Sharp estimates for the first eigenvalue of Schrödinger operator in the unit sphere

Yin, Jiabin; Qi, Xuerong

doi:10.1090/proc/15860

Sharp estimates for the first eigenvalue of Schrödinger operator in the unit sphere
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by Jiabin Yin and Xuerong Qi PDF

Proc. Amer. Math. Soc. 150 (2022), 3087-3101 Request permission

Abstract:

Let $\mathbf {H}$ be the mean curvature vector of an $n$-dimensional submanifold $M^n$ in a Riemannan manifold. The critical submanifolds of the total mean curvature functional $\mathcal H=\int |\mathbf {H}|^n$ are called $\mathcal H$-submanifolds. In this note, we will prove that compact Legendrian $\mathcal H$-surfaces in the unit sphere $\mathbb {S}^{5}$ are minimal. We also investigate the first eigenvalue of the Schrödinger operator $L=-\Delta -q$ on $M^2$ and $M^3$, where $q$ is some potential function, and obtain a sharp estimate for the first eigenvalue of $L$.

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