51 research outputs found
A note on twisted Dirac operators on closed surfaces
We derive an inequality that relates nodal set and eigenvalues of a class of
twisted Dirac operators on closed surfaces and point out how this inequality
naturally arises as an eigenvalue estimate for the Dirac operator.
This allows us to obtain eigenvalue estimates for the twisted Dirac operator
appearing in the context of Dirac-harmonic maps and their extensions, from
which we also obtain several Liouville type results
An estimate on the nodal set of eigenspinors on closed surfaces
We use a modified Bochner technique to derive an inequality relating the
nodal set of eigenspinors to eigenvalues of the Dirac operator on closed
surfaces. In addition, we apply this technique to solutions of similar
spinorial equations
Nonlinear Dirac equations, Monotonicity Formulas and Liouville Theorems
We study the qualitative behavior of nonlinear Dirac equations arising in
quantum field theory on complete Riemannian manifolds. In particular, we derive
monotonicity formulas and Liouville theorems for solutions of these equations.
Finally, we extend our analysis to Dirac-harmonic maps with curvature term
On the evolution of regularized Dirac-harmonic Maps from closed surfaces
We study the evolution equations for a regularized version of Dirac-harmonic
maps from closed Riemannian surfaces. We establish the existence of a global
weak solution for the regularized problem, which is smooth away from finitely
many singularities. Moreover, we discuss the convergence of the evolution
equations and address the question if we can remove the regularization in the
end
A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies
We prove a Liouville-type theorem for biharmonic maps from a complete
Riemannian manifold of dimension that has a lower bound on its Ricci
curvature and positive injectivity radius into a Riemannian manifold whose
sectional curvature is bounded from above. Under these geometric assumptions we
show that if the -norm of the tension field is bounded and the -energy
of the map is sufficiently small then every biharmonic map must be harmonic,
where
A structure theorem for polyharmonic maps between Riemannian manifolds
We prove that polyharmonic maps of arbitrary order from complete nonparabolic
Riemannian manifolds to arbitrary Riemannian manifolds must be harmonic if
certain smallness and integrability conditions hold
Magnetic Dirac-harmonic maps
We study a functional, whose critical points couple Dirac-harmonic maps from
surfaces with a two form. The critical points can be interpreted as coupling
the prescribed mean curvature equation to spinor fields. On the other hand,
this functional also arises as part of the supersymmetric sigma model in
theoretical physics. In two dimensions it is conformally invariant. We call
critical points of this functional magnetic Dirac-harmonic maps. We study
geometric and analytic properties of magnetic Dirac-harmonic maps including
their regularity and the removal of isolated singularities
Energy methods for Dirac-type equations in two-dimensional Minkowski space
In this article we develop energy methods for a large class of linear and
nonlinear Dirac-type equations in two-dimensional Minkowski space. We will
derive existence results for several Dirac-type equations originating in
quantum field theory, in particular for Dirac-wave maps to compact Riemannian
manifolds
A vanishing result for the supersymmetric nonlinear sigma model in higher dimensions
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
The normalized second order renormalization group flow on closed surfaces
We study a normalized version of the second order renormalization group flow
on closed Riemannian surfaces. We discuss some general properties of this flow
and establish several basic formulas. In particular, we focus on surfaces with
zero and positive Euler characteristic
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