Related papers: From CT scans to 4-manifold topology
The paper concerns the problem for the ultrahyperbolic equation in the Euclidean space with data on a characteristic hyperplane. Smoothness and asymptotics of the solution along characteristic lines transversal to the initial hyperplane are…
We characterize which cobounded quasigeodesics in the Teichmueller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path gamma in T, we show that gamma is a quasigeodesic with…
The classical boundary-value problem of the Einstein field equations is studied with an arbitrary cosmological constant, in the case of a compact ($S^{3}$) boundary given a biaxial Bianchi-IX positive-definite three-metric, specified by two…
We study the Cauchy problem for general, nonlinear, strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of…
In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal…
We introduce a topological invariant, it a type of a graph-manifold, which takes natural values. For a 4-dimensional graph-manifold, whose type does not exceed two, it is proved that its universal cover is bi-Lipschitz equivalent to a…
We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions…
We consider the quasihyperbolic metric, and its generalizations in both the $n$-dimensional Euclidean space $R^n$, and in Banach spaces. Historical background, applications, and our recent work on convexity properties of these metrics are…
We discuss the well-posedness of the Cauchy problem for hyperbolic operators with double characteristics which changes from non-effectively hyperbolic to effectively hyperbolic, on the double characteristic manifold, across a submanifold of…
We consider cusped hyperbolic $n-$manifolds, and compute \v{C}ech cohomology groups of the Morse boundaries of their fundamental groups. In particular, we show that the reduced \v{C}ech cohomology with real coefficients vanishes in…
Chernov-Nemirovski observed that the existence of a globally hyperbolic Lorentzian metric on a (3 + 1)-spacetime pins down a smooth structure on the underlying 4-manifold. In this paper, we point out that the diffeomorphism type of a…
This paper contains some more results on the topology of a nondegenerate action of $\mathbb{R}^n$ on a compact connected $n$-manifold $M$ when the action is totally hyperbolic (i.e. its toric degree is zero). We study the…
In this paper, for each finite group $G$, we construct explicitly a non-compact complete finite-volume arithmetic hyperbolic $4$-manifold $M$ such that $\mathrm{Isom}\,M \cong G$, or $\mathrm{Isom}^{+}\,M \cong G$. In order to do so, we use…
This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity…
We investigate the collapsing geometry of hyperk\"ahler 4-manifolds. As applications we prove two well-known conjectures in the field. (1) Any collapsed limit of unit-diameter hyperk\"ahler metrics on the K3 manifold is isometric to one of…
We construct stable minimal hypersurfaces with simple topology in certain compact $4$-manifolds $X$ with boundary, where $X$ embeds into a smooth manifold homeomorphic to $S^4$. For example, if $X$ is equipped with a Riemannian metric $g$…
This is a short survey on finite-volume hyperbolic four-manifolds. We describe some general theorems and focus on the concrete examples that we found in the literature. The paper contains no new result.
In this paper we obtain higher order asymptotic profilles of solutions to the Cauchy problem of the linear damped wave equation in $\textbf{R}^n$ \begin{equation*} u_{tt}-\Delta u+u_t=0, \qquad u(0,x)=u_0(x), \quad u_t(0,x)=u_1(x),…
In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild…
This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold M with toroidal boundary. We also survey a number of applications of this technique. The method involves…