Dirac-harmonic maps are a mathematical version (with commuting variables only) of the solutions of the field equations of the non-linear supersymmetric sigma model of quantum field theory. We explain this structure, including the appropriate boundary conditions, in a geometric framework. The main results of our paper are concerned with the analytic regularity theory of such Dirac-harmonic maps. We study Dirac-harmonic maps from a Riemannian surface to an arbitrary compact Riemannian manifold. We...

We present explicit expressions of the Poisson kernels for geodesic balls in the higher dimensional spheres and real hyperbolic spaces. As a consequence, the Dirichlet problem for the projective space is explicitly solved. Comparison of different expressions for the same Poisson kernel lead to interesting identities concerning special functions.

In this paper Bessel potentials on $C^{\infty }$-Riemannian manifolds (open or bordered) are studied. Let $\mathbf{M}$ be an $n$-dimensional manifold, and $\mathbf{N}$ a submanifold of $\mathbf{M}$ of dimension $k$. Sufficient conditions are given for: 1) the restriction to $\mathbf{N}$ of any potential of order $\alpha$ on $\mathbf{M}$ to be a potential of order $\alpha -\frac{n-k}{2}$ on $\mathbf{N}$ ; 2) any potential of order $\alpha -\frac{n-k}{2}$ on $\mathbf{N}$ to be extendable to a potential of order $\alpha$ on $\mathbf{M}$. It is also proved that for a bordered manifold $\mathbf{M}$ the restriction to its interior ${\mathbf{M}}^i$ is an isometric isomorphism between the...