Riemannian manifolds with curvature bounds

Problem's Description

The following problem appears in : [1]

Proposed by Lashi Bandara: For every $\ell,k > 0$, there exist $C,L,K > 0$ with the following effect. Let $(M,g)$ be a complete Riemannian manifold with injectivity radius $inj(M,g)\geq \ell$ and Ricci curvature $Ric(g)\geq k$. Then there exists a metric $h$ on $M$ with $inj(M,h)\geq L$ and $|Ric(h)|\leq K$ such that $$ \frac{1}{C}\,g\leq h\leq C\,g. $$ In other words: Is it possible to approximate a complete metric with Ricci curvature bounded below using a metric with two-sided Ricci curvature bounds in the $L^\infty$ sense?

Related discussions can be found in [2], [3] where the authros demonstrate that for metrics possessing both upper and lower bounds on Ricci curvature, as well as injectivity radius, this problem can be resolved. Subsequently, in a subsequent paper, the stability of the solution under $L^\infty$ perturbations is established, as described in the original question.

1. ArticleIs an originA list of open problems in differential geometry

   • Frank Morgan,
   • Pierre Pansu

   São Paulo Journal of Mathematical Sciences 15, 305-321, 2021fulltext

2. Article The Kato square root problem on vector bundles with generalised bounded geometry

   • Lashi Bandara,
   • Alan McIntosh

   Journal of Geometric Analysis 26, 428-462, 2016fulltext

3. Article Continuity of solutions to space-varying pointwise linear elliptic equations

   • Lashi Bandara

   Publicacions Matemàtiques 61, 239-258, 2017fulltext