arXiv subjects

Guangwen Zhao

Publications and source records attributed to Guangwen Zhao.

5 recordsLinked to original sources

arXiv2026

A refined Schwarz lemma for $V$-harmonic maps

In this note, we establish a refined Schwarz lemma for $V$-harmonic maps. Specifically, we prove that if $u$ is a $V$-harmonic map of generalized dilatation of order $β$ from a complete Riemannian manifold with Bakry--Émery Ricci curvature bounded below by a constant $-A$ to a Riemannian manifold with sectional curvature bounded above by a negative constant $-B$, then $$ u^*h\le \frac{Ac(β)}{BD(β)}g \le \frac{Aβ^2}{B}g, $$ where $c(β)=β^2/(1+β^2)$ and $D(β)=1-\lfloor 1/c(β)\rfloor c(β)^2-(1-\lfloor 1/c(β)\rfloor c(β))^2$. Equality in the second inequality holds if and only if $β=1,\ 1/\sqrt{2},\ 1/\sqrt{3}, \cdots $. Our result improves the previous bounds obtained by Shen (J. Reine Angew. Math., 1984) for harmonic maps and by Chen--Li--Qiu (Nonlinear Anal., 2022) for $f$-harmonic maps. We also present some applications of our main theorem.

math.DG↗

arXiv2026

$L^p$ Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons

We study Liouville-type theorems for real-valued pluriharmonic functions on complete gradient Kähler-Ricci solitons under gradient integrability assumptions. For a complete gradient Kähler-Ricci soliton $(M,g,J,f)$ and a real-valued pluriharmonic function $u$, we investigate conditions under which $u$ must be constant. By introducing a globally defined holomorphic quantity induced by the soliton potential, we obtain new Liouville-type results beyond the range available for harmonic functions. In the steady case, we prove that $u$ is constant whenever $$ \int_M|\nabla u|^p\mathrm{d}v<\infty $$ for some $0<p<\infty$. In the shrinking case, we prove the same conclusion for $0<p\leq 2$. Finally, we construct a complete Kähler example showing that the extension to the range $0<p<1$ relies essentially on the soliton structure and does not hold on general complete Kähler manifolds.

math.DG↗

arXiv2024

Hermitian pluriharmonic maps between almost Hermitian manifolds

In the case where both the domain and target manifolds are almost Hermitian, we introduce the concept of Hermitian pluriharmonic maps. We prove that any holomorphic or anti-holomorphic map between almost Hermitian manifolds is Hermitian pluriharmonic. We also establish some monotonicity formulae for the partial energies of Hermitian pluriharmonic maps into Kähler manifolds. As an application, under appropriate assumptions on the growth of the partial energies, some holomorphicity results are proven.

math.DG↗

arXiv2024

Gradient estimates for positive eigenfunctions of $ \mathcal{L} $-operator on conformal solitons and its applications

We prove a local gradient estimate for positive eigenfunctions of $ \mathcal{L} $-operator on conformal solitons given by a general conformal vector field. As an application, we obtain a Liouville type theorem for $ \mathcal{L} u = 0 $, which improves the one of Li--Sun (Acta Math. Sin. (Engl. Ser.), 37(11): 1768--1782, 2021.). We also consider applications where manifolds are special conformal solitons. Especially in the case of self-shrinkers, a better Liouville type theorem is obtained.

math.DG↗

arXiv2019

Gradient estimates and Harnack inequalities of a parabolic equation under geometric flow

In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equation \[ (Δ-q(x,t)-\partial_t)u(x,t)=A(u(x,t)),\quad (x,t)\in M\times [0,T]. \] We establish space-time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Finally, as applications, we give gradient estimates of some specific parabolic equations.

math.DG↗