We study finite action anti-self-dual Yang–Mills connections on the multi-Taub-NUT space. Under a technical assumption of generic asymptotic holonomy, we establish the curvature and the harmonic spinor decay rates and compute the index of the associated Dirac operator.

This is the first in a series of papers proving the completeness of the bow construction of instantons on multi-Taub-NUT spaces and exploring it in detail.

We study the moduli space $\mathsf{T}$ of the Calabi–Yau $n$‑folds arising from the Dwork family and enhanced with bases of the $n$‑th de Rham cohomology with constant cup product and compatible with Hodge filtration. We also describe a unique vector field $\mathsf{R}$ in $\mathsf{T}$ which contracted with the Gauss–Manin connection gives an upper triangular matrix with some non-constant entries which are natural generalizations of Yukawa couplings. For $n=1,2$ we compute explicit expressions of $\mathsf{R}$ and give a solution of $\mathsf{R}$ in terms of quasi-modular forms. The moduli space $\mathsf{T}$ is an affine variety and for $n = 4$ we give explicit coordinate system for $\mathsf{T}$ and compute the vector field $\mathsf{R}$ and the $q$-expansion of its solution.

We prove that a geodesic net with three boundary (= unbalanced) vertices on a non-positively curved plane has at most one balanced vertex. We do not assume any a priori bound for the degrees of unbalanced vertices.

The result seems to be new even in the Euclidean case.

We demonstrate by examples that the result is not true for metrics of positive curvature on the plane, and that there are no immediate generalizations of this result for geodesic nets with four unbalanced vertices.

We study compact and simply-connected Riemannian manifolds $(M,g)$ with positive sectional curvature $K \geq 1$. For a nontrivial homology class of lowest positive dimension in the space of loops based at a point $p \in M$ or in the free loop space one can define a critical length $\mathsf{crl}_p (M,g)$ resp. $\mathsf{crl} (M,g)$. Then $\mathsf{crl}_p (M,g)$ equals the length of a geodesic loop with base point $p$ and $\mathsf{crl} (M,g)$ equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and by Lusternik & Fet in the general case. It is the main result of the paper that the numbers $\mathsf{crl}_p (M,g)$ resp. $\mathsf{crl} (M,g)$ attain its maximal value $2\pi$ only for the round metric on the $n$-sphere.

Under the additional assumption $K \leq 4$ this result for $\mathsf{crl} (M,g)$ follows from results by Sugimoto in even dimensions and Ballmann, Thorbergsson & Ziller in odd dimensions.

In this paper we prove: if a bounded domain with $C^2$ boundary covers a manifold which has finite volume with respect to either the Bergman volume, the Kähler–Einstein volume, or the Kobayashi–Eisenman volume, then the domain is biholomorphic to the unit ball. This answers a question attributed to Yau. Further, when the domain is convex we can assume that the boundary only has $C^{1,\epsilon}$ regularity.