Related papers: Liouville type theorem for several generalized map…

This article provides an overview on various conservation laws for polyharmonic maps between Riemannian manifolds. Besides recalling that the variation of the energy for polyharmonic maps with respect to the domain metric gives rise to the…

We prove an extension of Eells and Sampson's rigidity theorem for harmonic maps from a closed manifold of non-negative Ricci curvature to a manifold of non-positive sectional curvature. We give an application of our result in the setting of…

In this paper, we study functional and geometric inequalities on complete Finsler measure spaces under the condition that the weighted Ricci curvature ${\rm Ric}_\infty$ has a lower bound. We first obtain some local uniform Poincar\'{e}…

We present a new, short and independent proof of the Liouville-type theorem for entire and subharmonic functions of finite order bounded outside some set of zero planar density.

Fifty years ago, Eells and Sampson have proved a famous theorem in which they argued that any harmonic mapping $f:(M,g) \rightarrow (\bar{M},\bar{g})$ is totally geodesic if $(M, g)$ is a compact manifold with the nonnegative Ricci tensor…

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the…

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(\Omega, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of…

In this paper, we study Riemannian maps whose base manifolds admit a Ricci soliton and give a non-trivial example of such a Riemannian map. First, we find Riemannian curvature tensor for the base manifolds of Riemannian map $F$. Further, we…

We prove a quantification result for harmonic maps with free boundary from arbitrary Riemannian surfaces into the unit ball of ${\mathbb R}^{n+1}$ with bounded energy. This generalizes results obtained by Da Lio on the disc.

A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed, these disparities can now be resolved by…

In this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion.…

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These…

The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at…

We investigate monotonicity properties of $p$-harmonic vector bundle-valued $k$-forms by studying the energy-momentum tensor associated with such a form. As a consequence, we obtain a unified proof of the monotonicity formul{\ae} for…

We study harmonic mappings from a Riemannian manifold $N$ into a principal $G$-bundle $P$ endowed with a $G$-invariant Riemannian metric (i.e. a Kaluza-Klein metric). These morphisms are called Kaluza-Klein harmonic maps and naturally lead…

We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields $a$ in the context of stochastic homogenization. The large-scale regularity of $a$-harmonic functions is…

Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and…

We propose global surjectivity theorems of differentiable maps based on second order conditions. Using the homotopy continuation method, we demonstrate that, for a $C^2$ differentiable map from a Hilbert space to a finite-dimensional…

Consider the following nonlinear Neumann problem \[ \begin{cases} \text{div}\left(y^{a}\nabla u(x,y)\right)=0, & \text{for }(x,y)\in\mathbb{R}_{+}^{n+1}\\ \lim_{y\rightarrow0+}y^{a}\frac{\partial u}{\partial y}=-f(u), & \text{on…

In this paper, we study a general class of Hessian elliptic equations, including the Monge-Amp\`ere equation, the $k$-Hessian equation and $p$-Monge-Amp\`ere equations. We propose new additional condition on the solution and prove Liouville…