A strict second-eigenvalue estimate for the scalar stability operator on surfaces in $\mathbb{RP}^{3}$
Year: 2026
Type: preprint
Abstract: Let $φ:Σ^2\looparrowright\mathbb{RP}^3$ be a closed immersed surface, with no orientability or, equivalently, two-sidedness assumptions. We establish the sharp quantitative estimate $$ λ_2(Δ+|σ|^2+2)\leq2-\frac{2}{\operatorname{Area}(Σ)}\int_ΣH^2dΣ+\frac{4πχ(Σ)}{\operatorname{Area}(Σ)}. $$ Our main contribution is the analysis of the borderline case, which rules out equality when $χ(Σ)\leq0$. More precisely, $$ λ_2(Δ+|σ|^2+2)<2\quad\text{whenever}\quadχ(Σ)\leq0. $$ 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 $\mathbb{RP}^3$.
Source: arXiv (Cornell University)
Authors: Batista, Márcio, Mendes, Abraão
Institution: Universidade Federal de Alagoas
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3405523112, G4587991357
OpenAlex ID: W7201867537
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