Back to harmonic mappings of compact Riemannian manifolds

Sergey Stepanov; Irina Tsyganok

Home/Archive/arXiv

arXiv · 2507.08105

Back to harmonic mappings of compact Riemannian manifolds

Sergey Stepanov·Irina Tsyganok

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we address several interconnected problems in the theory of harmonic maps between Riemannian manifolds. First, we present necessary background and establish one of the main results of the paper: a criterion characterizing when a smooth submersion or diffeomorphism between Riemannian manifolds is harmonic. This result provides a useful analytic condition for verifying the harmonicity of geometric mappings. Second, we investigate the L2-orthogonal decomposition of the pullback metric associated with a harmonic map. We analyze the structure of this decomposition and discuss its geometric implications, particularly in the context of the energy density and trace conditions. Finally, we study harmonic symmetric bilinear forms and harmonic Riemannian metrics. Special attention is given to their role in the theory of harmonic identity maps. We derive new results that link these notions and demonstrate how they contribute to the broader understanding of harmonicity in geometric analysis.

Explore related subjects

math.DG

Keep this discovery

BibTeX RIS

Sergey Stepanov, Irina Tsyganok. 2025-07-10. Back to harmonic mappings of compact Riemannian manifolds. https://arxiv.org/abs/2507.08105

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

KEEP EXPLORING

Related papers

arXiv2026

The $q$-deformed cross-ratio: modular invariants and Coxeter friezes

We introduce and study a scalar $q$-deformation of the cross-ratio on $\mathbb P^1(\mathbb Q)$. Our construction is based on the notion of $q$-deformed rational numbers due to Morier-Genoud and the author. The $q$-cross-ratio is invariant under $\mathrm{PSL}(2,\mathbb{Z})$, while elements of determinant $-1$ of $\mathrm{PGL}(2,\mathbb{Z})$ act by $q\mapsto q^{-1}$. A principal result is its relation to $q$-deformed Coxeter friezes associated with rational polygons. The expansion at $q=e^h$ yields an algebraically independent sequence of modular invariants and relative invariants, although this sequence does not separate modular orbits. We compute the first two nonconstant coefficients of this expansion explicitly.

math.DG↗

arXiv2026

The Cartan-Hadamard conjecture in dimension five

We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, in the spirit of Banchoff-Pohl, together with an estimate for Jacobi fields along geodesic chords. The inequality persists for boundaries of isoperimetric regions in geodesic balls, whose mean curvature is constant only on the free part. An isoperimetric-profile argument, after Kleiner, completes the proof. Our method also gives a new proof in dimension $3$.

math.DG↗

arXiv2026

On static manifolds with boundary admitting a nowhere-vanishing static potential

We study complete static manifolds with boundary admitting a nowhere-vanishing static potential. Our main result shows that, under a natural lower bound relating the scalar curvature and the boundary mean curvature, a simple static manifold with boundary must in fact have positive scalar curvature, negative boundary mean curvature, and be compact; we also obtain explicit relations and estimates involving the volume of the manifold and the geometry of its boundary. In the scalar-flat case, we prove global splitting and Ricci-flat rigidity results, including for disconnected boundary, while in the negative scalar curvature case we establish a sharp mean-curvature bound and characterize the equality case by an exponential warped-product structure. The proofs rely essentially on the study of the associated Einstein manifold. In appendix we derive several identities for static manifolds with boundary and discuss the associated Einstein manifold technique in the boundaryless setting.

math.DG↗