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In two-dimensional chiral metal-halide perovskites, chiral organic spacers endow structural and optical chirality to the metal-halide sublattice, enabling exquisite control of light, charge, and electron spin. The chiroptical properties of metal-halide perovskites have been measured by transmissive circular dichroism spectroscopy, which necessitates thin-film samples. Here, by developing a reflection-based approach, we characterize the intrinsic, circular polarization-dependent complex refractive index for a prototypical two-dimensional chiral lead-bromide perovskite and report large circular dichroism for single crystals. Comparison with ab initio theory reveals the large circular dichroism arises from the inorganic sublattice rather than the chiral ligand and is an excitonic phenomenon driven by electron-hole exchange interactions, which breaks the degeneracy of transitions between Rashba-Dresselhaus-split bands, resulting in a Cotton effect. Our study suggests that previous data for spin-coated films largely underestimate the optical chirality and provides quantitative insights into the intrinsic optical properties of chiral perovskites for chiroptical and spintronic applications.

For a given finite subsetSof a compact Riemannian manifold (M,g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric onM\Ssuch that each point ofScorresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by‐product, we define a purely local notion of Ricci lower bounds for continuous metrics that are conformal to smooth metrics and prove a corresponding volume comparison theorem. © 2022 The Authors.Communications on Pure and Applied Mathematicspublished by Wiley Periodicals LLC.

A classical theorem of A. D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to some results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.

We study the regularity of the viscosity solution of the ‐Loewner–Nirenberg problem on a bounded smooth domain for . It was known that is locally Lipschitz in . We prove that, with being the distance function to and sufficiently small, is smooth in and the first derivatives of are Hölder continuous in . Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of and its covariant derivatives and vanishes if and only if is smooth in . Using a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when contains more than one connected components, is not differentiable in .