Preprint
Inserted: 23 oct 2026
Last Updated: 23 oct 2026
Pages: 21
Year: 2026
Doi: 10.5281/zenodo.23605781
Notes:
Version 2, updated 24 October 2026. First published 8 September 2026. MSC 2020: 53C42, 58J50, 53E10, 35J10.
Links: Zenodo — version 2
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 $\Delta+\lvert h\rvert^2\ge n$ in quadratic-form sense on $H^1(M)$. In particular, $\int_M \lvert h\rvert^2\,dv_g\ge n\operatorname{Vol}(M)$, 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 $U=\lvert h\rvert^2+\langle Q\nu,\nu\rangle$. 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 $\{x+y:x\in E,\ \lvert x\rvert=\sqrt{\frac{p}{a}},\ y\in E^\perp\}$, 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.
Keywords: rigidity, Minimal hypersurface, self-shrinker, contact curvature, Schrödinger operator, Gaussian density
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