Polyharmonic hypersurfaces into pseudo-Riemannian space forms

In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form N-t(m+1)(c) of dimension m + 1 and index t (m >= 2 and 1 <= t <= m). We shall study hypersurfaces M-t'(m) which are polyharmonic of order r (briefly, r-harmonic), where r >= 3 and either t' = t or t' = t - 1. Let A denote the shape operator of M-t'(m). Under the assumptions that M-t'(m) is CMC and TrA(2) is a constant, we shall obtain the general condition which determines that M-t'(m) is r-harmonic. As a first application, we shall deduce the existence of several new families of proper r-harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper r-harmonic hypersurfaces (r >= 3). Finally, we shall obtain the complete classification of proper r-harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form

Impure Aspects of Supersymmetric Wilson Loops

We study a general class of supersymmetric Wilson loops operator in N = 4 super Yang-Mills theory, obtained as orbits of conformal transformations. These loops are the natural generalization of the familiar circular Wilson-Maldacena operator and their supersymmetric properties are encoded into a Killing spinor that is not pure. We present a systematic analysis of their scalar couplings and of the preserved supercharges, modulo the action of the global symmetry group, both in the compact and in the non-compact case. The quantum behavior of their expectation value is also addressed, in the simplest case of the Lissajous contours: explicit computations at weak-coupling, through Feynman diagrams expansion, and at strong-coupling, by means of AdS/CFT correspondence, suggest the possibility of an exact evaluation.Comment: 40 pages, 4 figure

Generalized quark-antiquark potential at weak and strong coupling

We study a two-parameter family of Wilson loop operators in N=4 supersymmetric Yang-Mills theory which interpolates smoothly between the 1/2 BPS line or circle and a pair of antiparallel lines. These observables capture a natural generalization of the quark-antiquark potential. We calculate these loops on the gauge theory side to second order in perturbation theory and in a semiclassical expansion in string theory to one-loop order. The resulting determinants are given in integral form and can be evaluated numerically for general values of the parameters or analytically in a systematic expansion around the 1/2 BPS configuration. We comment about the feasibility of deriving all-loop results for these Wilson loops.Comment: 43 pages: 15 comprising the main text and 25 for detailed appendice

BPS Wilson loops in N=4 SYM: Examples on hyperbolic submanifolds of space-time.

In this paper we present a family of supersymmetric Wilson loops of N=4 supersymmetric Yang-Mills theory in Minkowski space. Our examples focus on curves restricted to hyperbolic submanifolds, H_3 and H_2, of space-time. Generically they preserve two supercharges, but in special cases more, including a case which has not been discussed before, of the hyperbolic line, conformal to the straight line and circle, which is half-BPS. We discuss some general properties of these Wilson loops and their string duals and study special examples in more detail. Generically the string duals propagate on a complexification of AdS_5 x S^5 and in some specific examples the compact sphere is effectively replaced by a de-Sitter space.Comment: 28 pages, 7 figues, pdfte

Unique continuation properties for polyharmonic maps between Riemannian manifolds

Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher-order functionals which extend the classical energy functional for maps between Riemannian manifolds. The main aim of this paper is to investigate the so-called unique continuation principle. More precisely, assuming that the domain is connected, we shall prove the following extensions of results known in the harmonic and biharmonic cases: (i) if a k-harmonic map is harmonic on an open subset, then it is harmonic everywhere; (ii) if two k-harmonic maps agree on an open subset, then they agree everywhere; and (iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of the domain is mapped into the equator, then all the domain is mapped into the equator

Higher order energy functionals

The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called ES −r-energy functionals EES (φ) = (1/2) |(d∗ + rM d)r(φ)|2dV, where φ : M → N is a map between two Riemannian manifolds. In the initial part of this paper we shall clarify some relevant issues about the definition of an ES − r-harmonic map, i.e., a critical point of EES (φ). That r seems important to us because in the literature other higher order energy functionals have been studied by several authors and consequently some recent examples need to be discussed and extended: this shall be done in the first two sections of this work, where we obtain the first examples of proper critical points of EES(φ) when N = Sm (r ≥ 4, m ≥ 3), and r we also prove some general facts which should be useful for future developments of this subject. Next, we shall compute the Euler-Lagrange system of equations for EES(φ) for r = 4. We shall apply this result to the study of maps into space forms and to rotationally symmetric maps: in particular, we shall focus on the study of various family of conformal maps. In Section 4, we shall also show that, even if 2r > dim M , the functionals EES(φ) may not satisfy the classical Palais-Smale r Condition (C). In the final part of the paper we shall study the second variation and compute index and nullity of some significant examples