arXiv subjects

Chang-Yu Guo

Publications and source records attributed to Chang-Yu Guo.

At least 19 recordsLinked to original sources

arXiv2026

Uniformizing non-proper Gromov Hyperbolic Spaces

In this paper, we extend a large part of the uniformization theory of Bonk-Heinonen-Koskela [Asterisque 2001] to length spaces that are not necessarily proper or geodesic. Among other things, we show that there is a one-to-one correspondence between the quasiisometry classes of complete roughly starlike Gromov hyperbolic spaces and the quasisimilarity classes of bounded uniform spaces, which provides an affirmative solution to an open question of Bonk-Heinonen-Koskela. Our approach relies crucially on the work of Väisälä [Expo. Math. 2005], who investigated in depth Gromov hyperbolic spaces that are not necessarily proper or geodesic. One key new ingredient is to use the so-called (quasihyperbolic) $(c,μ)$-quasigeodesic as a suitable substitute for quasihyperbolic geodesic.

math.CV↗

arXiv2026

Energy identity for Intrinsic Stationary Biharmonic Mappings into Homogeneous Spaces in Supercritical Dimensions

In this paper, we consider energy identity for intrinsic stationary biharmonic maps into homogeneous spaces in supercritical dimensions, extending the corresponding result of Hornung-Moser [Anal. PDE. 2012] in critical dimension. The proof follows a similar strategy as that of Lin-Rivière [Duke Math. J. 2002]. A key ingredient is a conservation law for intrinsic biharmonic maps into homogeneous spaces, which allow us to derive higher regularity of the map.

math.AP↗

arXiv2026

Dimension-free inner uniform estimates for quasigeodesics

In this paper, we establish a dimension-free inner uniform estimate for quasigeodesics. More precisely, we prove that a $c_0$-quasigeodesic in a $δ$-Gromov hyperbolic $c$-John domain in $\mathbb{R}^n$ is $b$-inner uniform, for some constant $b$ depending only on $c_0$, $δ$ and $c$, but not on the dimension $n$. The proof relies crucially on the techniques introduced by Guo-Huang-Wang in their recent work [arXiv:2502.02930, 2025]. In particular, we actually show that the above result holds in general Banach spaces, which answers affirmatively an open question of J. Väisälä in [Analysis, 2004] and partially addresses the open question of Bonk-Heinonen-Koskela in [Asterisque, 2001]. As a byproduct of our main result, we obtain that a $c_0$-quasigeodesic in a $δ$-Gromov hyperbolic $c$-John domain in $\mathbb{R}^n$ is a $b$-cone arc with a dimension-free constant $b=b(c_0,δ,c)$. This resolves an open problem of J. Heinonen in [Rev. Math. Iberoam., 1989].

math.CV↗

arXiv2026

The dimension-free Gehring-Hayman inequality for quasigeodesics

A well-known theorem of J. Heinonen and S. Rohde in 1993 states that if $D\subset \mathbb{R}^n$ is quasiconformally equivalently to an uniform domain, then the Gehring-Hayman inequality holds in $D$: quasihyperbolic geodesics in $D$ minimizes the Euclidean length among all curves in $D$ with the same end points, up to a universal dimension-dependent multiplicative constant. In this paper, we develop a new approach to strengthen the above result in the following three aspects: 1) obtain a dimension-free multiplicative constant in the Gehring-Hayman inequality; 2) relax the class of quasihyperbolic geodesics to more general quasigeodesics; 3) relax the quasiconformal equivalence to more general coarsely quasihyperbolic equivalence. As a byproduct of our general approach, we are able to prove that the above improved Gehring-Hayman inequality indeed holds in Banach spaces. This answers affirmatively an open problem raised by J. Heinonen and S. Rohde in 1993 and reformulated by J. Väisälä in 2005.

math.CV↗

arXiv2026

Energy identity for stationary biharmonic mappings into spheres in supercritical dimensions

Energy identity for harmonic type maps in supercritical dimensions is an important and difficult problem. For sphere-valued harmonic maps, the first breakthrough was achieved by Lin-Rivière [Duke Math. J. 2002]. In this paper, by adapting their strategy, we establish the energy identity for stationary biharmonic maps into spheres in supercritical dimensions $n\ge 5$.

math.AP↗

arXiv2026

Global weak solution of 3-D focusing energy-critical nonlinear Schrödinger equation

In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schrödinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some class of initial data. The main ingredient of our new approach is to use solutions of an energy-critical Ginzburg-Landau equation as approximations for the corresponding nonlinear Schördinger equation. In our proofs, we first show the dichotomy of global well-posedness versus finite time blow-up of energy-critical Ginzburg-Landau equation in $\dot{H}^1( \mathbb{R}^d)$ for $d = 3,4 $ when the energy is less than the energy of the stationary solution $W$. We follow the strategy of C. E. Kenig and F. Merle [25,26], using a concentration-compactness/rigidity argument to reduce the global well-posedness to the exclusion of a critical element. The critical element is ruled out by dissipation of the Ginzburg-Landau equation, including local smoothness, backwards uniqueness and unique continuation. The existence of global weak solution of the three dimensional focusing energy-critical nonlinear Schrödinger equation in the non-radial case then follows from the global well-posedness of the energy-critical Ginzburg-Landau equation via a limitation argument. We also adapt the arguments of M. Struwe [37,38] to prove the weak-strong uniqueness when the $\dot{H}^1$-norm of the initial data is bounded by a constant depending on the stationary solution $W$.

math.AP↗

arXiv2025

Characterizations of quasihyperbolic John domains and uniform domains in metric spaces

In a recent work of Zhou and Ponnusamy [Ann. Sc. Norm. Super. Pisa Ci. Sci. 2025], the authors studied the following natural question: find sufficient and necessary conditions for a domain $Ω$ in a metric space $X$ to be quasihyperbolic John. It was proved that Gromov hyperbolic John domains are quasihyperbolic John, quantitatively. As an application, they obtained a characterization of uniform domains in Ahlfors regular spaces. In a recent work, using a deep improved characterization of Gromov hyperbolicity, Guo, Huang and Wang [arXiv 2025] proved the quantitative equivalence bteween inner uniformity and the quasihyperbolic John condition in metric doubling spaces. However, the proof does not yield a similar characterization for uniform domains. In this article, we find a new elementary approach to successfully extend the above characterization to uniform domains: a domain $Ω$ in a doubling length space $X$ is uniform if and only if it is linearly locally connected (LLC) and satisfies the ball separation condition, if and only if it is LLC-1 and quasihyperbolic John, quantitatively. This substantially improved the corresponding results of Zhou and Ponnusamy. Our new approach also allows us to give an alternative proof of the inner uniformity result of Guo-Huang-Wang without using the improved characterization on Gromov hyperbolicity.

math.CV↗

arXiv2025

Gromov hyperbolicity III: an improved geometric characterization and its applications

In the seminal work of Balogh-Buckley [Invent. Math. 2003], the authors asked the following fundamental open problem: for proper subdomains in the Euclidean space $\mathbb{R}^n$, does the ball separation condition alone imply the Gehring-Hayman inequality? In this paper, via a completely new measure-independent approach, we establish the following geometric characterization of Gromov hyperbolicity in a fairly general setting: The Gromov hyperbolicity of a proper subdomain in a doubling metric space is quantitatively equivalent to the geometric ball separation condition, with explicit dependence on the coefficients. In the special case of Euclidean spaces, it affirmatively solves the above Balogh-Buckely problem. Our result also significantly improves the main result of Koskela-Lammi-Manojlović [Ann. Sci. Éc. Norm. Supér. 2014]. As applications, we obtain the quasiconformal invariance of ball separation condition, a geometric characterization of inner uniformity in terms of ball separation condition, and the Gromov hyperbolicity of quasihyperbolic John length spaces.

math.CV↗

arXiv2025

Quantitative stratification and optimal regularity for harmonic almost complex structures

In a recent interesting work [15], W.Y. He established the important partial regularity theory and the almost optimal higher regularity theory for energy minimizing harmonic almost complex structures. Based on a new observation on the structure of equations, we give an easier new proof of the partial regularity theorem, and adapting the powerful quantitative stratification method of Naber-Valtorta [22], we further prove the rectifiability of singular stratum of energy minimizing harmonic almost complex structures. Based on this, we establish an optimal regularity theory, which improves the corresponding result of He.

math.AP↗

arXiv2025

A note on weak compactness of $Ω$-Yang-Mills connections

In this note, applying a compensation compactness argument developped by Chen and Giron (arXiv.2108.13529) on Yang-Mills fields, we extends their weak continuity result to the more general class of $Ω$-Yang-Mills connections on principle bundles over compact Riemannian manifold.

math.DG↗

arXiv2025

The Lamm-Rivière system II: energy identity

In this paper, we establish an angular energy quantization for the following fourth order inhomogeneous Lamm-Rivière system $$ Δ^2u=Δ(V\cdot\nabla u)+\text{div}(w\nabla u)+W\cdot\nabla u+f $$ in dimension four, with an inhomogeneous term $f\in L\log L$.

math.AP↗

arXiv2024

$L^p$-regularity of a geometrically nonlinear system in supercritical dimensions

In a recent work, Gastel and Neff introduced an interesting system from a geometrically nonlinear flat cosserat micropolar model and established interior regularity in the critical dimension. Inspired by their work on this flat Cosserat model, in this article, we establish both interior regularity and sharp $L^p$ regularity for their system in supercritical dimensions.

math.AP↗

arXiv2024

Global well-posedness of the energy-critical complex Ginzburg-Landau equation in exterior domains

In this article, we consider an energy-critical complex Ginzburg-Landau equation in the exterior of a smooth compact strictly convex obstacle. We prove the global well-posedness of energy-critical complex Ginzburg-Landau equation in an exterior domain by the concentration-compactness/rigidity theorem method. As corollaries of our main result, we establish both the existence of global weak solutions to energy-critical defocusing nonlinear Schrödinger equations and the global well-posedness theory for energy-critical semi-linear heat equation in exterior domains.

math.AP↗

arXiv2024

Quasiregular mappings between equiregular SubRiemannian manifolds

In this paper, we provide an alternative appraoch to an expectaion of Fässler et al [J. Geom. Anal. 2016] by showing that a metrically quasiregular mapping between two equiregular subRiemannian manifolds of homogeneous dimension $Q\geq 2$ has a negligible branch set. One main new ingredient is to develop a suitable extension of the generalized Pansu differentiability theory, in spirit of earlier works by Margulis-Mostow, Karmanova and Vodopyanov. Another new ingredient is to apply the theory of Sobolev spaces based on upper gradients developed by Heinonen, Koskela, Shanmugalingam and Tyson to establish the necessary analytic foundations.

math.CV↗

arXiv2024

Optimal higher regularity for biharmonic maps via quantitative stratification

This little note is devoted to refining the almost optimal regularity results of Breiner and Lamm \cite{Breiner-Lamm-2015} on minimizing and stationary biharmonic maps via the powerful quantitative stratification method introduced by Cheeger and Naber \cite{Cheeger-Naber-2013} and further developed by Naber and Valtorta \cite{Naber-V-2017,Naber-V-2018} for harmonic maps. In particular, we obtain an optimal regularity results for minimizing biharmonic maps.

math.AP↗

arXiv2023

Sharp Morrey regularity for an even order elliptic system

In this short note, we establish a sharp Morrey regularity theory for an even order elliptic system of Rivière type: \begin{equation*} Δ^{m}u=\sum_{l=0}^{m-1}Δ^{l}\left\langle V_{l},du\right\rangle +\sum_{l=0}^{m-2}Δ^{l}δ\left(w_{l}du\right)+f\qquad \text{in} B^{2m} \end{equation*} under minimal regularity assumptions on the coefficients functions V^l, w^l and that f belongs to certain Morrey space. This can be regarded as a further extension of the recent L^p-regularity theory obtained by Guo-Xiang-Zheng [15], and generalizes [7, 27] for second and fourth order elliptic systems.

math.AP↗

arXiv2023

Conservation law of harmonic mappings in supercritical dimensions

In this short note, we provide a partial extension of Rivière's convervation law in higher dimensions under certain Lorentz integrability condition for the connection matrix. As an application, we obtain a conservation law for weakly harmonic mappings around regular points in supercritical dimensions.

math.AP↗

arXiv2023

A note to "Radial limits of quasiregular Local Homeomorphisms''

In this short note, we consider quasiregular local homeomorphisms on uniform domains. We prove that such mappings always can be extended to some boundary points along John curves, which extends the corresponding result of Rajala [Amer. J. Math. 2008].

math.CV↗