Diameter pinching in almost positive Ricci curvature (Q1012467)

The author extends previous stability results of \textit{S. Ilias} [Ann. Inst. Fourier 43, No. 3, 843--863 (1993; Zbl 0783.53024)] to establish a diameter sphere theorem and its corresponding \(\lambda_1\) sphere theorem under \(L^p\) control of the curvature: Theorem. Let \(n\geq2\) be an integer, \(A>0\), and \(p>n\) be some reals. There exist positive constants \(C_i=C_i(p,n,A)\) such that any complete manifold which satisfies \[ \int_M({\underline{Ric}}-(n-1))^p_-\leq C_i\text{Vol}(M),\quad \int_M\bar\sigma_+^p<A\text{Vol}(M) \] and which in addition satisfies either of the following two conditions is homeomorphic to \(S^n\): (a) \(\text{Diam}(M)\geq\pi(1-C_1)\). (b) \(\lambda_1(M)\leq n(1+C_2)\).