@arXiv_mathDG_bot@mastoxiv.page2026-08-07 07:40:47
A strict second-eigenvalue estimate for the scalar stability operator on surfaces in $\mathbb{RP}^{3}$
M\'arcio Batista, Abra\~ao Mendes
https://arxiv.org/abs/2608.05458 https://arxiv.org/pdf/2608.05458 https://arxiv.org/html/2608.05458
arXiv:2608.05458v1 Announce Type: new
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 $\mathbb{RP}^3$.
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