[Paper Review] Hermitian Harmonic maps and non-degenerate curvatures

Kefeng Liu, Xiaokui Yang|arXiv (Cornell University)|Feb 15, 2014

Geometric Analysis and Curvature Flows24 references3 citations

TL;DR

This paper establishes rigidity and analyticity results for Hermitian harmonic and pluri-harmonic maps from compact Hermitian manifolds to Kähler or Riemannian target manifolds with non-degenerate curvature. Using refined Bochner formulas on non-Kähler Hermitian manifolds, it proves that such maps are holomorphic, anti-holomorphic, or constant under curvature and rank conditions, generalizing classical results of Siu, Jost-Yau, and Sampson to non-Kähler settings.

ABSTRACT

In this paper, we study the existence of various harmonic maps from Hermitian manifolds to Kaehler, Hermitian and Riemannian manifolds respectively. By using refined Bochner formulas on Hermitian (possibly non-Kaehler) manifolds, we derive new rigidity results on Hermitian harmonic maps from compact Hermitian manifolds to Riemannian manifolds, and we also obtain the complex analyticity of pluri-harmonic maps from compact complex manifolds to compact Kaehler manifolds (and Riemannian manifolds) with non-degenerate curvatures, which are analogous to several fundamental results in [28,Siu], [14,Jost-Yau] and [26, Sampson].

Motivation & Objective

• To extend classical rigidity theorems for harmonic maps to Hermitian (non-Kähler) manifolds.
• To establish conditions under which Hermitian harmonic or pluri-harmonic maps are holomorphic, anti-holomorphic, or constant.
• To generalize Siu’s and Jost-Yau’s results on harmonic maps to non-Kähler domains using curvature and rank constraints.
• To investigate the role of non-degenerate curvature and the $δ$-condition ($\partial\overline{\partial}\omega^{m-2}_h = 0$) in map rigidity.
• To clarify distinctions between various harmonic map types (classical, $\partial$-, $\overline{\partial}$-, Hermitian) on Hermitian manifolds.

Proposed method

• Derives refined Bochner formulas for Hermitian manifolds with non-trivial torsion, adapting techniques from Eells-Sampson and Jost-Yau.
• Uses the Hermitian harmonic map equation: $-h^{\alpha\overline{\beta}}\left(\frac{\partial^2 f^i}{\partial z^\alpha \partial \overline{z}^\beta} + \Gamma_{jk}^i \frac{\partial f^j}{\partial \overline{z}^\beta} \frac{\partial f^k}{\partial z^\alpha}\right) = 0$.
• Applies generalized divergence-free structures: $\left(\overline{\partial}_E - 2\sqrt{-1}\partial^*\omega_h\right)^*\left(\overline{\partial}f\right) = 0$.
• Introduces the concept of non-degenerate curvature (Definition 4.1), generalizing Siu’s strongly positive curvature.
• Applies Stokes’ Theorem and $(1,1)$-form integration to show $\omega_0 = 0$ under topological or curvature constraints.
• Uses curvature contraction identities: $R_{ijk\ell} \left( h^{\alpha\overline{\beta}} \frac{\partial f^i}{\partial z^\alpha} \frac{\partial f^\ell}{\partial \overline{z}^\beta} \right) \left( h^{\gamma\overline{\delta}} \frac{\partial f^j}{\partial z^\gamma} \frac{\partial f^k}{\partial \overline{z}^\delta} \right) = 0$ to constrain rank.

Experimental results

Research questions

• RQ1 Under what conditions are Hermitian harmonic maps from compact Hermitian manifolds to Riemannian manifolds holomorphic or constant?
• RQ2 Can the holomorphicity rigidity result of Siu be extended to non-Kähler Hermitian domains?
• RQ3 What role does the $\partial\overline{\partial}\omega^{m-2}_h = 0$ condition (astheno-Kähler) play in map rigidity?
• RQ4 How does non-degenerate curvature on the target manifold constrain the rank of the differential of pluri-harmonic maps?
• RQ5 When are pluri-harmonic maps from compact complex manifolds to Kähler or Riemannian manifolds necessarily constant?

Key findings

• If $f: (M,h) \to (N,g)$ is a Hermitian harmonic map with $\partial\overline{\partial}\omega_h^{m-2} = 0$ and $(N,g)$ has strongly Hermitian-negative curvature, then $\text{rank}_{\mathbb{R}} df \leq 2$, and no immersion exists into constant negative curvature manifolds if $\dim_{\mathbb{C}} M > 1$.
• Pluri-harmonic maps $f: M \to (N,g)$ from compact complex manifolds to compact Kähler manifolds with non-degenerate curvature are holomorphic or anti-holomorphic if $\text{rank}_{\mathbb{R}} df \geq 4$.
• If $\text{rank}_{\mathbb{R}} df < 2m$, then any pluri-harmonic map to a Kähler or Riemannian manifold with non-degenerate curvature is constant.
• For Calabi-Eckmann manifolds $\mathbb{S}^{2p+1} \times \mathbb{S}^{2q+1}$, any pluri-harmonic map to $N(c)$ is constant if $p+q \geq n$.
• Pluri-harmonic maps from $\mathbb{CP}^m$ to $\mathbb{CP}^n$ are constant if $m > n$, and from $\mathbb{CP}^n$ to real space form $N(c)$ are constant if $n \geq 2$.
• If the target manifold has non-degenerate Hermitian curvature at a point, then $\text{rank}_{\mathbb{R}} df(p) \leq 2$ for any pluri-harmonic map $f: M \to (N,g)$.