Harmonic Mappings into non-negatively curved Riemannian manifolds

Abstract: Fifty years ago, Eells and Sampson have proved a famous theorem in which they argued that any harmonic mapping $f:(M,g) \rightarrow (\bar{M},\bar{g})$ is totally geodesic if $(M, g)$ is a compact manifold with the nonnegative Ricci tensor and the section curvature of $(\bar{M},\bar{g})$ is nonpositive. Moreover, other main results of the theory of harmonic mappings "in the large" are the results on harmonic maps into nonpositively curved Riemannian manifolds. In our paper we develop a theory of harmonic mappings into Riemannian manifolds with nonnegative sectional curvature. In particular, we will prove that any harmonic map between Riemannian manifolds $f:(M,g) \rightarrow (\bar{M},\bar{g})$ is totally geodesic if the section curvature of $(\bar{M},\bar{g})$ is nonnegative and $(M, g)$ is a compact manifold with the Ricci tensor $Ric \geq f^{*}\bar{Ric}$ for the pullback $f^{*}\bar{Ric}$ of the Ricci tensor $\bar{Ric}$ by $f$. The above scheme will be extended to a harmonic mapping of a complete manifold to a manifold with the nonnegative sectional curvature. Moreover, we will obtain interesting corollaries from our results.

• The paper establishes that harmonic maps from compact Riemannian manifolds into targets with non-negative curvature are totally geodesic under specific Ricci curvature conditions.
• It applies the Bochner technique and Weitzenböck formula to derive novel vanishing and rigidity theorems for both harmonic and holomorphic maps.
• The unique results on the Ricci tensor and prescribed curvature problem offer new insights applicable to Einstein manifolds and geometric flows like Ricci flow.

Harmonic Maps into Non-Negatively Curved Riemannian Manifolds

Introduction

This work by Stepanov and Tsyganok systematically develops the theory of harmonic mappings between Riemannian manifolds with an explicit focus on the case when the target manifold admits non-negative sectional curvature, a significant contrast to the classical setting where the target manifold has non-positive curvature. The paper further leverages the Bochner technique to establish vanishing and rigidity theorems, yielding novel insights into the structure of harmonic and holomorphic maps, particularly with respect to almost Kählerian and nearly-Kählerian manifolds. The theoretical results also provide progress on uniqueness and non-existence questions in the prescribed Ricci curvature problem.

Bochner Technique and Main Results

The analytic foundation is given by the Bochner method, applied to the Weitzenböck formula for harmonic maps. Classically, vanishing theorems for harmonic maps critically depend on non-positive curvature assumptions for the target manifold. Stepanov and Tsyganok, however, demonstrate that analogous vanishing and rigidity theorems can be obtained in the opposite regime (non-negative sectional curvature) under suitable Ricci curvature conditions on the domain.

Their primary result establishes that a harmonic mapping $f : (M,g) \rightarrow (N,h)$ from a compact Riemannian manifold with Ricci tensor $\mathrm{Ric} \geq f^*\mathrm{Ric}$ into a target $(N,h)$ with $\mathrm{sec}_h \geq 0$ is necessarily totally geodesic with constant energy density. Furthermore, if there exists a point where $\mathrm{Ric} > f^*\mathrm{Ric}$, such a mapping must be constant. This provides a direct analogue and, under certain conditions, a reversal of the celebrated Eells–Sampson theorem, which applies to the non-positive curvature case.

Harmonic Contractions and Isometric Immersions

The investigation of harmonic contraction mappings (i.e., weakly length-decreasing maps) yields strong rigidity. For instance, any harmonic contraction from a compact Riemannian manifold with $\mathrm{Ric} > 0$ into a target of non-negative sectional curvature is shown to be totally geodesic, and under strict inequalities constant. This includes the case of mappings into the standard sphere. The results generalize to isometric immersions: if the target has positive curvature at infinity, harmonic immersions of compact manifolds are precluded.

Holomorphic Maps and Almost Kählerian Manifolds

The theory is extended to the holomorphic category. Any holomorphic map from a compact almost semi-Kählerian manifold with $\mathrm{Ric} \geq f^*\mathrm{Ric}$ into a nearly-Kählerian target with $\mathrm{sec} \geq 0$ is totally geodesic if the energy is finite; it is constant when the Ricci inequality is strict at some point. This underlines the rigidity of holomorphic and harmonic maps in positive curvature settings, providing new structural results for maps between almost Hermitian and nearly-Kählerian manifolds.

Uniqueness Theorems for the Ricci Tensor

In tackling the prescribed Ricci curvature problem, the authors show the Levi-Civita connection is uniquely determined by the Ricci tensor under non-negative curvature and $\mathrm{Ric} = g$. Existence and uniqueness results follow, as does non-existence under $\mathrm{sec} < n-1$. In dimension 3, if $0 \leq \mathrm{Ric} \leq \frac{1}{2}sg$ (with $s$ the scalar curvature and $g$ the metric), the same uniqueness holds, and no second metric with prescribed Ricci $g$ exists if $s < 2$ everywhere. These extend classical results of De Turck, Koiso, and Hamilton.

Implications and Prospects

The theoretical contributions of this paper decisively extend the landscape of harmonic map theory beyond the classical non-positive curvature terrain. The vanishing and rigidity results proven here have far-reaching implications, such as providing strong obstructions against the existence of non-trivial harmonic maps in the presence of non-negative (or positive) sectional curvature. This has direct impact on the geometry and topology of Riemannian manifolds, the study of minimal submanifolds, and the theory of Kähler and almost Kählerian structures.

The uniqueness results for metrics with prescribed Ricci tensors are particularly notable for the study of Einstein manifolds and for geometric flows, such as Ricci flow, which relies on the interplay between curvature conditions and the rigidity of the geometric structure. The rigidity for holomorphic mappings has potential applications in complex geometry, particularly in moduli problems and in understanding superrigidity-type phenomena.

Future research may focus on refining the Bochner method to handle more general curvature conditions, exploring the sharpness of the Ricci curvature bounds, and investigating analogous vanishing phenomena in the presence of lower regularity or more general geometric structures (e.g., in Lorentzian geometry or calibrated geometries).

Conclusion

This work advances the analytic and geometric understanding of harmonic mappings and their rigidity under non-negative curvature assumptions. By systematically developing the corresponding vanishing theorems and uniqueness results, it establishes deep connections between curvature, analysis, and the topology of mappings. The results serve as an important counterpart to the classical theory and open several avenues for future research in geometric analysis on Riemannian manifolds.