Let $\left( {{M}_{n}},\,g \right)$ be a strictly convex riemannian manifold with ${{C}^{\infty }}$ boundary. We prove the existence of classical solution for the nonlinear elliptic partial differential equation of Monge-Ampère: $\det (-u\delta _{j}^{i}\,+\,\nabla _{j}^{i}u)\,=\,F(x,\,\nabla u;\,u)$ in $M$ with a Neumann condition on the boundary of the form $\frac{\partial u}{\partial v}=\varphi (x,u)$, where $F\in {{C}^{\infty }}(TM\times \mathbb{R})$ is an everywhere strictly positive function satisfying some assumptions, $v$ stands for the unit normal vector field and $\varphi \in {{C}^{\infty }}(\partial M\times \mathbb{R})$ is a non-decreasing function in $u$.

Let M, N be Riemannian manifolds, f: M → N a harmonic map with potential H, namely, a smooth critical point of the functional EH(f) = ∫M[e(f)−H(f)], where e(f) is the energy density of f. Some results concerning the stability of these maps between spheres and any Riemannian manifold are given. For a general class of M, this paper also gives a result on the constant boundary-value problem which generalizes the result of Karcher-Wood even in the case of the usual harmonic maps. It can also be applied to the static Landau-Lifshitz equations.

Let ${{\mathcal{S}}_{r}}$ be the collection of all axially symmetric functions $f$ in the Sobolev space ${{H}^{1}}\left( {{\mathbb{S}}^{2}} \right)$ such that $\int_{{{\mathbb{S}}^{2}}}{{{x}_{i}}{{e}^{2f\left( x \right)}}\,dw\left( \text{x} \right)}$ vanishes for $i\,=\,1,\,2,\,3$. We prove that

$$\underset{f\in {{\mathcal{S}}_{r}}}{\mathop \inf }\,\frac{1}{2}\int_{{{\mathbb{S}}^{2}}}{{{\left| \nabla f \right|}^{2}}\,dw\,+\,2\,\int_{{{\mathbb{S}}^{2}}}{f\,dw\,-\,\log \,\int_{{{\mathbb{S}}^{2}}}{{{e}^{2f}}\,dw\,>\,-\infty ,}}}$$

and that this infimum is attained. This complements recent work of Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang concerning the Moser-Aubin inequality.

In [1], Jean-Philippe Anker conjectures an upper bound for the heat kernel of a symmetric space of noncompact type. We show in this paper that his prediction is verified for the space of positive definite n × n real matrices.

In this note, we propose an extension of the compactness property for Kähler-Einstein metrics to critical metrics of Weyl functional on compact Kähler surfaces.

The main topic of this paper is the study of the fundamental solution of the heat equation for the symmetric spaces of positive definite matrices, Pos(n,R).

Our first step is to develop a “False Abel Inverse Transform” which transforms functions of compact support on an euclidean space into integrable functions on the symmetric space. The transform is shown to satisfy the relation is the usual Laplacian with a constant drift).

Using this transform, we find explicit formulas for the heat kernel in the cases n = 2 and n = 3. These formulas allow us to give the asymptotic development for the heat kernel as t tends to infinity. Finally, we give an upper and lower bound of the same type for the heat kernel.