Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.05181

Schwarz lemmas and rigidity of \(\bar\partial_b\)-harmonic maps

Xin Huang·Hui Liu
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove Schwarz lemmas for \(\bar\partial_b\)-harmonic maps from complete pseudo-Hermitian manifolds into K"ahler manifolds of negative curvature. For maps of bounded generalized dilatation we bound the full and the horizontal differential. For CR maps a coupled Bochner identity yields rank-dependent horizontal estimates and a bound for the Reeb differential. On Sasakian sources and complex hyperbolic targets we classify the maps attaining the horizontal bound, bound the deviation from equality in integral form by the deficit at one point, and prove compactness of almost extremal sequences. As an application, a Siu-type divergence identity yields foliation, pluriharmonicity and rank rigidity under volume growth conditions.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xin Huang, Hui Liu. 2026-10-04. Schwarz lemmas and rigidity of \(\bar\partial_b\)-harmonic maps. https://arxiv.org/abs/2610.05181

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

KEEP EXPLORING

Related papers

arXiv2026

The Lipschitz-volume rigidity problem for metric manifolds

We prove a Lipschitz-volume rigidity result for $1$-Lipschitz maps of non-zero degree between metric manifolds (metric spaces homeomorphic to a closed oriented manifold) and Riemannian manifolds. The proof is based on degree theory and recent developments of Lipschitz-volume rigidity for integral currents.

math.DG↗
arXiv2026

Positive sectional curvature and non-isometric circle actions on a family of eleven-spheres

We study the construction of positively curved Riemannian metrics by non-isometric circle actions. For a family of homotopy eleven-spheres whose Eells--Kuiper invariants form the even subgroup of $\mathbb{Z}/992$, we construct, on each member, a smooth background metric $q$ and three effective circle actions with generators $W_1,W_2,W_3$ such that the metric determined by $g^{-1}=q^{-1}+\sum_{a=1}^3W_a\otimes W_a$ has positive sectional curvature. Each action is non-isometric for every partial metric, including its incoming metric and the final metric. We first describe the sphere by gauge transformations of the quaternionic Hopf bundle. We then construct compatible metrics on two disks and smooth their inverse metrics while preserving the action formula. A local conjugation makes the circle actions non-isometric. For each fixed member, we obtain an explicit positive lower bound $2^{-54}(1+M_{0,k}+M_{1,k})^{-28}$, where $M_{0,k}$ and $M_{1,k}$ are norms of the curvature and its first covariant derivative for its fixed connection. The bound may depend on the member of the family.

math.DG↗
arXiv2026

A solution to Lu's second gap conjecture

Let $M^n\to\mathbb{S}^{n+q}(1)$ be a closed connected minimal immersion, where $n\ge3$, and set $Q=S+λ_2$, with $S=|h|^2$ and $λ_2$ the second largest eigenvalue of Lu's fundamental matrix. We determine the sharp codimension range for Lu's second-gap conjecture. For every $2\le q\le n$, there exists $γ_{n,q}>0$ such that, if $Q$ is constant and $Q>n$, then $Q\ge n+γ_{n,q}$. Conversely, for every $q\ge n+1$, we construct closed connected homogeneous minimal embeddings, followed when necessary by totally geodesic inclusions, with constant scalar curvature and constant $Q$-values dense in $(n,2n)$. Thus, in every dimension $n\ge3$, Lu's conjecture holds precisely for $q\le n$. Combined with the theorem of Peng-Terng for hypersurfaces and the recent resolution of the two-dimensional case, this gives a complete resolution of Lu's second-gap conjecture: for every $n\ge2$, the conjecture holds exactly when $q\le n$ and fails when $q\ge n+1$. In codimension two we further obtain the explicit admissible gap $γ_{n,2}=\exp(-10^{16}n^2)$.

math.DG↗