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Xiangzhi Cao

Publications and source records attributed to Xiangzhi Cao.

At least 19 recordsLinked to original sources
arXiv2025

A remark on Symphonic maps

In this paper, we are devoted to define p symphonic morphism and characterize it partially as in the case of harmonic morphism.

math.DG↗
arXiv2025

Symphonic map from ellipsoid to ellipsoid

In this paper, we proved the existence of Symphonic map from ellipsoid to ellipsoid. We also geive give Hopf construction of Symphonic map from ellipsoid to ellipsoid.

math.DG↗
arXiv2025

Heat flow for VT harmonic map from compact manifold

In this paper, we obtain the existence of Dirichlet problem for VT harmonic map from compact Riemannian manifold with or without boundary into compact manifold via the heat flow method. We also obtain the existence of V T geodesics uncer certain conditions on T.

math.DG↗
arXiv2025

Schwarz Lemma for VT harmonic map

In this paper, we obtained Schwarz Lemma of $ VT $ harmonic map including distance decreasing property and volume decreasing property under some conditions about the eigenvalue of $ du^{+} \circ du $, $ T $ and the lower bound of $ Ric_f^N $ or $ Ric_V^N $. We generalized Schwarz lemma of $ V $ harmonic map.

math.DG↗
arXiv2025

A Liouville theorem of VT-harmonic map heat flow

We proved an Liouville theorem for Backward V T-harmonic map heat flow from evolution manifolds into generalized regular ball. Among others, we also proved an Liouville theorem for V T-harmonic map heat flow from complete manifolds into generalized regular ball.

math.DG↗
arXiv2025

Stability for $ Φ_{S, F,H} $ harmonic map and $ Φ_{T, F,H} $ harmonic map

In this paper, we mainly consider the stability of $ Φ_{S, F,H} $ harmonic map and $ Φ_{T,F,H} $ harmonic map from or into $ Φ$-SSU manifold. We mainly consider the stability of $ Φ_{S, F,H} $ harmonic map and $ Φ_{T,F,H} $ harmonic map from or into compact convex hypersurface. We also give some Theorems to know when a manifold is $ Φ_{S, F,H} $ -stable or $ Φ_{S, F,H} $ -unstable.

math.DG↗
arXiv2024

Gradient estimates of nonlinear equation on complete noncompact metric measure space with compact boundary

In this paper, firstly, we study gradient estimates for positive solution of the following equation \begin{equation*} Δ_ξ(u)-\partial_t u- q u =A(u),t\in (-\infty,\infty) \end{equation*} on metric measure space $ (M,g,e^{-ξ}\mathrm{d} v_g)$ with boundary , where $ Δ_ξ=Δ+\left \langle \nabla\cdot , \nabla ξ\right \rangle $. For this equation, we derive Li-Yau type gradient estimates and Hamilton's type gradient estimates. Secondly, we obtain gradient estimates for positive solution of the following elliptical type equation \begin{equation} Δ_ξ(u)- q u =A(u)\end{equation} on complete noncompact metric measure space $(M,g,e^{-ξ}\mathrm{d} v_g)$ with boundary.

math.DG↗
arXiv2024

Gradient estimates for the porous medium type equations and fast diffusion type equations on complete noncompact metric measure space with compact boundary

In the paper, we derive Li-Yau gradient estimates and Souplet Zhang type estimates of the following equation \begin{equation*} \begin{split} u_t= Δ_ξp+λu+A(u) , \end{split} \end{equation*} on complete noncompact metric measure space $ (M, g,e^{-ξ}dv_g) $ with compact boundary. We will also give the local gradient estimates of the equation \begin{equation*} Δ_ξ(u^p)+λu+A(u)=0, \end{equation*} on complete noncompact manifold with compact boundary.

math.DG↗
arXiv2023

A remark of Ricci-Bourguignon harmonic soliton

In this paper, we investigate the triviality of Ricci-Bourguignon harmonic solitons. We also use the results of V-harmonic map to investigate the property of Ricci harmonic soliton.

math.DG↗
arXiv2022

Liouville type theorem for several generalized maps between Riemannian manifold

In this paper, we mainly derive monotonicity formula of generalized map using conservation law, including $ϕ$-$F$ harmonic map coupled with $ϕ$-$F$ symphonic map with $m$ form and potential from metric measure space, $ p $ harmonic map with potential , $ V $ harmonic map with potential. As an corollary, we can derive Liouville theorem for these maps under some finite energy conditons. We also get Liouville type theorem for $ϕ$-$F$ harmonic map coupled with $ϕ$-$F$ symphonic map under asymptotic conditon on metric measure space. We also get Liouville theorem for $ϕ$-$F$-$V$-harmonic maps in terms of the upper bound of Ricci curvature and the bound about sectional curvature on metric measure space. We also get Liouville theorem for $ϕ$-$ F $-harmonic map without using monotonicity formula on metric measure space.

math.DG↗
arXiv2022

Vanishing theorem and finiteness theorem for $p$-harmonic $1$ form

In this paper, we will show vanishing theorem of $p$ harmonic $1$ form on submanifold $M$ in $ \bar{M} $ whose BiRic curvature satisfying $ \overline{\mathrm{BiRic}}^a \geq Φ_a(H,S) $. As an corollary, we can get the corresponding theorem for $ p $ harmonic function and $ p $ harmonic map. We also investigate the finiteness problem of $p$ harmonic $1$ form on submanifold $M$ in $ \bar{M} $ whose BiRic curvature satisfying $ \overline{\mathrm{BiRic}}^a \geq -k^2 $.

math.DG↗