Sharp Schrödinger inequalities for embedded hypersurfaces

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.

First published 8 September 2026. Version 2 revised 24 October 2026. Version 3, revised 10 November 2026 (23 pages).