Scalar and mean curvature comparison via $$\mu $$-bubbles

Nonnegative scalar curvature on manifolds with at least two ends

“…By construction, we also have $$\mathrm{scal}_g \geqslant \kappa n(n-1) > 0$$ on $$X$$. But this contradicts Corollary 3.7 or even the usual band width estimate on compact bands [15, Section 3.6][25] of dimension $$\leqslant 7$$.$$\Box$$…”

“…Remark This notion of a model space arises in the scalar and mean curvature comparison theory surrounding Conjecture 1.3. In [3, 25], a compact Riemannian band $$(X,g)$$ is compared, in scalar curvature, mean curvature and width, to a warped product $$(M,g_\varphi )$$ over an arbitrary scalar flat manifold $$(N,g_N)$$ with strictly $$\log$$‐concave warping function. It turns out that, if $$X$$ has Property A, $$\mathrm{scal}(X,g)\geqslant \mathrm{scal}(M,g_\varphi )$$ and $$\mathrm{H}(\partial X,g)\geqslant \mathrm{H}(\partial M,g_\varphi )$$, then $$\mathrm{width}(X,g)\leqslant \mathrm{width}(M,g_\varphi )$$.…”

“…The key ingredient for the corresponding functional is the potential function $$h \colon X\rightarrow \mathbb {R}$$. We use the ideas from [25, Section 3] for each band $$(V_j,g)$$ and model space $$(M_j,g_{\varphi _j})$$ separately to produce $$h_j \colon V_j\rightarrow \mathbb {R}$$ as the concatenation of a strictly 1‐Lipschitz band map $$(V_j,g)\rightarrow (M_j,g_{\varphi _j})$$ and the function $$h_{\varphi _j}:M_j\rightarrow \mathbb {R}$$. Subsequently, we use a gluing construction to paste all of the $$h_j$$ together to obtain a smooth function $$h\colon X\rightarrow \mathbb {R}$$ which is suitable for our purposes.…”

“…In [15, Sections 3.6, 5], Gromov proposes a different approach toward Conjecture 1.3 using a modified version of the minimal hypersurface method involving so‐called $$\mu$$‐bubbles. Following this idea, Räde [25] proved Conjecture 1.3 and generalizations thereof in case that $$Y$$ is orientable and $$n\leqslant 7$$ 1.1, 1.2. have not been directly approached via minimal hypersurface techniques so far, in particular, due to the noncompactness inherent to the problem.…”

“…The main objective of this article is to combine the ideas from [36] with $$\mu$$‐bubble methods, in particular from [25], to prove a generalization of Conjecture 1.1 in case $$Y$$ is orientable and of dimension $$\leqslant 6$$. By a reduction argument to a codimension 1 situation (compare [13, Theorem 7.5][17]), we also establish Conjecture 1.2 in case $$Y$$ is orientable and of dimension $$\leqslant 5$$.…”