A strict second-eigenvalue estimate for the scalar stability operator on surfaces in $\mathbb{RP}^{3}$

Abstract

Let $\varphi:\Sigma^2\looparrowright\mathbb{RP}^3$ be a closed immersed surface, with no orientability or, equivalently, two-sidedness assumptions. We establish the sharp quantitative estimate $$ \lambda_2(\Delta+|\sigma|^2+2)\leq2-\frac{2}{\operatorname{Area}(\Sigma)}\int_\Sigma H^2d\Sigma+\frac{4\pi\chi(\Sigma)}{\operatorname{Area}(\Sigma)}. $$ Our main contribution is the analysis of the borderline case, which rules out equality when $\chi(\Sigma)\leq0$. More precisely, $$ \lambda_2(\Delta+|\sigma|^2+2)<2\quad\text{whenever}\quad\chi(\Sigma)\leq0. $$ The proof combines the canonical Veronese embedding of $\mathbb{RP}^3$ into $\mathbb{S}^8$, the conformal test function method, an Obata-type rigidity argument, and the classification of closed flat minimal surfaces in $\mathbb{RP}^3$.