Pith. sign in

REVIEW 2 major objections 5 minor 45 references

Complete two-sided $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read For dimensions 3, 4, and 5, complete two-sided δ-stable minimal hypersurfaces in Euclidean space have Euclidean volume growth once δ exceeds δ0(n), and are hyperplanes once δ exceeds the slightly larger δ1(n).

desk verdict New Euclidean volume growth and rigidity theorems for δ-stable minimal hypersurfaces, with a real but likely repairable gap in the key spectral estimate. read the letter →

arxiv 2507.00342 v1 pith:AHLTWKFO submitted 2025-07-01 math.DG

Qing-Ming Cheng , Guoxin Wei This is my paper
classification math.DG MSC 53C4253A1049Q05
keywords δ-stableminimalhypersurfaceEuclideanvolumegrowthstableBernsteinproblembi-RiccicurvatureGulliver–Lawsonconformalmetricwarpedμ-bubblerigidity
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

This paper proves that in dimensions n=3,4,5, a complete two-sided minimal hypersurface in $\mathbb{R}^{n+1}$ that is $\delta$-stable cannot grow faster than Euclidean space once $\delta$ is above the thresholds $\delta_0(3)=1/3$, $\delta_0(4)=1/2$, and $\delta_0(5)=21/22$: geodesic balls satisfy $\operatorname{vol}\{B_R(p_0)\}\le \Lambda R^n$. With the slightly larger thresholds $\delta_1(3)=3/8$, $\delta_1(4)=2/3$, and $\delta_1(5)=21/22$, the same hypotheses force the hypersurface to be a hyperplane. Since ordinary stability is $\delta=1$ and all the $\delta_1(n)$ are at most $1$, this gives a new proof of the stable Bernstein rigidity in these dimensions. The argument works by transplanting the problem to the Gulliver--Lawson conformal metric and using a warped $\mu$-bubble to produce separating hypersurfaces with a spectral Ricci lower bound, which then yields the volume estimate.

What carries the argument

The machinery has three interacting pieces. The $(\alpha,\beta)$-bi-Ricci curvature $Bi_{(\alpha,\beta)}\mathrm{Ric}_{12}=\beta\sum_i R_{1i1i}+\alpha\sum_{j\ge3}R_{2j2j}$ is a weighted sum of sectional curvatures in two distinguished directions; Lemma 3.2 uses it, together with the Gauss and Codazzi equations, to lower-bound the squared norm $S=|A_M|^2$ by a quadratic form in the eigenvalues plus a term depending on the height function. The Gulliver--Lawson conformal metric $\tilde{g}=r^{-2}g$, where $r=|X|$, converts the $\delta$-stability inequality into a spectral inequality $b\int_N|\tilde\nabla\phi|^2\,dv_{\tilde g}\ge\int_N V\phi^2\,dv_{\tilde g}$ with potential $V\ge\varepsilon(n)-\widetilde{\Lambda}(\alpha,\beta)$. Finally, the warped $\mu$-bubble functional $\mathcal{A}(\Omega)=\int_{\partial^*\Omega}w^q\,dA-\int_\Omega w^qh\,dv$, minimized over domains separating two prescribed sets, manufactures a hypersurface $\Sigma$ with a spectral Ricci curvature lower bound; an area estimate for such $\Sigma$, combined with a bound on the Euclidean distance over the relevant conformal neighborhood, yields the Euclidean volume growth.

What would settle it

Compute $F(n,b,\alpha,\beta,t)$ for the three sets of constants in (3.6) over $t\in[0,1]$ with exact arithmetic or interval arithmetic and compare the minimum to the claimed $\varepsilon(n)$; a single value of $t$ where the expression drops below the claimed $\varepsilon(n)$, or below zero, would invalidate Lemma 3.3 and Theorem 4.2. It would also be worth checking the substitution 'with $b\delta=a$' in the proof of Theorem 3.1 against Lemma 3.3's assumption $a=b\delta_0(n)$, since the lemma's thresholds must align with the theorem's $\delta$ range.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that $\delta$-stability controls the global geometry of complete two-sided minimal hypersurfaces in Euclidean space far better than a naive reading of the stability inequality suggests. Theorem 4.2 says that for $3\le n\le 5$ and $\delta>\delta_0(n)$ every such hypersurface has Euclidean volume growth, $\operatorname{vol}\{B_R(p_0)\}\le \Lambda R^n$; Theorem 1.3 then says that when $\delta>\delta_1(n)$ the only such hypersurface is the hyperplane $\mathbb{R}^n$. The proof is quantitative: it produces explicit positive constants $\varepsilon(n)$ in a lower bound $V\ge \varepsilon(n)-\widetilde{\Lambda}(\alpha,\beta)$ for a Schr\"odinger operator on the conformally changed manifold, and the rigidity comes from combining that bound with the paper's $L^p$ estimates on the second fundamental form.

Load-bearing premise

Everything rests on the positivity asserted in Lemma 3.3: for the dimension-dependent constants chosen in (3.6), the algebraic expression $F(n,b,\alpha,\beta,t)$ stays at least the stated positive value $\varepsilon(n)$ for every $t\in[0,1]$; the paper states the numerical values $\varepsilon(3)=9/11$, $\varepsilon(4)=377/5260$, and $\varepsilon(5)\approx0.014999$ without displaying the computation, and if the inequality fails on the intended $\delta$ ranges the spectral bound $V\ge\varepsilon(n)-\widetilde{\Lambda}(\alpha,\beta)$, and with it the volume-growth argument, collapses.

Editorial extensions

If this is right

  • For each $n\in\{3,4,5\}$ and each $\delta>\delta_1(n)$, a complete two-sided $\delta$-stable minimal hypersurface in $\mathbb{R}^{n+1}$ is the hyperplane $\mathbb{R}^n$.
  • For each $\delta>\delta_0(n)$, the same hypersurface satisfies $\operatorname{vol}\{B_R(p_0)\}\le \Lambda R^n$ for all $R$, so Euclidean volume growth follows without assuming any growth condition in advance.
  • Since ordinary stability is $\delta=1$ and $1\ge \delta_1(n)$ for $n=3,4,5$, the stable Bernstein rigidity for these dimensions is a special case.
  • The $L^p$ estimates on $\sqrt{S}$ from Theorem 1.2, valid for $p=4k+2$ in a specified range of $k$, are what turn Euclidean volume growth into the vanishing of the second fundamental form when $\delta$ is large enough.

Reading between the lines

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

  • The algebraic positivity of Lemma 3.3 is asserted with explicit numbers but no displayed verification; checking $F(n,b,\alpha,\beta,t)$ over $t\in[0,1]$ by exact or interval arithmetic is the cheapest independent test of the whole theorem.
  • The thresholds $\delta_0(n)$ and $\delta_1(n)$ likely reflect the particular choice of $(\alpha,\beta)$ in (3.6) rather than a sharp boundary for $\delta$-stability; optimizing those constants could lower the thresholds, while the still-open $n=6$ case would probably need a different curvature combination.
  • Because $\delta$-stability is the stability condition for anisotropic area functionals, Theorem 1.3 implies corresponding rigidity for the associated complete two-sided anisotropic minimal hypersurfaces in these dimensions, a connection the paper notes but does not develop.
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

2 major / 5 minor

Summary. The paper studies complete two-sided δ-stable minimal hypersurfaces in R^{n+1}. It claims Euclidean volume growth for 3≤n≤5 and δ>δ0(n), with δ0(3)=1/3, δ0(4)=1/2, and δ0(5)=21/22, and then, via an L^p estimate for |A|, that for δ>δ1(n) with δ1(3)=3/8, δ1(4)=2/3, and δ1(5)=21/22, the hypersurface is a hyperplane. The proofs use the Gulliver-Lawson conformal metric, an (α,β)-bi-Ricci curvature Simons-type inequality, and μ-bubble techniques together with an isoperimetric estimate of Antonelli-Xu. The paper also contains two independent estimates, Theorem 1.1 and Theorem 1.2, which may be of independent interest.

Significance. If the results are correct, they give a unified proof of the stable Bernstein theorem in dimensions 3, 4, and 5 as a corollary and provide quantitative δ thresholds. A strength of the paper is that the numerical constants and positivity lower bounds are stated explicitly, which makes the main algebraic claims verifiable. The overall strategy is credible and builds on established tools. However, the proof as written contains a substitution mismatch in the key spectral estimate and one omitted verification in the μ-bubble inequality; both issues are localized and appear repairable.

major comments (2)
  1. [§3, Lemma 3.3 and Theorem 3.1] In the proof of Theorem 3.1, immediately before (3.12), the paper states 'with bδ=a'. This identifies the parameter a of Lemma 3.2 with bδ. But Lemma 3.3 is proved only for a=bδ0(n), and the constants in (3.6) indeed satisfy a=bδ0(n) (for n=3, a=10/11 and b=30/11 with δ0=1/3). Since Theorem 3.1 assumes δ>δ0(n), the substitution a=bδ makes a strictly larger than the value for which the positivity (3.7) is established. The subsequent lower bound (3.13) therefore does not follow from Lemma 3.3 as written. This is load-bearing because (3.13) is the only source of the spectral inequality (4.1) used in the μ-bubble volume-growth proof. The gap is repairable: apply Lemma 3.2 with a=bδ0(n) and use δr^2S ≥ δ0r^2S (since S≥0), and replace the phrase 'with bδ=a' by that argument; alternatively prove the positivity for all a≥bδ0(n). As printed, Theorem 3.1, and consequently Theorems 4.2 and 1.3, are not proved for the asserted δ range.
  2. [§4, equations (4.8)–(4.10)] In passing from (4.8) to (4.9), the term involving H̄² is removed without explanation. The displayed inequality becomes a spectral bound on the hypersurface Σ only if the coefficient of H̄² is nonnegative. The values of L(n) appear to be chosen so that c+1/q-1 = L(n)|1/2-1/q| exactly (for n=3, with c = [4β²-(n-2)α²]/[4β((n-1)β-(n-2)α)], this identity holds), but this computation is not shown. Please state that L(n) is chosen to make the H̄² coefficient vanish, and display the verification. As written, the step is not self-evident and is central to the spectral Ricci bound (4.10) that is later used in the area estimate.
minor comments (5)
  1. [§5, Proof of Theorem 1.2] The Simons inequality is written with a factor 4/(S+ε) in one line and with 1/4 in the next; the correct factor is 1/[4(S+ε)]. The subsequent algebra uses the 1/4 form, so the final coefficient is unaffected, but the displayed formula should be corrected.
  2. [§1, Proof of Corollary 1.1] The displayed inequality '2k > (n-2)^2' is not correct; the computation gives 2k > (n-2)/2, which is still sufficient for p=4k+2>n. Please correct this algebraic claim.
  3. [§3, Lemma 3.3] The statement says 'with δ>δ0(n)' although the inequality (3.1) contains no δ. Clarify that the constants are chosen at δ0 and that the condition δ>δ0 is used later in Theorem 3.1.
  4. [§5, Proof of Theorem 1.1] After setting k=1/R^{(n-2)/n}, the R-powers in the two terms are treated as reciprocals in the bound for C0, but the cancellation with the assumed bound on ∫_{B_R}S^{qn/(n-2)} is not shown explicitly. Please spell out the exponent bookkeeping so that C0^{n/2}S1 is uniformly small in R.
  5. [References and miscellaneous] There are several typographical issues, including 'hyperusfraces' in Section 1 and an incomplete URL in reference [2]; a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the δ-stability derivation and μ-bubble volume-growth proof are self-contained; the δ/a mismatch is a technical gap, not a circular reduction.

full rationale

The paper's claimed hierarchy is: δ-stability leads to a weighted Poincaré inequality in the Gulliver–Lawson conformal metric (Theorem 3.1), which feeds a μ-bubble spectral estimate (Theorem 4.1), which yields Euclidean volume growth (Theorem 4.2), and finally a hyperplane rigidity conclusion via Corollary 1.1 (Theorem 1.3). Each step is an actual derivation rather than a restatement of the hypothesis. Theorem 3.1 is obtained by substituting f = r^{(2-n)/2}φ into the δ-stability inequality; the lower bound V ≥ ε(n) − Λ̃ comes from an algebraic pointwise lemma (Lemma 3.3). The constants δ0(n), ε(n), α, β, b, q are chosen to satisfy explicit algebraic inequalities, not fitted to the conclusion. The μ-bubble estimate follows the standard Chodosh–Li/Zhu second-variation argument, and the volume growth uses the external Antonelli–Xu area bound and isoperimetry. Citations to prior work are independent: Fischer-Colbrie–Schoen, Chodosh–Li, Zhu, Antonelli–Xu, and Hong–Li–Wang are external and proved elsewhere, and no load-bearing step reduces to a self-citation. The apparent mismatch between the proof's phrase "with bδ = a" in (3.12) and Lemma 3.3's condition a = bδ0 is a technical gap, not a circular step, because the intended argument is repairable using δ0-stability and the fact that S ≥ 0; it does not make the theorem equivalent to its assumptions by construction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on hand-picked constants (a, b, α, β) used in a key algebraic inequality, plus standard geometric analysis tools. No new physical entities or ad hoc axioms were introduced. The Antonelli-Xu theorem is a load-bearing external preprint assumption.

free parameters (5)
  • a (per dimension) = 10/11 (n=3), 24/25 (n=4), 10/11 (n=5)
    Chosen by hand in eq. (3.6) to make Lemma 3.3's quadratic inequality positive.
  • b (per dimension) = 30/11 (n=3), 48/25 (n=4), 20/21 (n=5)
    Hand-picked with a, α, β so that a = bδ0(n) and the ε(n) lower bound is positive.
  • α (per dimension) = 18/11 (n=3), 51/50 (n=4), 31/40 (n=5)
    Hand-picked in eq. (3.6) to satisfy the constraints of Lemmas 3.1 and 3.3.
  • β (per dimension) = 3/2 (n=3), 5/4 (n=4), 207/250 (n=5)
    Hand-picked in eq. (3.6); also sets q = b/β in Theorem 4.1.
  • q in Theorem 1.1 = any value with (n-2)/n < q < δ
    Free parameter in the curvature-decay condition; the theorem requires existence, not a specific value.
assumptions (5)
  • standard math Simons identity 1/2 ΔS = |∇A|² − S² and the sharp inequality |∇A|² ≥ (1+2/n)|∇S|²/(4S) for minimal hypersurfaces
    Used throughout Section 5 to derive the L^p estimates in Theorem 1.2.
  • domain assumption Existence and regularity of warped μ-bubbles with prescribed boundary
    Invoked in Section 4 to construct the minimizing domain Ω* and the hypersurface Σ.
  • domain assumption Antonelli-Xu spectral Bishop-Gromov and isoperimetric estimate (arXiv:2405.08918)
    Used in Theorem 4.2 to bound Area(Σ0); this is a recent preprint not independently verified in this paper.
  • standard math Michael-Simon Sobolev inequality for submanifolds
    Used in eq. (5.8) of Theorem 1.1 to pass from Dirichlet energy to L^{2n/(n-2)} bounds.
  • standard math Gulliver-Lawson conformal metric formulas for the bi-Ricci curvature
    Background for Lemma 2.2; derived in the paper from the Gauss and Codazzi equations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Complete two-sided $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$." pith.science (2026). https://pith.science/paper/AHLTWKFO

@misc{pith2026250700342,
  author       = {Pith},
  title        = {Pith review of: Complete two-sided $\delta$-stable minimal hypersurfaces in $\mathbf R^n+1$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHLTWKFO}},
  note         = {Machine review of arXiv:2507.00342}
}
abstract

In this paper, we study complete $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$. We prove that complete two-sided $\delta$-stable minimal hypersurfaces have Euclidean volume growth if $3\leq n\leq 5$ and $\delta>\delta_0(n)$, where $\delta_0(3)=1/3$, $\delta_0(4)=1/2$ and $\delta_0(5)=21/22$. We also give a sufficient condition such that complete two-sided $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$ is the hyperplane. Furthermore, we prove that a complete two-sided $\delta$-stable minimal hypersurface is the hyperplane if $3\leq n\leq 5$ and $\delta>\delta_1(n)$, where $\delta_1(3)=3/8$, $\delta_1(4)=2/3$ and $\delta_1(5)=21/22$.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

45 extracted references · 40 canonical work pages

  1. [1]

    F. J. Jr. Almgren, Some interior regularity theorems for m inimal surfaces and an extension of Bern- stein’s theorem, Ann.of Math., 84(1966), 277-292

    work page 1966

  2. [2]

    Anderson, The compactfication of a minimal submanifold by its Gauss map

    M. Anderson, The compactfication of a minimal submanifold by its Gauss map. http://www.math.sunysb.edu/5anderson/papers.html 2

    work page

  3. [3]

    Antonelli, G.; Xu, K., New spectral Bishop-Gromov and Bon net-Myers theorems and application to isoperimetry, arXiv:2405.08918

    arXiv

  4. [4]

    Extensions of Schoen--Simon--Yau and Schoen--Simon theorems via iteration \`{a} la De Giorgi

    C. Bellettini, Extensions of Schoen-Simon-Yau and Schoe n-Simon theorems via iteration á la De Giorgi, to appear in Invent. Math., arXiv:2310.01340

    work page Pith review arXiv

  5. [5]

    Über ein geometrisches Theorem und sein e Anwendung auf die partiellen Diferential- gleichungen vom elliptischen Typus

    Serge Bernstein. Über ein geometrisches Theorem und sein e Anwendung auf die partiellen Diferential- gleichungen vom elliptischen Typus. Math. Z., 26(1927), 55 1-558

    work page 1927

  6. [6]

    Bombieri, E

    E. Bombieri, E. De Giorgi, E. Giusti, Minimal cones and the Bernstein problem, Invent. Math., 7(1969), 243-268

    work page 1969

  7. [7]

    S. Brendle. The isoperimetric inequality for a minimal su bmanifold in Euclidean space. J. Amer. Math. Soc., 34(2021), 595-603

    work page 2021

  8. [8]

    H.-D. Cao, Y. Shen, S. Zhu, The structure of stable minimal hypersurfaces in Rn+1, Math. Res. Lett. 4 (1997) , 637-644

    work page 1997

Show all 45 references
  1. [9]

    Cheng, On constant mean curvature hypersurfaces with fi nite index, Arch

    X. Cheng, On constant mean curvature hypersurfaces with fi nite index, Arch. Math., 86 (2006), 365- 374

    2006

  2. [10]

    Xu Cheng and Detang Zhou, Manifolds with weighted Poinca rè inequality and uniqueness of minimal hypersurfaces, Comm. Anal. Geom., 17( 2009),139-154

    2009

  3. [11]

    Giovanni Catino, Luciano Mari, Paolo Mastrolia, and Alb erto Roncoroni, Criticality, splitting theorems under spectral Ricci Bounds and the topology of stable minim al hypersurfaces, arXiv: 2412.12631

    arXiv

  4. [12]

    Giovanni Catino, Paolo Mastrolia, and Alberto Roncoron i, Two rigidity results for stable minimal hypersurfaces, Geom. Funct. Anal., 34(2024), 1-18. δ-STABLE MINIMAL HYPERSURF ACES 23

    2024

  5. [13]

    Generalized soap bubbles and th e topology of manifolds with positive scalar curvature, Ann

    Otis Chodosh and Chao Li. Generalized soap bubbles and th e topology of manifolds with positive scalar curvature, Ann. of Math., 199(2024), 707-740

    2024

  6. [14]

    Chodosh, C

    O. Chodosh, C. Li, Stable minimal hypersurfaces in R4, Acta Math., 2024

    2024

  7. [15]

    Stable anisotropic minimal hyp ersurfaces in R4, Forum Math

    Otis Chodosh and Chao Li. Stable anisotropic minimal hyp ersurfaces in R4, Forum Math. Pi, 11:Paper No. e3, 22, 2023

    2023

  8. [16]

    Complete sta ble minimal hypersurfaces in positively curved 4-manifolds, to appear in J

    Otis Chodosh, Chao Li, and Douglas Stryker. Complete sta ble minimal hypersurfaces in positively curved 4-manifolds, to appear in J. Eur. Math. Soc

  9. [17]

    Otis Chodosh, Chao Li, Paul Minter, and Douglas Stryker, Stable minimal hypersurfaces in R5, arXiv:2401.01492

    arXiv

  10. [18]

    Colding and William P

    Tobias H. Colding and William P. Minicozzi, II, The space of embedded minimal surfaces of fixed genus in a 3-manifold. II. Multi-valued graphs in disks, Ann . of Math. (2), 160(2004), 69-92

    2004

  11. [19]

    Scuola Norm

    Ennio De Giorgi, Una estensione del teorema di Bernstein , Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1965), 79-85

    1965

  12. [20]

    do Carmo and C

    M. do Carmo and C. K. Peng, Stable complete minimal surfac es in R3 are planes, Bull. Amer. Math. Soc. (N.S.), 1(1979), 903-906

    1979

  13. [21]

    Math., 82(1985), 121-132

    Doris Fischer-Colbrie, On complete minimal surfaces wi th finite Morse index in three-manifolds, Invent. Math., 82(1985), 121-132

    1985

  14. [22]

    Pure Appl

    Doris Fischer-Colbrie and Richard Schoen, The structur e of complete stable minimal surfaces in 3- manifolds of nonnegative scalar curvature, Comm. Pure Appl . Math., 33(1980), 199-211

    1980

  15. [23]

    Fleming, On the oriented Plateau problem, Ren d

    Wendell H. Fleming, On the oriented Plateau problem, Ren d. Circ. Mat. Palermo, 11 (1962), 69-90

    1962

  16. [24]

    Fu and Z

    H. Fu and Z. Li, The structure of complete manifolds with w eighted Poincare¡¯ inequalities and minimal hypersurfaces, International J. Math. 21 (2010), 1-8

    2010

  17. [25]

    Robert Gulliver and H. Blaine Lawson, The structure of st able minimal hypersurfaces near a singularity, In Geometric measure theory and the calculus of variations ( Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math., pages 213-237. Amer. Math. Soc., Provid ence, RI, 1986

    1984

  18. [26]

    A. J. Hoffman and H.W. Wielandt, The variation of the spect rum of a normal matrix, Duke Math. J., 20(1953), 37-39

    1953

  19. [27]

    H. Hong, H. Li, G. Wang, On δ-stable minimal hypersurfaces in Rn+1, arXiv:2407.03222

    arXiv

  20. [28]

    Han Hong and Zeitan Yan, Rigidity and nonexistence of CMC hypersurfaces in 5-manifolds, arXiv:2405.06867

    arXiv

  21. [29]

    Operator ∆ − aK on surfaces

    Shigeo Kawai. Operator ∆ − aK on surfaces. Hokkaido Math. J., 17(1988), 147-150

    1988

  22. [30]

    Laurent Mazet, Stable minimal hypersurfaces in R6, arXiv:2405.14676

    arXiv

  23. [31]

    Meeks, J

    W. Meeks, J. Pérez and A. Ros, Liouville-type properties for embedded minimal surfaces, Comm. Analy. Geom. 2006, 703-723

    2006

  24. [32]

    J. H. Michael and L. M. Simon. Sobolev and mean-value ineq ualities on generalized submanifolds of Rn, Comm. Pure Appl. Math., 26(1973), 361-379

    1973

  25. [33]

    Amer Math

    Frank Morgan, Regularity of isoperimetric hypersurfac es in Riemannian manifolds, Trans. Amer Math. Soc., 355(2003), 5041-5052

    2003

  26. [34]

    A. V. Pogorelov, On the stability of minimal surfaces, Do kl. Akad. Nauk SSSR, 260(1981), 293-295

    1981

  27. [35]

    Schoen, L

    R. Schoen, L. Simon, and S. T. Yau, Curvature estimates fo r minimal hypersurfaces, Acta Math., 134(1975) 275-288,

    1975

  28. [36]

    Richard Schoen and Leon Simon, Regularity of stable mini mal hypersurfaces. Comm. Pure Appl. Math., 34(1981), 741-797

    1981

  29. [37]

    Ying Shen and Rugang Ye, On the geometry and topology of ma nifolds of positive bi-ricci curvature, arXiv:9708014

  30. [38]

    Y. Shen, S. Zhu, Rigidity of stable minimal hypersurface s, Math. Ann. 309 (1997),107-116

    1997

  31. [39]

    Shen, X.-H

    Y.-B. Shen, X.-H. Zhu, On stable complete minimal hypers urfaces in Rn+1., Amer. J. Math. 120 (1998), 103-116

    1998

  32. [40]

    L.-F. Tam, D. Zhou, Stability properties for the higher d imensional catenoid in Rn+1, Proc. Amer. Math. Soc. 137 (2009), 3451-3461

    2009

  33. [41]

    James Simons, Minimal varieties in riemannian manifold s, Ann. of Math. (2), 88(1968), 62-105

    1968

  34. [42]

    G. Wei, W. Wylie, Comparison geometry for the Bakry-Emer y Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377-405

    2009

  35. [43]

    White, Introduction to minimal surface theory, Geome tric analysis, IAS/Park City

    B. White, Introduction to minimal surface theory, Geome tric analysis, IAS/Park City. Mathematics Series vol. 22. 2015

    2015

  36. [44]

    Kai Xu, Dimension Constraints in Some Problems Involvin g Intermediate Curvature, arXiv:2301.02730

    arXiv

  37. [45]

    δ-STABLE MINIMAL HYPERSURF ACES 24 Qing-Ming Cheng Ma thema tical Science Research Center, Chongqing University of Technology, Chongqing 400054, P

    Jintian Zhu, Width estimate and doubly warped product,T rans.Amer.Math.Soc., 374 (2021), 1497- 1511. δ-STABLE MINIMAL HYPERSURF ACES 24 Qing-Ming Cheng Ma thema tical Science Research Center, Chongqing University of Technology, Chongqing 400054, P. R . China qingmingcheng@y ah...

    2021

Pith tools

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