Sharp Schrödinger inequalities for embedded hypersurfaces

Nifa, Anass

doi:10.5281/zenodo.24120103

Preprint · v3

Sharp Schrödinger inequalities for embedded hypersurfaces

Anass NifaPublished: 8 September 2026Updated: 10 November 2026

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Abstract

Let $n\ge2$ , let $h$ denote the second fundamental form, and put $\Delta=-\operatorname{div}\nabla$ . Every closed connected smooth embedded minimal hypersurface $M^n\subset\mathbb S^{n+1}$ which is not totally geodesic satisfies $\Delta+|h|^2\ge n$ in quadratic-form sense on $H^1(M)$ . In particular, $\int_M |h|^2\,dv_g\ge n\operatorname{Vol}(M)$ , as conjectured by Perdomo, with equality precisely for minimal Clifford hypersurfaces.

Let $Q>0$ be self-adjoint on $\mathbb R^{n+1}$ , and put $a=\lambda_{\min}(Q)$ and $F(x)=\frac12\langle Qx,x\rangle$ . On a hypersurface with unit normal $\nu$ , write $d\mu_F=e^{-F}dv_g$ , $\Delta_F=\Delta+\langle\nabla F,\nabla\rangle$ and $U=|h|^2+\langle Q\nu,\nu\rangle$ . Every connected complete smooth properly embedded hypersurface $M^n\subset\mathbb R^{n+1}$ without boundary, stationary for the weighted area $e^{-F}dv_g$ and not a hyperplane, satisfies $\Delta_F+U\ge2a$ in quadratic-form sense on $C_c^\infty(M)$ in $L^2(M,d\mu_F)$ . Its weighted volume is finite, and $\int_M U\,d\mu_F\ge2a\mu_F(M)$ . Equality in this average inequality holds precisely for the cylinders $\{x+y:x\in E,\ |x|=\sqrt{p/a},\ y\in E^\perp\}$ , where $1\le p\le n$ , $E$ is a linear subspace of $\ker(Q-a\operatorname{Id})$ and $\dim E=p+1$ . For $Q=\frac12\operatorname{Id}$ , this gives the Gaussian average-curvature inequalities conjectured by Li and Zhao.

In the spherical case, Morse index $n+3$ characterizes the minimal Clifford hypersurfaces. Every smooth entire graph stationary for the quadratic density in a $Q$ -eigendirection with eigenvalue $\beta\le2a$ is a hyperplane through the origin.

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.24120103
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, v3 (2026). doi:10.5281/zenodo.24120103

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References

As listed in the manuscript bibliography.

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DOI
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arXiv v1
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arXiv v3

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arXiv v3

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DOI

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arXiv v1