Sharp Schrödinger inequalities for embedded hypersurfaces

Nifa, Anass

doi:10.5281/zenodo.22999024

Preprint · v1

Sharp Schrödinger inequalities for embedded hypersurfaces

Anass Nifa8 September 2026

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Abstract

Let $\Delta=-\operatorname{div}\nabla$ . For a closed connected embedded minimal hypersurface $M^n\subset\mathbb{S}^{n+1}$ , $n\ge2$ , which is not totally geodesic, we prove the quadratic-form inequality $\Delta+|h|^2\ge n$ . Taking the test function equal to one gives Perdomo’s average-curvature inequality, with equality precisely for minimal Clifford hypersurfaces. Let $Q$ be a positive-definite self-adjoint endomorphism of $\mathbb{R}^{n+1}$ and let $F(X)=\frac12\langle QX,X\rangle$ . For a connected complete properly embedded hypersurface without boundary satisfying $H=\langle QX,\nu\rangle$ , where $H=\operatorname{tr} h$ , and which is not a hyperplane, we prove $\Delta_F+|h|^2+\langle Q\nu,\nu\rangle\ge2\lambda_{\min}(Q)$ . Properness implies finite weighted volume. Equality in the corresponding average inequality characterizes generalized cylinders whose spherical factor lies in the least eigenspace of $Q$ . In dimensions $n\ge2$ , the isotropic case gives the sharp Gaussian average inequalities conjectured by Li and Zhao. The proofs construct a positive supersolution from the sum of two tangent-ball curvature functions. The singular parts of their distributional Hessians are retained in the weak inequality. Consequences include compact spectral comparisons, the embedded low-index classification, and a Bernstein theorem in specified eigendirections.

MSC 2020

53C42: Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J50: Spectral problems; spectral geometry; scattering theory on manifolds
53E10: Flows related to mean curvature
35J10: Schrödinger operator, Schrödinger equation

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22999024
All versions: 10.5281/zenodo.22999023
Subjects: Minimal hypersurface; self-shrinker; contact curvature; Schrödinger operator; Gaussian density; rigidity
Language: English

Cite this paper

Anass Nifa. Sharp Schrödinger inequalities for embedded hypersurfaces. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22999024

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References

As listed in the manuscript bibliography.

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C. Mantegazza, G. Mascellani and G. Uraltsev, On the distributional Hessian of the distance function, Pacific J. Math. 270 (2014), 151–166.
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O. M. Perdomo, On the average of the scalar curvature of minimal hypersurfaces of spheres with low stability index, Illinois J. Math. 48 (2004), 559–565.
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Y. Pinchover and K. Tintarev, A ground state alternative for singular Schrödinger operators, J. Funct. Anal. 230 (2006), 65–77; arXiv:math/0411658v1.
arXiv v1
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J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105.
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F. Urbano, Minimal surfaces with low index in the three-dimensional sphere, Proc. Amer. Math. Soc. 108 (1990), 989–992.
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L. Wang, A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata 151 (2011), 297–303.
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C. Wang and P. Wang, On embedded minimal hypersurfaces in Sⁿ⁺¹ with symmetries, preprint, arXiv:2010.16261v1 (2020).
arXiv v1

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B. Andrews, Noncollapsing in mean-convex mean curvature flow, Geom. Topol. 16 (2012), 1413–1418.

[reference-2] [2]
B. Andrews, X. Han, H. Li and Y. Wei, Non-collapsing for hypersurface flows in the sphere and hyperbolic space, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), 331–338.

[reference-3] [3]
B. Andrews and M. Langford, Two-sided non-collapsing curvature flows, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 543–560; arXiv:1310.0717v2.
arXiv v2

[reference-4] [4]
B. Andrews, M. Langford and J. McCoy, Non-collapsing in fully non-linear curvature flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013), 23–32.

[reference-5] [5]
B. Andrews and H. Li, Embedded constant mean curvature tori in the three-sphere, J. Differential Geom. 99 (2015), 169–189; arXiv:1204.5007v3.
arXiv v3

[reference-6] [6]
S. Brendle, Embedded minimal tori in S³ and the Lawson conjecture, Acta Math. 211 (2013), 177–190.

[reference-7] [7]
S. Brendle, A sharp bound for the inscribed radius under mean curvature flow, Invent. Math. 202 (2015), 217–237; arXiv:1309.1459.
arXiv

[reference-8] [8]
S. Brendle and R. Tsiamis, Eigenvalue estimates on shrinkers, to appear in Amer. J. Math.; arXiv:2402.11803v3 (2026).
arXiv v3

[reference-9] [9]
H.-D. Cao and H. Li, A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension, Calc. Var. Partial Differential Equations 46 (2013), 879–889; arXiv:1101.0516.
arXiv

[reference-10] [10]
I. Chavel, Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics 115, Academic Press, Orlando, 1984.

[reference-11] [11]
H. Chen and P. Wang, On the index of minimal hypersurfaces in Sⁿ⁺¹ with λ₁ < n, preprint, arXiv:2405.10843v1 (2024).
arXiv v1

[reference-12] [12]
X. Cheng, T. Mejia and D. Zhou, Stability and compactness for complete f-minimal surfaces, Trans. Amer. Math. Soc. 367 (2015), 4041–4059; arXiv:1210.8076v1.
arXiv v1

[reference-13] [13]
X. Cheng, T. Mejia and D. Zhou, Simons-type equation for f-minimal hypersurfaces and applications, J. Geom. Anal. 25 (2015), 2667–2686; arXiv:1305.2379.
arXiv

[reference-14] [14]
S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, in Functional Analysis and Related Fields, Springer-Verlag, Berlin, 1970, pp. 59–75.

[reference-15] [15]
X. Cheng and D. Zhou, Volume estimate about shrinkers, Proc. Amer. Math. Soc. 141 (2013), 687–696.

[reference-16] [16]
T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow I; generic singularities, Ann. of Math. (2) 175 (2012), 755–833.

[reference-17] [17]
Q. Ding and Y. L. Xin, Volume growth, eigenvalue and compactness for self-shrinkers, Asian J. Math. 17 (2013), 443–456.

[reference-18] [18]
Q. Ding and Y. L. Xin, The rigidity theorems of self shrinkers, Trans. Amer. Math. Soc. 366 (2014), 5067–5085; arXiv:1105.4962.
arXiv

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L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, revised edition, CRC Press, Boca Raton, 2015.

[reference-20] [20]
J. Ge and F. Li, A lower bound for L₂ length of second fundamental form on minimal hypersurfaces, Proc. Amer. Math. Soc. 150 (2022), 2671–2684; arXiv:2103.07747v3.
arXiv v3

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M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer-Verlag, New York, 1976.

[reference-22] [22]
A. Hurtado, V. Palmer and C. Rosales, Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights, Nonlinear Anal. 192 (2020), 111681; arXiv:1805.10055.
arXiv

[reference-23] [23]
D. Impera and M. Rimoldi, Stability properties and topology at infinity of f-minimal hypersurfaces, Geom. Dedicata 178 (2015), 21–47.

[reference-24] [24]
T. Kato, Perturbation Theory for Linear Operators, second edition, Grundlehren der mathematischen Wissenschaften 132, Springer-Verlag, Berlin, 1976.

[reference-25] [25]
H. B. Lawson, Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197.

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F. Li and Y. Zhao, Gaussian-weighted curvature gaps for self-shrinkers, preprint, arXiv:2606.14360v1 (2026).
arXiv v1

[reference-27] [27]
F. Li and Y. Zhao, The rigidity theorems for self-shrinkers in the mean curvature flow, preprint, arXiv:2607.04297v3 (2026).
arXiv v3

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C. Mantegazza, G. Mascellani and G. Uraltsev, On the distributional Hessian of the distance function, Pacific J. Math. 270 (2014), 151–166.

[reference-29] [29]
O. M. Perdomo, On the average of the scalar curvature of minimal hypersurfaces of spheres with low stability index, Illinois J. Math. 48 (2004), 559–565.

[reference-30] [30]
Y. Pinchover and K. Tintarev, A ground state alternative for singular Schrödinger operators, J. Funct. Anal. 230 (2006), 65–77; arXiv:math/0411658v1.
arXiv v1

[reference-31] [31]
J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105.

[reference-32] [32]
F. Urbano, Minimal surfaces with low index in the three-dimensional sphere, Proc. Amer. Math. Soc. 108 (1990), 989–992.

[reference-33] [33]
L. Wang, A Bernstein type theorem for self-similar shrinkers, Geom. Dedicata 151 (2011), 297–303.

[reference-34] [34]
C. Wang and P. Wang, On embedded minimal hypersurfaces in Sⁿ⁺¹ with symmetries, preprint, arXiv:2010.16261v1 (2020).
arXiv v1