Publications Mathématiques de l'Institut des Hautes Études Scientifiques

We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements in higher homotopy and homology groups of these spaces, which, in contrast to previous approaches, are of infinite order and survive in the (observer) moduli space of such metrics. Along the way we construct smooth fiber bundles over spheres whose total spaces have non-vanishing $\hat{A}$ -genera, thus establishing the non-multiplicativity of the $\hat{A}$ -genus in fiber bundles with simply connected base.

We study Liouville first passage percolation metrics associated to a Gaussian free field $h$ mollified by the two-dimensional heat kernel $p_{t}$ in the bulk, and related star-scale invariant metrics. For $\gamma \in (0,2)$ and $\xi = \frac{\gamma }{d_{\gamma }}$ , where $d_{\gamma }$ is the Liouville quantum gravity dimension defined in Ding and Gwynne (Commun. Math. Phys. 374:1877–1934, 2020), we show that renormalized metrics $(\lambda _{t}^{-1} e^{ \xi p_{t} * h} ds)_{t \in (0,1)}$ are tight with respect to the uniform topology. We also show that subsequential limits are bi-Hölder with respect to the Euclidean metric, obtain tail estimates for side-to-side distances, and derive error bounds for the normalizing constants $\lambda _{t}$ .