Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.35589

Harmonic maps from spheres with lowest possible index and their properties

Surkov Egor
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this note, we prove that for n > 2 a harmonic map $\mathbb{S}^n \to \mathbb{CP}^m$, of index n+1 is pseudo horizontally weakly conformal, that $n$ must be odd. In particular, for even n every non-constant harmonic map $\mathbb{S}^n \to \mathbb{CP}^m$ has index at least n+2. We also present a harmonic map from $\mathbb{S}^4$ to $\mathrm{Gr}_{\mathbb{C}}(2,4)$ of index 5 that is not pseudo horizontally weakly conformal, so the statement does not extend to Grassmannians.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Surkov Egor. 2026-09-28. Harmonic maps from spheres with lowest possible index and their properties. https://arxiv.org/abs/2609.35589

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Miyaoka-Yau inequality for minimal Kähler klt spaces

Let $X$ be a compact Kähler space with quotient singularities in codimension $2$. In terms of orbifold Chern classes, we prove a generalized Bogomolov-Gieseker inequality for a reflexive coherent sheaf $\mathcal{E}_X$ on $X$ equipped with a Higgs field $θ_{X_{reg}}$ defined on the regular locus of $X$. As a direct application, we prove the Miyaoka-Yau inequality for minimal Kähler spaces. The proof relies on two main ingredients. The first is the existence of $L^p$-approximate critical Hermitian structures for orbifold Higgs bundles over compact Gauduchon orbifolds. The second is the construction of an orbifold Higgs bundle on a suitable orbifold modification $f:W\to X$, agreeing with the reflexive pullback of $(\mathcal{E}_X,θ_{X_{reg}})$ over $f^{-1}(X_{reg})$.

math.DG↗
arXiv2026

Length spectrum rigidity and flexibility of spheres of revolution with one equator

We define a notion of marked length spectrum for $S^1$-symmetric Riemannian metrics on the two-sphere having only one equator. We prove that isospectral metrics in this class have conjugate geodesic flows. Under a further $\mathbb{Z}_2$-symmetry assumption, we show that the marked length spectrum determines the metric. Finally, we show that every isospectral class of metrics contains a unique $\mathbb{Z}_2$-symmetric metric and give an explicit description of this isospectral class as an infinite dimensional convex set, generalizing the known description of $S^1$-symmetric Zoll metrics. This paper contains also two appendices, in which we provide an elementary proof of the fact that a $C^2$ real valued function on an interval is determined by the set of tangent lines to its graph, and we classify a class of $S^1$-invariant contact forms on three-manifolds.

math.DG↗
arXiv2026

Finite-energy equivariant minimal immersions in $\mathbb{CH}^2$ with fixed-lift proper cusp ends

Let $Σ_0\,=\,Σ\setminus\{z_1,\,\ldots,\, z_n\}$ be a finite-type hyperbolic Riemann surface, with its universal cover being identified with $\mathbb H$. We prove that a finite-energy equivariant harmonic map \[ f\colon\mathbb H\;\longrightarrow\;\mathbb{CH}^2 \] is fixed-lift proper at every puncture --- that is, its restriction to a fixed lifted cusp eventually leaves every compact subset of the target --- if and only if its peripheral holonomy is parabolic. For conformal immersions, parabolic holonomy at every puncture also implies completeness of the descended induced metric. We further show that if a polystable parabolic $\mathrm{PU}(2,\,1)$-Higgs bundle has local data $α_p\,=\,s_p\,=\,0$ and $Y_p\,\ne\,0$, and if the associated harmonic map is weakly conformal, then the corresponding end is unbranched, complete, and has finite energy. We also show that, for mixed conformal data, stability is equivalent to two explicit parabolic slope inequalities. Finally, for every $n\;\ge\;5$, we construct explicit Higgs data producing mixed unbranched $\mathbb{CH}^2$-$n$-noids with nonidentity unipotent peripheral monodromy, finite total energy, complete induced metric, and fixed-lift proper ends.

math.DG↗