Deforming metrics with curvature and injectivity radius bounded from below (Q1323134)

We begin with the following problem: Let \((M,g)\) be a complete Riemannian \(n\)-manifold satisfying the bounds (1) \(\text{sec} (M)\geq -1\), \(\text{inj}(M)\geq i_ 0\). Does there exist a uniform upper bound \(C(n,K,i_ 0)\) on the sectional curvature? This is in fact a local geometric problem. As remarked in [\textit{M. T. Anderson} and \textit{J. Cheeger}, J. Differ. Geom. 35, No. 2, 265-281 (1992; Zbl 0774.53021)], the answer is no. D. Gromoll and P. Petersen have constructed manifolds satisfying the bounds (1), but without a uniform upper bound on the sectional curvature [cf. Anderson and Cheeger (loc. cit.)]. Our main result is that in an arbitrarily small \(C^ 0\) neighborhood \({\mathcal U}\) of \(g\), there exists a metric whose sectional curvature and injectivity radius can be controlled by the bounds (1) and the size of \({\mathcal U}\). More precisely, we obtain Theorem 1. Let \((M,g)\) be a complete \(n\)-manifold satisfying the bounds (1). There exist positive numbers \(\varepsilon(n)<1\) and \(C(n)>1\) such that for any \(\varepsilon< \varepsilon(n) i_ 0\) there is a complete Riemannian metric \(g_ \varepsilon\) which has the following properties: (a) \((1- {{C(n)}\over {i_ 0}} \varepsilon) g\leq g_ \varepsilon\leq (1+ {{C(n)} \over {i_ 0}} \varepsilon)g\), (b) \(\| \nabla- \nabla^ \varepsilon \|_ g\leq C(n) \varepsilon^{-1}\), (c) \(| \text{sec}(M,g_ \varepsilon)|\leq C(n)^ 2 \varepsilon^{ _ - 2}\), (d) \(\text{inj} (M,g_ \varepsilon)\geq C(n)^{-1} \varepsilon\). Moreover, at every point \(p\in M\), the values of \(g_ \varepsilon\) depends only on \(g|_{B(p,{1\over 2}i_ 0)}\). Here \(\nabla\), \(\nabla^ \varepsilon\) denote the Levi-Civita connections of \(g\) and \(g_ \varepsilon\), respectively.