Uniform curvature-radius lower bound on complete Ricci shrinkers

Establish, for every complete gradient Ricci shrinker and every fixed constant C>0, the uniform estimate \(\sqrt{\lambda}\,\inf_{\{V_f<C\lambda\}}\rho\to\infty\) as \(\lambda\to\infty\), where \(V_f=\frac14(f+R-n)\) is the Schrödinger potential and \(\rho\) is the regularity radius controlling curvature, its first two derivatives, and injectivity radius.

Background

For a normalized complete gradient Ricci shrinker $(M,g,f)$, the weighted Laplacian is unitarily equivalent to the Schrödinger operator $H_f=\Delta+V_f$, with $V_f=\frac14(f+R-n)$. A general criterion for the classical Weyl law would require the regularity scale $\rho$ to satisfy a uniform lower bound throughout the classically allowed region $\{V_f<C\lambda\}$. After rescaling by the spectral scale $\sqrt{\lambda}$, this is the condition $\sqrt{\lambda}\inf_{\{V_f<C\lambda\}}\rho\to\infty$.