mathbb{R}$, or $\{p\}\times\mathbb{R}$; a horizontal boundary component is a complete geodesic. The horizontal statement is obtained from Theorem 2, which says that any isolated properly embedded arc in the horizontal asymptotic boundary of a translator, with the arc and surface forming a continuous surface with boundary, must itself be a geodesic arc. In other words, translators cannot approach infinity along curved or finite boundary configurations; the only allowed profiles at infinity are exactly the ones already exhibited by the symmetric examples.

Load-bearing premise

The classification depends on the asymptotic boundary being a proper family of disjoint immersed continuous curves that, together with the surface, forms a continuous surface with boundary; without this regularity the proof cannot isolate one boundary component to apply the tangency principle.

Editorial extensions

If this is right

  • If the paper is right, every complete properly immersed translator with the stated boundary regularity has vertical asymptotic boundary components that are either whole vertical lines or vertical half-lines; no finite or curved vertical boundary component can occur.
  • Every horizontal asymptotic boundary component is a complete geodesic; in particular, compact horizontal boundary arcs and non-geodesic curves are impossible for such translators.
  • Known symmetric examples are sharp: the parabolic bowl has vertical boundary $\{p\}\times\mathbb{R}$, parabolic translating catenoids realize $\{p\}\times[T,\infty)$, and the hyperbolic catenoid with parameter $r_0=0$ realizes a horizontal boundary $E_0\times\{+\infty\}$, so the classification covers the full range of examples.
  • As corollaries of the proof, no translator can be contained in a vertical cylinder over a compact domain or in a horocylinder, and a vertical plane $\Gamma\times\mathbb{R}$ is the unique translator in any region bounded by two equidistant curves that contains $\Gamma$.

Reading between the lines

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

  • The boundary regularity hypothesis—disjoint immersed continuous curves plus a continuous surface-with-boundary structure—is strong; a natural test is whether it follows from the translator equation or can be relaxed, since without it the tangency-based isolation of a single boundary component may fail.
  • The same barrier-and-tangency strategy should extend to translators in $\mathbb{H}^n\times\mathbb{R}$ or to other translation directions in $\mathbb{H}^2\times\mathbb{R}$, wherever enough symmetric translator barriers are available; the classification of boundary components would then follow the same vertical-line/geodesic pattern.
  • Because the horizontal boundary must be geodesic, the theorem suggests that entire-graph translators in $\mathbb{H}^2\times\mathbb{R}$ with asymptotic boundary data that is not a union of geodesics cannot exist; this is a non-existence statement of the Jenkins-Serrin type that the authors do not spell out.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies complete properly immersed translators of the mean curvature flow in H2 × R moving in the vertical direction. Under regularity assumptions on the asymptotic boundary (a proper family of disjoint immersed continuous curves such that M ∪ ∂∞M is a continuous surface with boundary), it classifies the possible components of the vertical and horizontal asymptotic boundaries: a vertical component must be a vertical line or a vertical half-line {p} × [T, ∞), and a horizontal component must be a complete geodesic. The proofs use symmetric translators (parabolic, rotational, and hyperbolic bowls/catenoids) as barriers, together with the tangency principle. The horizontal boundary statement is deduced from a more technical theorem (Theorem 2) via a lemma asserting unboundedness of certain pieces of the horizontal asymptotic boundary.

Significance. If the proof can be completed, the result is a natural and valuable rigidity theorem for translators in H2 × R, extending earlier asymptotic boundary results for minimal and constant mean curvature surfaces. A notable strength is that the paper uses concrete barrier translators whose asymptotic boundaries are known from prior work (including de Lima–Pipoli), so the argument is not circular: the barriers are examples, not the classification being proved. The conclusions are sharp, matching the known parabolic bowl and parabolic translating catenoid. However, the proof as written contains several load-bearing gaps in the barrier arguments: the key limiting step in Claim 7 is not justified, and the first-contact arguments in Claims 8, 9, Lemma 11, and Theorem 2 rely on unproved compactness and intersection assertions. These gaps are likely fixable with additional detail, but they are substantial enough that the central claim is not yet rigorously established in the submitted manuscript.

major comments (4)
  1. [Section 5, Claim 7] The proof of Claim 7 contains an unsupported implication. From 'Mε ∩ [H2 × {t + δ}] ⊂ eS(p, ε)' the text concludes 'by definition' that (p, t+δ) is an asymptotic boundary point of M. This is not a definition and is not proved. What is needed is the observation that the sets eS(p, ε) form a fundamental system of neighbourhoods of p in the closed disk, so that any sequence of points in Mε at height t+δ with ε→0 has horizontal coordinates converging to p. Without this limiting argument the contradiction with the isolation of (p,t) on the vertical line is not established. In addition, the reduction 'without loss of generality' to the case that the violation of the slab always occurs at height t+δ (rather than t−δ) is not justified and would need a separate symmetric argument. Since Claim 7 supplies the height bounds used in Claims 8 and 9, this gap is load-bearing.
  2. [Section 5, Claims 8 and 9] The first-contact arguments in Claims 8 and 9 are not fully justified. In Claim 8, the proof asserts that some translated copy of C_T 'must obtain its first point of contact with Mε at an interior point' without proving that (i) some translate actually intersects Mε, (ii) the first-contact parameter is attained rather than approached only at infinity, and (iii) contact cannot occur on ∂Mε. In Claim 9, the assertion 'C^0 ∩ T_{n_μ}(Mα) ≠ ∅' is unsupported: T_{n_μ}(Mα) contains the point (0, t_n) with t_n ∈ (−δ, δ), while C^0 is the graph of the rotational bowl with lowest point (0, −2δ), so (0, t_n) is not on C^0 and a separate argument is needed to show the two surfaces intersect. These details are necessary before the tangency principle (Theorem 3) can be applied, and they are not merely cosmetic.
  3. [Section 6, Lemma 11] The first bullet in the positive case of Lemma 11, 'for any ν ≥ ρ/10 + ν, M^σ_τ ∩ [Eν × R] = ∅', does not follow from the boundedness of ∆ alone. The absence of points of ∂+∞M over Eν for ν > ν does not rule out finite-height points of M over Eν whose heights tend to infinity while their horizontal coordinates escape along Eν. A compactness or limiting argument using properness and the asymptotic boundary hypothesis is required. Similarly, the conclusion that the family W^μ 'must intersect M^σ_τ a first time at an interior point' needs a rigorous proof excluding contact at ∂M^σ_τ or at infinity. These gaps affect the validity of Lemma 11, which is the main tool for the horizontal boundary classification.
  4. [Section 6, Proof of Theorem 2] The proof of Theorem 2 is too sketchy to verify. The passage 'after applying a sequence of downward translations, and possibly replacing Γ with a nearby geodesic, we can assume that β is the only asymptotic boundary component of a connected component of Σ ∩ H2 × [0, +∞)...' is vague: the symbol Σ is not defined, the limiting procedure is not described, and it is not shown that the resulting component satisfies the hypotheses of Lemma 11, in particular ∂v∞Σ = ∅ and ∂Σ ⊂ [Γ × R] ∪ [H2 × {0}]. Since Theorem 2 is the basis for the horizontal boundary component classification in Theorem 1 and Corollary 14, this is a load-bearing gap.
minor comments (5)
  1. [Section 6, Proof of Theorem 2] The expression 'H2 × 0' should read 'H2 × {0}', and the symbol Σ is used before being introduced.
  2. [Section 5] The symbol δ is used with two different meanings in Claim 7 and Claim 8; this should be clarified with distinct notation.
  3. [Section 5, Claim 9] The symbol C denotes both the full rotational bowl and its truncation C ∩ [H2 × [−2δ, 2δ]]; use different names for these two objects.
  4. [Section 6, Lemma 11] The parameter ν is overloaded: it is used for the distance of the equidistant curves and also in the condition 'for any ν > ν'. Rename one of these to avoid confusion.
  5. [References] In reference [14], 'pp. 7780-7812' is missing a space after 'pp.'; also verify that reference [6] belongs in the H3 list rather than the H2 × R list in the introduction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the symmetric translators used as barriers are concrete examples, not the classification being proved; the only flagged issue is a proof gap in Claim 7, not circular reasoning.

full rationale

This paper does not exhibit circularity. The main theorems use external inputs: the tangency principle (Theorem 3, from [2,15]) and explicit symmetric translators (parabolic, rotational, and hyperbolic bowls and catenoids) whose asymptotic boundaries are computed in prior work [5,8,14]. These barriers are concrete examples with known boundary geometry; invoking them to force an arbitrary boundary component to be a vertical line or half-line, or a geodesic, is a standard barrier argument, not a reduction of the theorem to itself. The fact that one of the barrier sources, [8], involves an author of the present paper does not make the step circular, because the same family is also documented in [5,14] and because the cited result is an explicit construction, not a classification theorem that already contains Theorem 1. The only substantive worry, noted in the reader's take, is Claim 7's assertion that nonempty intersection in eS(p, epsilon) with H2 x {t+delta} 'by definition' gives (p,t+delta) in the asymptotic boundary; this is a compactness or properness gap rather than a definitional equivalence or a fitted parameter disguised as a prediction. Such a gap may affect the correctness of Theorem 5, but it is not a circular step. The classification therefore does not reduce to its assumptions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities are introduced. The paper's conclusions are conditional on the regularity assumptions on the asymptotic boundary and on the availability of barrier translators from prior work.

assumptions (4)
  • standard math Geodesic compactification of H2 × R and the decomposition of its asymptotic boundary into vertical, horizontal, and corner parts (Section 2).
    Background recalled from Sa Earp-Toubiana [18]; no new content.
  • domain assumption Tangency principle for translators in H2 × R (Theorem 3), taken from Bueno [2, Theorem 2.2].
    Used repeatedly to force contradictions from first contact of the given surface with barrier translators.
  • domain assumption Existence, regularity, and asymptotic boundary behavior of symmetric translators: parabolic bowl/catenoids, rotational bowl/catenoids, hyperbolic bowl and family Ω of hyperbolic translating catenoids (Section 4).
    These barriers are cited from [2,5,8,14]; their boundary shapes are asserted and relied on in the proofs of Theorem 5 and Lemma 11.
  • domain assumption The set of hypotheses on ∂∞M (proper family of disjoint immersed continuous curves; M ∪ ∂∞M a continuous surface with boundary).
    These assumptions delimit the theorem; the proof uses them to ensure components can be separated and that barrier contact can be localized.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the geometry of the asymptotic boundary of translators in $\mathbb H^2\times \mathbb R$." pith.science (2026). https://pith.science/paper/IYVKUI4R

@misc{pith2026250521083,
  author       = {Pith},
  title        = {Pith review of: On the geometry of the asymptotic boundary of translators in $\mathbb H^2\times \mathbb R$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IYVKUI4R}},
  note         = {Machine review of arXiv:2505.21083}
}
abstract

In this work, we study complete properly immersed translators in the product space $\mathbb H^2\times\mathbb R$, focusing on their asymptotic behavior at infinity. We classify the asymptotic boundary components of these translators under suitable continuity assumptions. Specifically, we prove that if a boundary component lies in the vertical asymptotic boundary, it is of the form $\{p\}\times [T,\infty)$ or $\{p\}\times \mathbb R$, while if it lies in the horizontal asymptotic boundary, it is a complete geodesic. Our approach is inspired by earlier work on minimal and constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$, with a key ingredient being the use of symmetric translators as barriers.

Figures

Figures reproduced from arXiv: 2505.21083 by the authors.

Figure 1
Figure 1. Parabolic translating catenoid (left); Parabolic bowl (right) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Rotational translating catenoid (left); Rotational bowl (right) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Using the nomenclature of [8]: hyperbolic translating catenoid with [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: An element of Ω, r0 > 0 (left) and W ∩[c × R], W ∈ Ω, where c is the geodesic in H2 whose end points are (1, 0) and (−1, 0) and W is determined by the geodesic whose end points are (0, 1) and (0, −1), tangent to the horizontal plane H2 × {0} and to the vertical plane E…
Figure 5
Figure 5. Figure 5: S(p, ε), as in Definition 4 Given (p, t) ∈ ∂∞H2 × R, we denote by lp the vertical line {p} × R ⊂ ∂∞H2 × R. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: A vertical section of H2 × (t − δ, t + δ) describing Claim 7. Proof. Arguing by contradiction, assume that for any ε < ε1, Mε is not contained in H2 × (t − δ, t + δ). Without loss of generality, let’s assume that for any ε < ε1 we have that Mε ∩ [H 2 × {t + δ}] ̸= ∅. S…
Figure 7
Figure 7. Figure 7: On the left, Case 1, on the right Case 2. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: The constructions in the proof of Claim 8. On the left, front view; on the right, top [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 5
Figure 5. Figure 5: Let c2 ⊂ H2 × {0} be the complete geodesic with asymptotic boundary points q1 and q2, where q2 is the midpoint of β. Consider hyperbolic translations along c2. Recall that by Claim 7, ∂Mε ⊂ S(p, ε) ∩ [H 2 × [−δ, δ]] and observe that all translated copies of CT are cont…
Figure 9
Figure 9. Figure 9: Rotational bowl as barrier We next rule out Case 2. Claim 9. Case 2. does not hold. Proof. In this case, the vertical projection of γε on ∂∞H2 × {0} is an arc β with p as one of its interior points. See [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: The construction in the proof of Claim 9. [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: The stretching argument and the rotational bowl [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: The points (pn, tn) are “inside” Wε 2 By continuity, there exists ε1 < ε 2 such that (pn, tn) ∈ V ε1 in . However since (pn, tn) converges to (p, 0) ∈/ V ε1 in , then there are points (pn, tn) ∈/ V ε1 in for n large, giving that Wε1 ∩ Mfε ̸= ∅. Let Wτ ε1 be Wε1 vertic…
Figure 13
Figure 13. Figure 13: The constructions in the proof of Lemma 11. [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: A graphical representation of Proposition 12: in blue the empty regions. [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]
Figure 15
Figure 15. Figure 15: A graphical representation of Proposition 13. In blue two examples of domains [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]
Figure 16
Figure 16. Figure 16: The construction in the proof of Theorem 2. [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

18 extracted references · 16 canonical work pages

  1. [1]

    Extrinsic geomet- ric flows

    Ben Andrews, Bennett Chow, Christine Guenther, and Mat Langford. Extrinsic geomet- ric flows. Vol. 206. Graduate Studies in Mathematics. American Mathematical Soci- ety, Providence, RI, 2020, pp. xxviii+759. isbn: 978-1-4704-5596-5. doi:10.1090/gsm/206. url:https://doi.org/10.1090/gsm/206

    work page doi:10.1090/gsm/206 2020

  2. [2]

    Translating solitons of the mean curvature flow in the space H2 × R

    Antonio Bueno. Translating solitons of the mean curvature flow in the space H2 × R. J. Geom. 109.3 (2018), Paper No. 42, 26. issn: 0047-2468. doi: 10.1007/s00022-018-0447-x. url: https://doi.org/10.1007/s00022-018-0447-x

    work page doi:10.1007/s00022-018-0447-x 2018

  3. [3]

    Horo-shrinkers in the hyperbolic space

    Antonio Bueno and Rafael L´ opez. Horo-shrinkers in the hyperbolic space. (2024). arXiv: 2402.05527 [math.DG]. url: https://arxiv.org/abs/2402.05527

    work page Pith review arXiv 2024

  4. [4]

    A new family of translating solitons in hyperbolic space

    Antonio Bueno and Rafael L´ opez. A new family of translating solitons in hyperbolic space. (2024). arXiv: 2402.05533 [math.DG]. url: https://arxiv.org/abs/2402.05533

    arXiv 2024

  5. [5]

    Antonio Bueno and Rafael L´ opez.The class of grim reapers in H2×R. J. Math. Anal. Appl. 541.2 (2025), Paper No. 128730, 23. issn: 0022-247X. doi: 10.1016/j.jmaa.2024.128730. url: https://doi.org/10.1016/j.jmaa.2024.128730

    work page arXiv 2025

  6. [6]

    Ronaldo Freire de Lima and Alvaro Kr¨ uger Ramos and Jo˜ ao Paulo dos Santos.Solitons to Mean Curvature Flow in the hyperbolic 3-space. (2023). arXiv: 2307.14136 [math.DG]. url: https://arxiv.org/abs/2307.14136

    arXiv 2023

  7. [7]

    Rotators-translators to mean curvature flow in H2 × R

    Ronaldo Freire de Lima and Alvaro Kr¨ uger Ramos and Jo˜ ao Paulo dos Santos. Rotators-translators to mean curvature flow in H2 × R. Arch. Math. (Basel) 124.3 (2025), pp. 343–353. issn: 0003-889X,14208938. doi: 10.1007/s00013-024-02076-9. url: https://doi.org/10.1007/s00013-02402076-9

    work page doi:10.1007/s00013-024-02076-9 2025

  8. [8]

    Translators to Higher Order Mean Curvature Flows in Rn × R and Hn × R

    Ronaldo Freire de Lima and Giuseppe Pipoli. Translators to Higher Order Mean Curvature Flows in Rn × R and Hn × R. J. Geom. Anal. 35, 92 (2025). url: https://doi.org/10.1007/s12220-025-01925-5

    work page doi:10.1007/s12220-025-01925-5 2025

Show all 18 references
  1. [9]

    Totally umbilical hypersurfaces of product spaces

    Ronaldo Freire de Lima and Jo˜ ao Paulo dos Santos. Totally umbilical hypersurfaces of product spaces. Manuscripta Math. 169.3-4 (2022), pp. 649–666. issn: 0025-2611. doi: 10.1007/s00229-021-01339-x. url: https://doi.org/10.1007/s00229-021-01339-x. 15

    2022 doi

  2. [10]

    Gama, Esko Heinonen, Jorge H

    Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira, and Francisco Mart ´ ın. The Jenkins-Serrin problem for translating horizontal graphs in M × R. Rev. Mat. Iberoam. 37.3 (2021), pp. 1083–1114. issn: 0213-2230. doi: 10.4171/rmi/1221. url: https://doi.org/10.4171/rmi/1221

    2021 doi

  3. [11]

    Convexity estimates for mean curvature flow and singularities of mean convex surfaces

    Gerhard Huisken and Carlo Sinestrari. Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Mathematica 183.1 (1999), pp. 45–70. doi: 10.1007/BF02392946. url: https://doi.org/10.1007/BF02392946

    1999 doi

  4. [12]

    Soliton solutions for the mean curvature flow

    Norbert Hungerb¨ uhler and Knut Smoczyk. Soliton solutions for the mean curvature flow. Differential Integral Equations 13.10-12 (2000), pp. 1321–1345. issn: 0893-4983

    2000

  5. [13]

    Kloeckner and Rafe Mazzeo

    Beno ˆ ıt R. Kloeckner and Rafe Mazzeo. On the asymptotic behavior of minimal surfaces in H2 × R. Indiana Univ. Math. J. 66.2 (2017), pp. 631–658. issn: 0022-2518,1943-5258. doi: 10.1512/iumj.2017.66.6014. url: https://doi.org/10.1512/iumj.2017.66.6014

    2017 doi

  6. [14]

    Jorge H. S. de Lira and Francisco Mart ´ ın. Translating solitons in Riemannian prod- ucts. J. Differential Equations 266.12 (2019), pp. 7780-7812. issn: 0022-0396. doi: 10.1016/j.jde.2018.12.015. url:https://doi.org/10.1016/j.jde.2018.12.015

    2019 doi

  7. [15]

    On the topology of translating solitons of the mean curvature flow

    Francisco Mart ´ ın and Andreas Savas-Halilaj and Knut Smoczyk. On the topology of translating solitons of the mean curvature flow. Calc. Var. Partial Differential Equa- tions 54.3 (2015), pp. 2853–2882. issn: 0944-2669. doi: 10.1007/s00526-015- 0886- 2. url: https://doi.org/1...

    2015 doi

  8. [16]

    Uniqueness of H-surfaces in H2 × R, |H| ≤1/2, with boundary one or two parallel horizontal circles

    Barbara Nelli and Ricardo Sa Earp and Walcy Santos and Eric Toubiana. Uniqueness of H-surfaces in H2 × R, |H| ≤1/2, with boundary one or two parallel horizontal circles. Ann. Global Anal. Geom. 33.4 (2008), pp. 307–321

    2008

  9. [17]

    An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H2 × R

    Ricardo Sa Earp and Eric Toubiana. An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H2 × R. Math. Ann. 342.2 (2008), pp. 309–331

    2008

  10. [18]

    Concentration of total curvature of minimal sur- faces in H2 × R

    Ricardo Sa Earp and Eric Toubiana. Concentration of total curvature of minimal sur- faces in H2 × R. Math. Ann. 369.3-4 (2017), pp. 1599–1621. issn: 0025-5831,1432-1807. doi:10.1007/s00208-016-1508-9. url:https://doi.org/10.1007/s00208-016-1508-9. 16

    2017 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.