Quantitative implications of positive scalar curvature

Thomas Richard

Comparison inequalities (Toponogov, Myers, Bishop-Gromov) are essential tools in studying manifolds with positive sectional or Ricci curvature. Until recently, these tools had no analog in the context of manifolds with positive scalar curvature: all the results on these kinds of manifolds were topological in nature. We will present more or less recent progress in this direction due to Gromov using minimal hypersurfaces as well as what Gromov calls the “mu-bubbles.” In particular, we will show that a metric with scalar curvature greater than \( n(n-1)\) on \(\mathbb{T}^{n-1} \times [-1,1]\; (n ­\leq 7) \) cannot have its two boundary components too far apart. We will conclude by showing how the positive scalar curvature controls the size of minimal 2-spheres in \( \mathbb{S}_2 \times \mathbb{T}^{n-2}, \mathbb{S}_2 \times \mathbb{R}^2 \) and \( \mathbb{S}_2 \times \mathbb{S}_2\).

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