Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions

We consider a periodic Schroedinger operator and the composite Wannier functions corresponding to a relevant family of its Bloch bands, separated by a gap from the rest of the spectrum. We study the associated localization functional introduced by Marzari and Vanderbilt, and we prove some results about the existence and exponential localization of its minimizers, in dimension d < 4. The proof exploits ideas and methods from the theory of harmonic maps between Riemannian manifolds.Comment: 37 pages, no figures. V2: the appendix has been completely rewritten. V3: final version, to appear in Commun. Math. Physic

Polynomial cubic differentials and convex polygons in the projective plane

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with prescribed Pick differential, and can be seen as an analogue of the Labourie-Loftin parameterization of convex RP^2 structures on a compact surface by the bundle of holomorphic cubic differentials over Teichmuller space.Comment: 64 pages, 5 figures. v3: Minor revisions according to referee report. v2: Corrections in section 5 and related new material in appendix

Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere

We prove existence of a countable family of spherically symmetric self-similar wave maps from 3+1 Minkowski spacetime into the 3-sphere. These maps can be viewed as excitations of the ground state wave map found previously by Shatah. The first excitation is particularly interesting in the context of the Cauchy problem since it plays the role of a critical solution sitting at the threshold of singularity formation. We analyze the linear stability of our wave maps and show that the number of unstable modes about a given map is equal to its excitation index. Finally, we formulate a condition under which these results can be generalized to higher dimensions.Comment: 15 pages, extended version to appear in Comm. Math. Phy

Triunduloids: Embedded constant mean curvature surfaces with three ends and genus zero

In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; they are classified using their asymptotic necksizes. We work in a class slightly more general than embedded surfaces, namely immersed surfaces which bound an immersed three-manifold, as introduced by Alexandrov.Comment: LaTeX, 22 pages, 2 figures (8 ps files); full version of our announcement math.DG/9903101; final version (minor revisions) to appear in Crelle's J. reine angew. Mat

Effectively Closed Infinite-Genus Surfaces and the String Coupling

The class of effectively closed infinite-genus surfaces, defining the completion of the domain of string perturbation theory, can be included in the category $O_G$, which is characterized by the vanishing capacity of the ideal boundary. The cardinality of the maximal set of endpoints is shown to be 2^{\mit N}. The product of the coefficient of the genus-g superstring amplitude in four dimensions by $2^g$ in the $g\to \infty$ limit is an exponential function of the genus with a base comparable in magnitude to the unified gauge coupling. The value of the string coupling is consistent with the characteristics of configurations which provide a dominant contribution to a finite vacuum amplitude.Comment: TeX, 33 page