5,878 research outputs found
Remarks on the boundary curve of a constant mean curvature topological disc
We discuss some consequences of the existence of the holomorphic quadratic
Hopf differential on a conformally immersed constant mean curvature topological
disc with analytic boundary. In particular, we derive a formula for the mean
curvature as a weighted average of the normal curvature of the boundary curve,
and a condition for the surface to be totally umbilic in terms of the normal
curvature.Comment: 6 pages, 1 figure. Version 2: comments and references adde
Spin chain sigma models with fermions
The complete one-loop planar dilatation operator of N=4 supersymmetric
Yang-Mills is isomorphic to the hamiltonian of an integrable PSU(2,2|4) quantum
spin chain. We construct the non-linear sigma models describing the continuum
limit of the SU(1|3) and SU(2|3) sectors of the complete N=4 chain. We
explicitely identify the spin chain sigma model with the one for a superstring
moving in AdS_5xS^5 with large angular momentum along the five-sphere.Comment: 16 pages. Latex. v2: Misprints corrected and references adde
Existence and non existence results for the singular Nirenberg problem
In this paper we study the problem, posed by Troyanov (Trans AMS 324: 793–821, 1991), of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a nonlinear PDE with exponential type non-linearity admitting a variational structure. In particular, we are concerned with the case where the prescribed function K changes sign. When the surface is the standard sphere, namely for the singular Nirenberg problem, we give sufficient conditions on K, concerning mainly the regularity of its nodal line and the topology of its positive nodal region, to be the Gaussian curvature of a conformal metric with assigned conical singularities. Besides, we find a class of functions on S^2 which do not verify our conditions and which can not be realized as the Gaussian curvature of any conformal metric with one conical singularity. This shows that our result is somehow sharp
- …