Deformations of harmonic maps with conical singularities

Abstract

We study the deformation theory of harmonic maps with isolated singularities between compact Riemannian manifolds, in the case where all the tangent maps are smooth (away from the origin) and the decay to the tangents is polynomial. We will refer to these as conically singular harmonic maps. Under certain conditions on the Morse index and the nullity of the tangent maps and a non-degeneracy assumption on a weighted kernel of the Jacobi operator, we prove that such harmonic maps persist under a small C²-perturbation of the background metric. In the second part of the paper, we construct examples of conically singular harmonic maps satisfying the assumptions of our deformation theorem. We first prove that the desired conditions on the tangent maps are satisfied by the radial projections ℝ^m ∖ {0} → 𝕊^m−1 (m ≥ 4) and the complex Hopf fibrations ℂⁿ⁺¹ ∖ {0} → ℂℙⁿ (n ≥ 1). We then construct explicit examples of conically singular maps 𝕊^m ∖ {N,S} → 𝕊^m−1, 𝕊⁴ ∖ {N,S} → 𝕊² and ℂℙ² ∖ {[0:0:1]} → ℂℙ¹ which are harmonic and satisfy the assumptions of the deformation theorem with respect to an appropriate metric on the domain and the round metric on the target.

Isolated singularities of harmonic maps with generic boundary data
Xuanyu Li
arXiv24 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEs Geometry & topologyarXiv
Spaces with Multiple Conical Singularities and their Stratifications
Mohammad Alattar, Lewis Tadman
arXiv2 Sep 2026PreprintarXiv licenceFull textGeometry & topologyarXiv
Closed Ricci flows with singularities modeled on asymptotically conical shrinkers
Maxwell Stolarski
Geometry & Topology9 Jul 2026Peer-reviewedCC BYFull text6 citedGeometry & topologyOpenAlex
Deformations, local freeness, and base change for higher Du Bois singularities
Haoming Ning
arXiv7 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Deformations of Compact Calabi--Yau and Fano Varieties with Isolated Singularities
Yohsuke Imagi
arXiv1 Sep 2026PreprintCC BY 4.0Full textAlgebra & number theory Analysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{gutwein2026deformations,
  title = {{Deformations of harmonic maps with conical singularities}},
  author = {Dominik Gutwein and Thibault Langlais},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20588},
  eprint = {2609.20588},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20588},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Deformations of harmonic maps with conical singularities
AU  - Dominik Gutwein
AU  - Thibault Langlais
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20588
UR  - https://arxiv.org/abs/2609.20588
AB  - We study the deformation theory of harmonic maps with isolated singularities between compact Riemannian manifolds, in the case where all the tangent maps are smooth (away from the origin) and the decay to the tangents is polynomial. We will refer to these as conically singular harmonic maps. Under certain conditions on the Morse index and the nullity of the tangent maps and a non-degeneracy assumption on a weighted kernel of the Jacobi operator, we prove that such harmonic maps persist under a small C^2-perturbation of the background metric. In the second part of the paper, we construct examples of conically singular harmonic maps satisfying the assumptions of our deformation theorem. We first prove that the desired conditions on the tangent maps are satisfied by the radial projections ℝ^m ∖ {0} → 𝕊^m−1 (m ≥ 4) and the complex Hopf fibrations ℂ^n+1 ∖ {0} → ℂℙ^n (n ≥ 1). We then construct explicit examples of conically singular maps 𝕊^m ∖ {N,S} → 𝕊^m−1, 𝕊^4 ∖ {N,S} → 𝕊^2 and ℂℙ^2 ∖ {[0:0:1]} → ℂℙ^1 which are harmonic and satisfy the assumptions of the deformation theorem with respect to an appropriate metric on the domain and the round metric on the target.
ER  -

Abstract

Related

Cite