Existence and stability of weak critical points of $r$-energy functionals
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| Format: | Preprint |
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2026
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| _version_ | 1866918484424785920 |
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| author | Montaldo, Stefano Ratto, Andrea Sanna, Antonio |
| author_facet | Montaldo, Stefano Ratto, Andrea Sanna, Antonio |
| contents | The main aim of this paper is to prove the existence of certain proper weakly $r$-harmonic ($ES-r$-harmonic) maps. We construct critical points which belong to a family of rotationally symmetric maps $φ_a : B^n \to \mathbb{S}^n$, where $B^n$ and $\mathbb{S}^n$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. We find that the existence of solutions within this family is restricted to specific dimensions $n$. Next, we prove that our critical points are \textit{unstable}. In the course of this analysis we point out some specific differences between the $r$-harmonic and the $ES-r$-harmonic cases when $r \geq 4$. Next, we analyse two variants of the problem. First, we replace the target manifold $\mathbb{S}^n$ with a rotationally symmetric ellipsoid $E^n(b)$ and establish the existence of proper weakly biharmonic maps for all $n \geq 5$, as well as proper weakly triharmonic maps for all $n \geq 7$. Finally, we study a similar problem replacing the domain $B^n$ with a suitable warped product manifold. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_03532 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence and stability of weak critical points of $r$-energy functionals Montaldo, Stefano Ratto, Andrea Sanna, Antonio Differential Geometry The main aim of this paper is to prove the existence of certain proper weakly $r$-harmonic ($ES-r$-harmonic) maps. We construct critical points which belong to a family of rotationally symmetric maps $φ_a : B^n \to \mathbb{S}^n$, where $B^n$ and $\mathbb{S}^n$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. We find that the existence of solutions within this family is restricted to specific dimensions $n$. Next, we prove that our critical points are \textit{unstable}. In the course of this analysis we point out some specific differences between the $r$-harmonic and the $ES-r$-harmonic cases when $r \geq 4$. Next, we analyse two variants of the problem. First, we replace the target manifold $\mathbb{S}^n$ with a rotationally symmetric ellipsoid $E^n(b)$ and establish the existence of proper weakly biharmonic maps for all $n \geq 5$, as well as proper weakly triharmonic maps for all $n \geq 7$. Finally, we study a similar problem replacing the domain $B^n$ with a suitable warped product manifold. |
| title | Existence and stability of weak critical points of $r$-energy functionals |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.03532 |