Equivariant wave maps exterior to a ball

We consider the exterior Cauchy-Dirichlet problem for equivariant wave maps from 3+1 dimensional Minkowski spacetime into the three-sphere. Using mixed analytical and numerical methods we show that, for a given topological degree of the map, all solutions starting from smooth finite energy initial data converge to the unique static solution (harmonic map). The asymptotics of this relaxation process is described in detail. We hope that our model will provide an attractive mathematical setting for gaining insight into dissipation-by-dispersion phenomena, in particular the soliton resolution conjecture.Comment: 16 pages, 9 figure

Mixed quasi-\'etale surfaces, new surfaces of general type with pg=0p_g=0 and their fundamental group

We call a projective surface $X$ mixed quasi-\'etale quotient if there exists a curve $C$ of genus $g(C)\geq 2$ and a finite group $G$ that acts on $C\times C$ exchanging the factors such that $X=(C\times C)/G$ and the map $C\times C \rightarrow X$ has finite branch locus. The minimal resolution of its singularities is called mixed quasi-\'etale surface. We study the mixed quasi-\'etale surfaces under the assumption that $(C\times C)/G^0$ has only nodes as singularities, where $G^0\triangleleft G$ is the index two subgroup of the elements that do not exchange the factors. We classify the minimal regular surfaces with $p_g=0$ whose canonical model is a mixed quasi-\'etale quotient as above. All these surfaces are of general type and as an important byproduct, we provide an example of a numerical Campedelli surface with topological fundamental group \bbZ_4, and we realize 2 new topological types of surfaces of general type. Three of the families we construct are \bbQ-homology projective planes.Comment: 18 pages, 3 tables, v2: change title, exposition improved; v3: minor corrections, final version to be published in Collectanea Mathematic

A Comment on Quantum Distribution Functions and the OSV Conjecture

Using the attractor mechanism and the relation between the quantization of $H^{3}(M)$ and topological strings on a Calabi Yau threefold $M$ we define a map from BPS black holes into coherent states. This map allows us to represent the Bekenstein-Hawking-Wald entropy as a quantum distribution function on the phase space $H^{3}(M)$. This distribution function is a mixed Husimi/anti-Husimi distribution corresponding to the different normal ordering prescriptions for the string coupling and deviations of the complex structure moduli. From the integral representation of this distribution function in terms of the Wigner distribution we recover the Ooguri-Strominger-Vafa (OSV) conjecture in the region "at infinity" of the complex structure moduli space. The physical meaning of the OSV corrections are briefly discussed in this limit.Comment: 27 pages. v2:reference and footnote adde

Y -cone metric spaces and coupled common fixed point results with application to integral equation

This paper acquaints with a concept of Y -cone metric space and to study some topological properties of Y -cone metric space. We prove the coupled common fixed point results for mixed weakly monotone map in ordered Y -cone metric spaces. We give an example, which constitutes the main theorem.Publisher's Versio

Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one of the poles of the sphere. We show that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time $T_i$, and eventually the solution comes to rest at the zero energy constant map.Comment: 24 pages, 8 figures, minor corrections, matches published versio

Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds

The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at once, in a way that depends continuously on $f$ and leaves $f$ fixed if it is holomorphic to begin with. In other words, is \scrO(S,X) a deformation retract of \scrC(S,X)? We prove that it is if $S$ has a strictly plurisubharmonic Morse exhaustion with finitely many critical points; in particular, if $S$ is affine algebraic. The only property of $X$ used in the proof is the parametric Oka property with approximation with respect to finite polyhedra, so our theorem holds under the weaker assumption that $X$ is an Oka manifold. Our main tool, apart from Oka theory itself, is the theory of absolute neighbourhood retracts. We also make use of the mixed model structure on the category of topological spaces.Comment: Version 2: A few very minor improvements to the exposition. Version 3: Another few very minor improvements to the exposition. To appear in Proceedings AM