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

Abstract

Let $\phi :{\Sigma }^2\looparrowright {\mathbb{RP}}^3$ be a closed immersed surface, with no orientability or, equivalently, two-sidedness assumptions. We establish the sharp quantitative estimate $${\lambda }_2(\Delta +|\sigma {|}^2+2)\le 2-\frac{2}{\mathrm{Area}(\Sigma )}{\int }_{\Sigma }{H}^{2}d\Sigma +\frac{4\pi \chi (\Sigma )}{\mathrm{Area}(\Sigma )}.$$ Our main contribution is the analysis of the borderline case, which rules out equality when $\chi (\Sigma )\le 0$. More precisely, $${\lambda }_2(\Delta +|\sigma {|}^2+2)<2\text{whenever}\chi (\Sigma )\le 0.$$ 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 ${\mathbb{RP}}^3$.

16 pages. Comments welcome