Given a finite collection of lines $L_j \subset \mathbb{CP}^2$ together with real numbers $0 \lt \beta_j \lt 1$ satisfying natural constraint conditions, we show the existence of a Ricci–flat Kähler metric $g_{RF}$ with cone angle $2\pi\beta_j$ along each line $L_j$ asymptotic to a polyhedral Kähler cone at each multiple point. Moreover, we discuss a Chern–Weil formula that expresses the energy of $g_{RF}$ as a logarithmic Euler characteristic with points weighted according to the volume density of the metric.

We construct the (enhanced Rogers) dilogarithm function from the spin Chern–Simons invariant of $\mathbb{C}^\times$-connections. This leads to geometric proofs of basic dilogarithm identities and a geometric context for other properties, such as the branching structure.

This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents of Carnot geodesics are geodesics in some groups of lower nilpotency step. Namely, every blowup curve of every geodesic in every Carnot group is still a geodesic in the group modulo its last layer. Then as a consequence we get that in every sub-Riemannian manifold any $s$ times iterated tangent of any geodesic is a line, where $s$ is the step of the sub-Riemannian manifold in question. With a similar approach, we also show that blowdown curves of geodesics in sub-Riemannian Carnot groups are contained in subgroups of lower rank. This latter result is also extended to rough geodesics.

In the Euclidean unit three-ball, we construct compact, embedded, two-sided free boundary minimal surfaces with connected boundary and prescribed high genus, by a gluing construction tripling the equatorial disc. Aside from the equatorial disc itself, these are the first examples in the three-ball of compact free boundary minimal surfaces with connected boundary.

In this article, we give a generalized construction of the $2$-loop term of the $SU(2)$-Chern–Simons perturbation theory by using the technique developed in [12]. As an application, we give a Morse theoretic description of the $2$-loop term of the $SU(2)$-Chern–Simons perturbation theory at a non-trivial connection.