Related papers: Alday-Maldacena duality and AdS Plateau problem

It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL_2-type over \Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism…

In this note we solve a minimization problem arising in a recent work of Bolognesi and Tong on the determination of an AdS monopole wall. We show that the problem has a unique solution. Although the solution cannot be obtained explicitly,…

We consider the B2 and G2 Toda systems on compact surfaces. We attack the problem using variational techniques. We get existence and multiplicity of solutions under a topological assumption on the surface and some generic conditions on the…

In an earlier paper, Alvarez, Alvarez-Gaume, Barbon and Lozano pointed out, that the only way to "flatten" negative curvature by means of a T-duality is by introducing an appropriate, non-constant NS-NS B-field. In this paper, we are…

We study classical open string solutions with a null polygonal boundary in AdS_3 in relation to gluon scattering amplitudes in N=4 super Yang-Mills at strong coupling. We derive in full detail the set of integral equations governing the…

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used.

In this expository article we revisit the Bernstein problem for several geometric PDEs including the minimal surface, Monge-Amp\`{e}re, and special Lagrangian equations. We also discuss the minimal surface system where appropriate. The…

A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known…

We extend the classification of supersymmetric locally AdS$_3$ geometries, beyond BTZ black holes, to the Banados geometries, noting that supersymmetries are in one-to-one correspondence with solutions to the Hill differential condition. We…

Explicit Calabi-Yau metrics are derived that are argued to map to the Maldacena-Nu\~{n}ez AdS solutions of M-theory and IIB under geometric transitions. In each case the metrics are singular where a H^2 K\"{a}hler two-cycle degenerates but…

The AdS/CFT correspondence relates the expectation value of Wilson loops in N=4 SYM to the area of minimal surfaces in AdS_5 In this paper we consider minimal area surfaces in generic Euclidean AdS_{n+1} using the Pohlmeyer reduction in a…

We consider minimal hypersurfaces inside the unit ball whose boundary on the sphere is a small perturbation of the link of a minimizing quadratic cone. We show that such minimal surfaces are uniquely determined by their boundary condition.…

An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for…

This is a brief survey of recent works by Neil Trudinger and myself on the Bernstein problem and Plateau problem for affine maximal hypersurfaces.

We study minimal area world sheets ending on two concentric circumferences on the boundary of Euclidean $AdS_{3}$ with mixed R-R and NS-NS three-form fluxes. We solve the problem by reducing the system to a one-dimensional integrable model.…

The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work…

Towards identifying the number of minimal surfaces sharing the same boundary from the geometry of the boundary, we propose a numerical scheme with high speed and high accuracy. Our numerical scheme is based on the method of fundamental…

We formulate the Landau problem in the context of the noncommutative analog of a surface of constant negative curvature, that is $AdS_2$ surface, and obtain the spectrum and contrast the same with the Landau levels one finds in the case of…

Minimal area surfaces in AdS$_3$ ending on a given curve at the boundary are dual to planar Wilson loops in N=4 SYM. In previous work it was shown that the problem of finding such surfaces can be recast as the one of finding an appropriate…

We prove a compactness principle for the anisotropic formulation of the Plateau problem in codimension one, along the same lines of previous works of the authors [DGM14, DPDRG15]. In particular, we perform a new strategy for proving the…