Regularity of manifolds with integral scalar curvature bound and entropy lower bound (Q6607994)

Let \(\left( M,g\right) \) be an \(n\)-dimensional Riemannian manifold. In their recent paper [Geom. Topol. 27, No. 1, 227--350 (2023; Zbl 1522.53029)], \textit{M.-C. Lee} et al. introduced the \(d_{p}\)-distance for \(p>n\):\N\[\Nd_{p}\left( x,y\right) =d_{p,g}\left( x,y\right) =\sup\left\{ \left\vert f\left( x\right) -f\left( y\right) \right\vert :\int_{M}\left\vert \nabla f\right\vert ^{p}d\operatorname{Vol}_{g}\leq1\right\} \text{.}\N\]\NTheir main result establishes the \(\varepsilon\)-regularity relative to the \(d_{p}\)-distance for a complete \(n\)-dimensional Riemannian manifold \(\left( M,g\right) \) with almost non-negative scalar curvature and Perelman entropy \(\nu\).\N\NFor the scalar curvature \(R\), let \(R_{-}=\max\left\{ -R,0\right\} \). For \(q>n/2\), the author introduces the integral scalar curvature as \N\[\N\left\Vert R\right\Vert _{g,q,r}=\sup_{x\in M}\left\{ r^{2-\frac{n}{q}}\left( \int_{B_{r}\left( x\right) }\left\vert R_{-}\right\vert^{q}\right) ^{\frac{1}{q}}\right\} \text{,}\N\]\Nwhere \(B_{r}\left( x\right) \) is the geodesic ball centered at \(x\in M\) with radius \(r>0\). The author also introduces the notion of bounded capacity of \(\left( M,g\right) \) denoted \(\operatorname*{Cap}_{\left( M,g\right)\N}\left( r\right) \leq N\). Specifically, for a fixed positive \(r\), \(N\in\mathbb{N}^{+}\), and every \(x\in M\), there exists \(\left\{ x_{j}\right\} _{j=1} ^{N}\subset B_{2r}\left( x\right) \) such that \(\left\{ B_{r}\left(\Nx_{j}\right) \right\} _{j=1}^{N}\) forms a covering of \(B_{2r}\left(x\right) \).\N\NThe author's main result (Theorem 1.4) generalizes the \(\varepsilon \)-regularity theorem in [\textit{M.-C. Lee} et al., Geom. Topol. 27, No. 1, 227--350 (2023; Zbl 1522.53029)] to complete \(n\)-dimensional manifolds with bounded integral scalar curvature \(\left\Vert R\right\Vert _{g,q,r}\), bounded capacity \(\operatorname*{Cap}_{\left( M,g\right) }\left( r\right) \), and almost non-negative Perelman entropy \(\nu\). Specifically, the author proves that if \(\left( M,g\right) \) is a complete \(n\,\)-dimensional Riemannian manifold with bounded integral scalar curvature and if \(\varepsilon,r,N>0,p>n\), and \(q>n/2\) are fixed, then there exists \(\delta=\delta\left( n,\varepsilon\N,N,p,q\right) >0\) such that if \(\nu\left( g,2r^{2}\right) \geq -\delta,\left\Vert R\right\Vert _{g,q,r}\leq\delta,\) \(\operatorname*{Cap} _{\left( M,g\right) }\left( r\right) \leq N\), then for every \(x\in M,\)\N\[\Nd_{GH}\left( \left( B_{p,g}\left( x,r^{1-\frac{n}{p}}\right),d_{p,g}\right) ,\left( B_{p,g_{euc}}\left( 0^{n},r^{1-\frac{n}{p}}\right),d_{p,g_{euc}}\right) \right) \leq\varepsilon r^{1-\frac{n}{p}},\N\]\Nand for every \(0<s\leq r^{1-\frac{n}{p}}\),\N\[\N1-\varepsilon\leq\frac{\operatorname*{Vol}_{g}\left( B_{p,g}\left(x,s\right) \right) }{\operatorname*{Vol}_{g_{euc}}\left( B_{p,g_{euc}}\left( x,s\right) \right) }\leq1+\varepsilon.\N\]\NFor Riemannian manifolds with bounded integral scalar curvature, the author also presents results analogous to those in [loc. cit.] related to the structure of limit spaces with convergence in the \(d_{p}\)-Gromov-Hausdorff sense (Theorem 1.6), as well as results related to a priori \(L_{p}\)-bounds for scalar curvature for \(p<1\) (Theorem 1.7).