An attempt to solve question number 2 from here https://www.imo.universite-paris-saclay.fr/~pierre.pansu/problems_MTDG.pdf posed by Pierre pansu https://www.imo.universite-paris-saclay.fr/~pierre.pansu/
Hello, MathOverflow community,
I have been working on the L-infinity approximation of complete Riemannian manifolds with Ricci curvature bounded below. I would like to request a review of my attempt to construct a metric with two-sided Ricci curvature bounds that approximates the given manifold. Here is the outline of my approach:
Given a complete Riemannian manifold $(M, g)$ with injectivity radius $\operatorname{inj}(M, g) \geq \ell$ and Ricci curvature $\operatorname{Ric}(g) \geq k$, I first smooth the metric $g$ using a compactly supported smooth function $\varphi: M \to [0, \infty)$ and a small constant $\varepsilon > 0$. I define a new metric $g_\varepsilon$ on $M$ by $g_\varepsilon = g + \varepsilon \varphi g$.
Next, I approximate the smoothed metric $g_\varepsilon$ by a metric $h$ with two-sided Ricci curvature bounds. I define a new metric $h$ on $M$ by $h = g_\varepsilon + C_1 \chi g_\varepsilon$, where $\chi: M \to [0, \infty)$ is a smooth function chosen to control the Ricci curvature of $h$.
I show that for appropriately chosen $\chi$, the metric $h$ satisfies $|\operatorname{Ric}(h)| \leq K$ for some constant $K > 0$. This yields a metric $h$ that approximates the original manifold with the desired properties.
You can find the full LaTeX document containing the details of my approach here https://drive.google.com/file/d/1YHNYHyy9IHuZZ46Y_Gvv7cm84IrSwflg/view?usp=sharing. I would appreciate it if you could review my attempt and provide any feedback, corrections, or suggestions for improvement. In particular, I would like to know if there are any issues with the proof or if there are any more efficient methods for achieving the L-infinity approximation.
Thank you in advance for your valuable input!