Related papers: Self-dual metrics and twenty-eight bitangents
Following his discovery that finite metric spaces have injective envelopes naturally admitting a polyhedral structure, Isbell, in his pioneering work on injective metric spaces, attempted a characterization of cellular complexes admitting…
In this survey article, given a smooth closed manifold M we study the space of Riemannian metrics of positive scalar curvature on M. A long-standing question is: when is this space non-empty (i.e. when does M admit a metric of positive…
We construct the moduli space of r-jets at a point of Riemannian metrics on a smooth manifold. The construction is closely related to the problem of classification of jet metrics via differential invariants. The moduli space is proved to be…
We study the moduli space of solutions to the Seiberg-Witten equations with $N$ spinors on a compact Riemann surface. These moduli spaces arise in a program to define a new enumerative invariant of 3-manifolds. They are also of independent…
We study the asymptotics of the natural $L^2$ metric on the Hitchin moduli space with group $G = \mathrm{SU}(2)$. Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore \cite{gmn13}, is that on…
In a 2013 paper, Gromov proves that if smooth Riemannian metrics $g_i$ converge to a smooth Riemannian metric $g$ uniformly, and $g_i$ have scalar curvature uniformly bounded below, then $g$ shares the same scalar curvature lower bound. In…
We concern $C^2$-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are $4$, $5$ or $6$. By conducting a quantitative analysis of a linear…
The structure of simplicial manifolds in a model of Causal Dynamical Triangulations in 3+1 dimensions with the spatial topology of a 3-torus is analyzed with the help of topological observables, such as loops with nonzero winding numbers…
We classify $n$-dimensional geometric graph manifolds with nonnegative scalar curvature, and first show that if $n>3$, the universal cover splits off a codimension 3 Euclidean factor. We then proceed with the classification of the…
We discuss Cartan-Schouten metrics (Riemannian or pseudo-Riemannian metrics that are parallel with respect to the Cartan-Schouten canonical connection) on perfect Lie groups. Applications are foreseen in Information Geometry. Throughout…
We outline the construction of the Atiyah-Hitchin metric on the moduli space of SU(2) BPS monopoles with charge 2, first as an algebraic curve in C^3 following Donaldson and then as a solution of the Toda field equations in the continual…
We consider manifolds endowed with metric contact pairs for which the two characteristic foliations are orthogonal. We give some properties of the curvature tensor and in particular a formula for the Ricci curvature in the direction of the…
Anti-self-dual metrics in the $(++--)$ signature which admit a covariantly constant real spinor are studied. It is shown that finding such metrics reduces to solving a fourth order integrable PDE, and some examples are given. The…
It is first shown that the scalar product on any orthogonal space (V, g) allows one to define linear isomorphisms of the vector spaces of bivectors and 2-forms on V with the underlying vector spaces of the Lie algebra so(p, q) and its dual,…
We prove that a totally umbilical biharmonic surface in any $3$-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if…
We provide a general B\"ochner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature…
Consider a smooth manifold $M$ with a smooth cometric $g^{\ast}$ which changes the bilineal type by transverse way, on a hypersurface $D^{\infty}$. Suppose that the radical annihilator hyperplane is tangent to $D^{\infty}$. We examine the…
We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shear-free geodesic…
We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every…
The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article,…