A vanishing result for the supersymmetric nonlinear sigma model in higher dimensions

Abstract

We prove a vanishing result for critical points of the supersymmetric nonlinear sigma model on complete non-compact Riemannian manifolds of positive Ricci curvature that admit an Euclidean type Sobolev inequality, assuming that the dimension of the domain is bigger than two and that a certain energy is sufficiently small.

Harmonic maps with torsion

“…In this subsection, we establish a vanishing result for solutions of (1.3) on a large class of complete noncompact domain manifolds. For standard harmonic maps a corresponding result was obtained in [25] (see also [10] for further generalizations).…”

“…Such an inequality holds in R m and is well known as the Gagliardo-Nirenberg inequality. However, if one considers a non-compact complete Riemannian manifold of infinite volume one has to make additional assumptions to have an equality of the form (4.11) at hand (see the introduction of [10] for more details). We will make use of a cutoff function 0 η 1 on M that satisfies…”

Dirac-harmonic maps with potential

“…Besides the aforementioned existence results, several Liouville-type results have also been established [ 11 , 12 , 14 , 16 ]. These provide criteria under which a Dirac-harmonic map must be trivial, that is the map part maps to a point and the spinor vanishes identically.…”

Nonlinear Dirac Equations, Monotonicity Formulas and Liouville Theorems

“…The conformal invariance gives rise to a removable singularity theorem [ 10 ] and an energy identity [ 27 ]. Conservation laws for Dirac-harmonic maps with curvature term were established in [ 11 ] and a vanishing result for the latter under small-energy assumptions was derived in [ 13 ]. For Dirac-wave maps with curvature term (which are Dirac-harmonic maps with curvature term from a domain with Lorentzian metric) on expanding spacetimes an existence result could be achieved in [ 14 ].…”