Differentiable sphere theorems for Ricci curvature (Q1323441)

The purpose of this note is to prove the following results: Theorem 1.1. Given an integer \(n \geq 2\) and \(\rho_0 > 0\) there exists an \(\varepsilon = \varepsilon(n,\rho_0) > 0\) such that if \(M\) admits a metric \(g\) satisfying \[ \text{Ric}_{(M,g)} \geq (n - 1)g,\quad \text{inj}_{(M,g)} \geq \rho_0,\quad \text{diam}_{(M,g)} \geq \pi - \varepsilon \] then \(M\) is diffeomorphic to \(S^n(1)\) and the metric \(g\) is \(\varepsilon' = \varepsilon'(\varepsilon)\) close in the \(C^\alpha\) topology to the canonical metric \(g_{\text{can}}\) of curvature \(+1\) on \(S^n(1)\). Theorem 1.3. Given \(n \geq 2\) and \(\rho_0 > 0\) there exists an \(\varepsilon = \varepsilon(n,\rho_0) > 0\) such that if \(M\) admits a metric \(g\) satisfying \[ \text{Ric}_{(M,g)} \geq (n -1)g,\quad \text{inj}_{(M,g)} \geq \rho_0, \quad \lambda_1(M,g) \leq n + \varepsilon \] then \(M\) is diffeomorphic to \(S^n(1)\) and the metric \(g\) is \(\varepsilon' = \varepsilon'(\varepsilon)\) close in the \(C^\alpha\) topology to the canonical metric \(g_{\text{can}}\) of curvature \(+1\) on \(S^n(1)\).