Related papers: Alday-Maldacena duality and AdS Plateau problem
Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the…
We construct closed embedded minimal surfaces in the round three-sphere, resembling two parallel copies of the equatorial two-sphere, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of the…
A low-energy background field solution is presented which describes several D-membranes oriented at angles with respect to one another. The mass and charge densities for this configuration are computed and found to saturate the BPS bound,…
We study the duality between string theory on AdS_3 X S^3 X S^3 and two-dimensional conformal theories with large N=4 superconformal algebra A_gamma. We discuss configurations of intersecting branes which give rise to such near-horizon…
Using the classical approach we show the existence of disc type solutions to the asymptotic Plateau problem in certain Hadamard manifolds which may have arbitrarily strong curvature and volume growth.
We introduce the embedded Nash problem allowing singularities in the ambient space, and solve it in the case of surfaces, generalizing \cite[Theorem 1.22]{BdlB}.
We introduce a method-of-lines formulation of the closest point method, a numerical technique for solving partial differential equations (PDEs) defined on surfaces. This is an embedding method, which uses an implicit representation of the…
We obtain an AdS$_{2}$ solution to Type IIA supergravity with 4 Poincare supersymmetries, via non-Abelian T-duality with respect to a freely acting SL(2,$\mathbf{R}$) isometry group, operating on the AdS$_3\times$S$^3\times$CY$_2$ solution…
We introduce an iterative scheme to solve the Yamabe equation $ - a\Delta_{g} u + S u = \lambda u^{p-1} $ on small domains $(\Omega,g)\subset {\mathbb R}^n$ equipped with a Riemannian metric $g$. Thus $g$ admits a conformal change to a…
Building on work of Du, Gao, and Yau, we give a characterization of smooth solutions, up to normalization, of the complex Plateau problem for strongly pseudoconvex Calabi--Yau CR manifolds of dimension $2n-1 \ge 5$ and in the hypersurface…
We study the problem of finding the minimal (maximal) genus for a surface where a given four-valent graph with fixed opposite edge structure can be embedded into. We find several partial relations and give new reformulations in…
Two flat sub-Lorentzian problems on the Martinet distribution are studied. For the first one, the attainable set has a nontrivial intersection with the Martinet plane, but for the second one it does not. Attainable sets, optimal…
We construct necessary and sufficient geometric conditions for a class of AdS$_2$ solutions of M-theory with, at least, minimal supersymmetry to exist. We generalize previous results in the literature for ${\cal N}=(2,0)$ supersymmetry in…
In this paper, we study trigonal minimal surfaces in flat tori. First, we show a topological obstruction similar to that of hyperelliptic minimal surfaces. Actually, the genus of trigonal minimal surface in 3-dimensional flat torus must be…
We consider a Plateau problem in codimension $1$ in the non-parametric setting. A Dirichlet boundary datum is given only on part of the boundary $\partial \Omega$ of a bounded convex domain $\Omega\subset\mathbb{R}^2$. Where the Dirichlet…
We study the asymptotic Plateau problem in $\BHH$ for area minimizing surfaces, and give a fairly complete solution for finite curves.
Few vacua are known for the three tachyon-free non-supersymmetric string theories. We find new classes of AdS backgrounds by focusing on spaces where the equations of motion reduce to purely algebraic conditions. Our first examples involve…
We survey recent developments on rationality problems for algebraic varieties, with a particular emphasis on cycle-theoretic and combinatorial methods and their applications to hypersurfaces.
Classically, Plateau's problem asks to find a surface of the least area with a given boundary $B$. In this article, we investigate a version of Plateau's problem, where the boundary of an admissible surface is only required to partially…
We provide new general methods in the calculus of variations for the anisotropic Plateau problem in arbitrary dimension and codimension. A new direct proof of Almgren's 1968 existence result is presented; namely, we produce from a class of…