For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright\overline{\mathbb B}^{,N}(1)$ satisfies $\kappa(F)^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with $\kappa(F)\le\sqrt{3/2}$ is diffeomorphic to $S^3$, $\mathbb{R}\mathbb{P}^3$, or $S^2\times S^1$. All three possibilities occur, while $\kappa(F)<\sqrt{3/2}$ forces $X\cong S^3$. These results answer a question of Petrunin and prove a conjecture of Chodosh--Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar--systolic inequality \[ (\min_Y R_g)\text{sys}(g)^2<6\pi^2 \] for every spherical three-space form $Y$ with $|\pi_1(Y)|>2$. Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar--systolic inequality for $\mathbb{R}\mathbb{P}^3$ of Bray--Brendle--Eichmair--Neves.
Let $M$ be a connected smooth $n$-manifold without boundary, where $n\geq3$, and let $\kappa\in\mathbb{R}$, with $\kappa\leq0$ if $M$ is open. We prove that every smooth Riemannian metric $g_0$ with $\mathrm{Scal}_{g_0}\geq\kappa$ is a locally uniform limit of smooth Riemannian metrics $g_i$ with $\mathrm{Scal}_{g_i}=\kappa$ that are locally uniformly bounded in $W^{1,\infty}$. As a corollary, combining this with Gromov's $C^0$-stability theorem, we obtain the perhaps surprising identity \[ \overline{\{g:\mathrm{Scal}_g=\kappa\}}^{\,C^{0,\alpha}_{\mathrm{loc}}}=\{g:\mathrm{Scal}_g\geq\kappa\}, \quad \forall \alpha \in(0,1). \] The restriction $\alpha<1$ is sharp. At $\kappa=0$, this proves and strengthens the Riemannian reverse-Burnett conjecture of Huneau and Luk.
For a smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright \overline{\mathbb{B}}^{N}(1)$, we observe that the sharp systolic inequality forces a sharp lower bound on its maximal normal curvature $\kappa(F)$. In dimensions $m=2,3$, the sharp inequalities of Pu and Bray--Brendle--Eichmair--Neves therefore give $\kappa(F)^2\ge \frac{2m}{m+1}$. Equality holds precisely for the Veronese embedding. This recovers Petrunin's theorem for $\mathbb{R}\mathbb{P}^2$ and, for $\mathbb{R}\mathbb{P}^3$, confirms the first open case of his question for real projective spaces.
Tsz-Kiu Aaron Chow, Jingbo Wan· 0 citations
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