We consider manifolds with conic singularities that are isometric to $\mathbb{R}^n$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process.

The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author to establish a “very weak” Huygens’ principle, which may be of independent interest.

As applications of the estimate, we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schrödinger equation.

We verify a conjecture of Lutwak, Yang, and Zhang about the equality case in the Orlicz-Petty projection inequality, and provide an essentially optimal stability version.

We show that closed hypersurfaces in Euclidean space with non-negative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the $k$th mean curvature, for $k$ greater than 2, as we construct the counterexamples for all $k$ greater than 2. Our proof relies on a new geometric argument which relates the scalar curvature and mean curvature of a hypersurface to the mean curvature of the level sets of a height function. By extending the argument, we show that complete noncompact asymptotically flat hypersurfaces with non-negative scalar curvature are weakly mean convex and prove the positive mass theorem for such hypersurfaces in all dimensions.

In the general, this article addresses the nature of, and relation between, canonical and log-canonical singularities of foliations by curves. The case of equivariance under a finite group action has several peculiarities, amongst which are examples from dimension 3 onward where no birational modification can be both Gorenstein and log-canonical. The article in the specific, i.e. a functorial resolution algorithm for foliated 3-folds in the 2-category of Deligne-Mumford champs, is thus not only optimal but natural.

We show that a properly immersed minimal hypersurface in $M \times \mathbb{R}^+$ equals some $M \times\{c\}$ when $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with bounded curvature. If on the other hand, $M$ is not necessarily recurrent but has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over $M$.

Strongly pseudoconvex CR manifolds are boundaries of Stein varieties with isolated normal singularities. We prove that any non- constant CR morphism between two $(2n−1)$-dimensional strongly pseudoconvex CR manifolds lying in an $n$-dimensional Stein variety with isolated singularities are necessarily a CR biholomorphism. As a corollary, we prove that any nonconstant self map of $(2n − 1)$-dimensional strongly pseudoconvex CR manifold is a CR automorphism. We also prove that a finite étale covering map between two resolutions of isolated normal singularities must be an isomorphism.