Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.23648

Three-circle inequalities for almost harmonic functions on cylinders

Yuxiang Li·Wenjuan Qi
View original source ↗Read source document ↗Export metadata ↓

Abstract

The three-circle method was introduced into the blow-up analysis of harmonic maps by Qing and Tian [Comm. Pure Appl. Math. 1997]. We extend this method to a class of $(τ,ε)$-almost harmonic functions on long cylinders and establish three-circle inequalities for their weighted energies. These inequalities yield a dichotomy between exponential energy decay and asymptotic dominance of a critical Fourier mode, together with a characterization of the corresponding normalized limiting profiles. In the unweighted case, we prove convergence of the averaged neck curves and identify their limiting equations under additional structural assumptions. Applications include neck analysis and energy identities for harmonic maps, solutions of Rivière's equation, $α$-harmonic maps, biharmonic approximations of harmonic maps, and Willmore surfaces.

Explore related subjects

math.DGmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuxiang Li, Wenjuan Qi. 2026-10-16. Three-circle inequalities for almost harmonic functions on cylinders. https://arxiv.org/abs/2610.23648

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

KEEP EXPLORING

Related papers

arXiv2026

Shannon entropy for harmonic metrics on cyclic Higgs bundles

Let $X$ be a Riemann surface, $K_X$ its canonical bundle, and $T_X=K_X^{-1}$. A diagonal harmonic metric $h=(h_1,\dots,h_r)$ on a cyclic Higgs bundle of rank $r\geq 2$ associated with a holomorphic $r$-differential $q$ induces Hermitian metrics $H_j=h_j^{-1}\otimes h_{j+1}$ on $T_X$ for $j=1,\dots,r-1$, together with a possibly degenerate metric $H_r$ determined by $h_1,h_r$, and $q$. This construction depends on $q$ only through the subharmonic weight $ϕ_q=\frac{1}{r}\log|q|^2$ on $K_X$. In previous work, we extended harmonic metrics to solutions associated with arbitrary subharmonic weight functions $φ$, yielding analogous metrics $H_1,\dots,H_r$. For each nonzero real parameter $β$, we introduce a function on $X$, which we call entropy, that quantifies their mutual misalignment. By extending estimates of Dai--Li and Li--Mochizuki to general subharmonic weight functions, we obtain upper and lower bounds for $β>0$ when $H_1,\dots,H_{r-1}$ are complete and the weight is neither induced by a complete flat Kähler metric nor identically $-\infty$. The upper bound is $\log r$, the constant entropy associated with a complete flat Kähler metric, for which $H_1=\cdots=H_r$. The lower bound is the constant entropy associated with the weight $φ\equiv-\infty$. Furthermore, for every nonzero real $β$, we determine the asymptotic behavior, as $r\to\infty$, of the difference between these two entropy values, showing that it converges to a finite real number if and only if $β>-1$.

math.DG↗
arXiv2026

Complete harmonic metrics and subharmonic functions on the unit disc

Let $X$ be a Riemann surface with canonical bundle $K_X$ and holomorphic tangent bundle $T_X=K_X^{-1}$. For each integer $r\geq 2$, each $q\in H^0(K_X^r)$, and each choice of a square root of $K_X$, we obtain a cyclic Higgs bundle. A diagonal harmonic metric $h=(h_1,\ldots,h_r)$ induces Hermitian metrics $H_j=h_j^{-1}\otimes h_{j+1}$ on $T_X$ for $j=1,\ldots,r-1$, together with a possibly degenerate Hermitian metric $H_r$ determined by $h_1,h_r$, and $q$. We call $h$ complete if the Kähler metrics induced by $H_1,\ldots,H_{r-1}$ are all complete. Li--Mochizuki proved that if $q$ is nonzero unless $X$ is hyperbolic, there exists a unique complete harmonic metric on the corresponding cyclic Higgs bundle satisfying $\det(h)=1$. The Hitchin equation depends on $q$ only through the subharmonic weight $ϕ_q=\frac{1}{r}\log|q|^2$ on $K_X$. We establish uniqueness of complete solutions with determinant $1$ for general subharmonic weight functions $φ$ whose exponentials are $C^2$ outside a compact subset of $X$. On the unit disc $\mathbb D=\{z\in\mathbb C:|z|<1\}$, we also prove that a real complete solution exists for every subharmonic weight function. We further establish a monotone approximation theorem. If $φ_ε$ are smooth subharmonic weight functions on the discs $\mathbb D_ε=\{z\in\mathbb C:|z|<1-ε\}$ decreasing to $φ$ as $ε\searrow0$, the corresponding complete solutions $h_ε$, normalized by $\det(h_ε)=1$, converge monotonically to a complete solution $h$ associated with $φ$.

math.DG↗
arXiv2026

On $q$-convex hypersurfaces in Riemannian manifolds

We prove that any closed, convex hypersurface in an $(n+1)$-dimensional Riemannian manifold with $\lceil \frac{n}{2} \rceil$-positive curvature operator is a rational homology sphere with finite fundamental group. The same conclusion holds for any $\lceil \frac{n}{2} \rceil$-convex hypersurface, provided that the mean curvature satisfies a sharp pinching condition. Both results follow from more general vanishing and estimation theorems for the Betti numbers of closed $q$-convex immersed hypersurfaces in $(n+1)$-dimensional Riemannian manifolds, under a lower bound on the average of the smallest $(n-p)$ eigenvalues of the curvature operator.

math.DG↗