Let us consider two closed surfaces $ℳ$, $𝒩$ of class $C^1$ and two functions $\varphi :ℳ\to ℝ$, $\psi :𝒩\to ℝ$ of class $C^1$, called measuring functions. The natural pseudodistance $d$ between the pairs $(ℳ,)$, $(𝒩,\psi )$ is defined as the infimum of $\Theta \left(f\right):={max}_{P\in ℳ}\left|\varphi \left(P\right)-\psi \left(f\left(P\right)\right)\right|$ as $f$ varies in the set of all homeomorphisms from $ℳ$ onto $𝒩$. In this paper we prove that the natural pseudodistance equals either $|c_1-c_2|$, $\frac{1}{2}\left|c_1-c_2\right|$, or $\frac{1}{3}\left|c_1-c_2\right|$, where $c_1$ and $c_2$ are two suitable critical values of the measuring functions. This shows that a previous relation between the natural pseudodistance and critical values...

We give some new methods to construct nonharmonic biharmonic maps in the unit n-dimensional sphere 𝕊ⁿ.

We compute numerically the minimizers of the Dirichlet energy$$E\left(u\right)=\frac{1}{2}{\int }_{B^2}{\left|\nabla u\right|}^2\mathrm{d}x$$among maps $u:B^2\to S^2$ from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous $P_1$ finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned...

We compute numerically the minimizers of the Dirichlet energy $$E\left(u\right)=\frac{1}{2}{\int }_{B^2}{\left|\nabla u\right|}^2\mathrm{d}x$$ among maps $u:B^2\to S^2$ from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which...