This overview is based on the talk [101] given at the mini-workshop 0648c" Dirac operators
in di? erential and non-commutative geometry", Mat-matisches Forschungsinstitut …

Let us fix a conformal class $[g_0] $ and a spin structure $\sigma $ on a compact manifold $
M $. For any $ g\in [g_0] $, let $\lambda^+ _1 (g) $ be the smallest positive eigenvalue of the …

We study some basic analytical problems for nonlinear Dirac equations involving critical
Sobolev exponents on compact spin manifolds. Their solutions are obtained as critical …

Abstract Let (M, g, σ) be a compact Riemannian spin manifold of dimension≥ 2. For any
metric ̃ g conformal to g, we denote by ̃ λ the first positive eigenvalue of the Dirac operator on …

We prove sharp pointwise decay estimates for critical Dirac equations on $\mathbb {R}^ n $
with $ n\geqslant 2$. They appear for instance in the study of critical Dirac equations on …

We consider super-linear and sub-linear nonlinear Dirac equations on compact spin
manifolds. Their solutions are obtained as critical points of certain strongly indefinite …

This paper is part of a program to establish the existence theory for the conformally invariant
Dirac equation\[D_g\psi= f (x)|\psi| _g^{\frac 2 {m-1}}\psi\] on a closed spin manifold $(M, g) …

We prove a classification result for ground state solutions of the critical Dirac equation on R^
n R n, n\geqslant 2 n⩾ 2. By exploiting its conformal covariance, the equation can be posed …

For m ≥ 2, we prove the existence of non-trivial solutions for a certain kind of nonlinear
Dirac equations with critical Sobolev nonlinearities on S^ m via a perturbative variational …

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact
Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we …