arXiv subjects

Nguyen Thac Dung

Publications and source records attributed to Nguyen Thac Dung.

18 recordsLinked to original sources

arXiv2025

Rigidity results with curvature conditions from Lichnerowicz Laplacian and applications

The Bochner technique is a classical tool in global differential geometry for proving vanishing and rigidity results by exploiting curvature conditions. Building on recent extensions of this method to complete non-compact settings by Petersen and Wink, we investigate $L^Q$-harmonic tensors with $Q>1$ governed by the Lichnerowicz Laplacian on complete Riemannian manifolds. Our results generalize Bochner-type theorems to the non-compact realm, revealing new geometric rigidity phenomena not visible in compact cases. We establish vanishing theorems under integral curvature bounds and weighted Poincaré inequalities, and derive conditions under which harmonic tensors must vanish. In particular, we show that on Ricci-flat or Einstein manifolds, curvature tensors such as $\mathrm{Rm}$ or the Weyl tensor $W$ vanish identically under natural $L^Q$-integrability and positivity assumptions on the curvature operator. These results imply strong rigidity: flatness in the Ricci-flat case and constant sectional curvature in the Einstein case. We further apply our framework to closed hypersurfaces in space forms and derive vanishing results for intermediate Betti numbers under positivity conditions on the second fundamental form. Finally, we extend our theory to asymptotically locally Euclidean (ALE) spaces, proving that harmonic Weyl tensors and Codazzi tensors must vanish under curvature positivity and decay conditions. Our analysis also links these results to ADM mass rigidity, establishing new obstructions to nontrivial decaying solutions on ALE 4-manifolds.

math.DG↗

arXiv2025

Refined Kato type inequalities and new vanishing theorems on complete Kähler and quaternionic Kähler manifolds

Given a complete Riemannian manifold satisfying a weighted Poincaré inequality and having a bounded below Ricci curvature, various vanishing theorems for harmonic functions and harmonic 1-forms have been published. We generalized these results to $L^p$-integrable pluriharmonic functions and harmonic 1-forms on complete Kähler and quaternionic Kähler manifolds respectively by utilizing the Böchner technique and several refined Kato type inequalities. Moreover, we also prove the vanishing property of pluriharmonic functions with finite $L^p$ energy on complete Kähler manifolds satisfying a Sobolev type inequality.

math.DG↗

arXiv2022

Vanishing results from Lichnerowicz Laplacian on complete Kähler manifolds and applications

In this paper, we show several rigidity results for harmonic $(p,q)$-forms in complete Kähler manifolds. We also give several applications to study non-compact Kähler manifolds with parallel Bochner tensor or quaternion Kähler manifolds. Our results are natural extensions of Petersen and Wink's results in \cite{PW21, PW} in the setting of complete, non-compact Kähler manifolds.

math.DG↗

arXiv2021

Rigidity and vanishing theorems for complete translating solitons

In this paper, we prove some rigidity theorems for complete translating solitons. Assume that the $L^q$-norm of the trace-free second fundamental form is finite, for some $q\in\mathbb{R}$ and using a Sobolev inequality, we show that translator must be hyperspace. Our results can be considered as a generalization of \cite{Ma, WXZ16, Xin15}. We also investigate a vanishing property for translators which states that there are no nontrivial $L_f^p\ (p\geq2)$ weighted harmonic $1$-forms on ${M}$ if the $L^n$-norm of the second fundamental form is bounded.

math.DG↗

arXiv2021

Gradient estimates for weighted harmonic function with Dirichlet boundary condition

We prove a Yau's type gradient estimate for positive $f$-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces with compact boundary when the infinite dimensional Bakry-Emery Ricci tensor and the weighted mean curvature are bounded below. As an application, we give a Liouville type result for bounded $f$-harmonic functions with the Dirichlet boundary condition. Our results do not depend on any assumption on the potential function $f$.

math.DG↗

arXiv2021

Sharp gradient estimates on weighted manifolds with compact boundary

In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.

math.DG↗

arXiv2020

Gradient estimates for weighted $p$-Laplacian equations on Riemannian manifolds with a Sobolev inequality and integral Ricci bounds

In this paper, we consider the non-linear general $p$-Laplacian equation $Δ_{p,f}u+F(u)=0$ for a smooth function $F$ on smooth metric measure spaces. Assume that a Sobolev inequality holds true on $M$ and an integral Ricci curvature is small, we first prove a local gradient estimate for the equation. Then, as its applications, we prove several Liouville type results on manifolds with lower bounds of Ricci curvature. We also derive new local gradient estimates provided that the integral Ricci curvature is small enough.

math.DG↗

arXiv2018

Sharp gradient estimates for a heat equation in Riemannian manifolds

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation $u_t=Δu+au\log u$ in complete noncompact Riemannian manifolds. As its application, we show that if $u$ is a positive solution of the equation $u_t=Δu$ and $\log u$ is of sublinear growth in both spatial and time directions then $u$ must be constant. This gradient estimate is sharp since it is well-known that $u(x,t)=e^{x+t}$ satisfying $u_t=Δu$. We also emphasize that our results are better than those given by Jiang (\cite{XJ16}), Souplet-Zhang (\cite{SZ06}), Wu (\cite{Wu15, Wu17}), and others.

math.DG↗

arXiv2018

Gradient estimates and Liouville type theorems for Poisson equations

In this paper, we will address to the following parabolic equation $$ u_t=Δ_fu + F(u) $$ on a smooth metric measure space with Bakry-Émery curvature bounded from below. Here $F$ is a differentiable function defined in $\mathbb{R}$. Our motivation is originally inspired by gradient estimates of Allen-Cahn and Fisher equations (\cite{Bai17, CLPW17}). In this paper, we show new gradient estimates for these equations. As their applications, we obtain Liouville type theorems for positive or bounded solutions to the above equation when either $F=cu(1-u)$ (the Fisher equation) or; $F=-u^3+u$ (the Allen-Cahn equation); or $F=au\log u$ (the equation involving gradient Ricci solitons).

math.DG↗

arXiv2017

Gradient estimates for some $f$-heat equations driven by Lichnerowicz's equation on complete smooth metric measure spaces

Given a complete, smooth metric measure space $(M,g,e^{-f}dv)$ with the Bakry-Émery Ricci curvature bounded from below, various gradient estimates for solutions of the following general $f$-heat equations $$ u_t=Δ_f u+au\log u+bu +Au^p+Bu^{-q} $$ and \[ u_t=Δ_f u+Ae^{pu}+Be^{-pu}+D \] are studied. As by-product, we obtain some Liouville-type theorems and Harnack-type inequalities for positive solutions of several nonlinear equations including the Schrödinger equation, the Yamabe equation, and Lichnerowicz-type equations as special cases.

math.DG↗

arXiv2017

Vanishing properties of $p$-harmonic $\ell$-forms on Riemannian manifolds

In this paper, we show several vanishing type theorems for $p$-harmonic $\ell$-forms on Riemannian manifolds ($p\geq2$). First of all, we consider complete non-compact immersed submanifolds $M^n$ of ${N}^{n+m}$ with flat normal bundle, we prove that any $p$-harmonic $\ell$-forms on $M$ is trivial if $N$ has pure curvature tensor and $M$ satisfies some geometric condition. Then, we obtain a vanishing theorem on Riemannian manifolds with weighted Poincaré inequality. Final, we investigate complete simply connected, locally conformally flat Riemannian manifolds $M$ and point out that there is no nontrivial $p$-harmonic $\ell$-form on $M$ provided that $\operatorname{Ric}$ has suitable bound.

math.DG↗

arXiv2017

The number of cusps of complete Riemannian manifolds with finite volume

In this paper, we will count the number of cusps of complete Riemannian manifolds $M$ with finite volume. When $M$ is a complete smooth metric measure spaces, we show that the number of cusps in bounded by the volume $V$ of $M$ if some geometric conditions hold true. Moreover, we use the nonlinear theory of the $p$-Laplacian to give a upper bound of the number of cusps on complete Riemannian manifolds. The main ingredients in our proof are a decay estimate of volume of cusps and volume comparison theorems.

math.DG↗

arXiv2016

Gradient estimates for some evolution equations on complete smooth metric measure spaces

In this paper, we consider the following general evolution equation $$ u_t=Δ_fu+au\log^αu+bu $$ on smooth metric measure spaces $(M^n, g, e^{-f}dv)$. We give a local gradient estimate of Souplet-Zhang type for positive smooth solution of this equation provided that the Bakry-Émery curvature bounded from below. When $f$ is constant, we investigate the gereral evolution on compact Riemannian manifolds with no nconvex boundary satisfying an "\emph{interior rolling $R$-ball}" condition. We show a gradient estimate of Hamilton type on such manifolds.

math.DG↗

arXiv2015

Local and global sharp gradient estimates for weighted $p$-harmonic functions

Let $(M^n, g, e^{-f}dv)$ be a smooth metric measure space of dimensional $n$. Suppose that $v$ is a positive weighted $p$-eigenfunctions associated to the eigenvalues $λ_{1,p}$ on $M$, namely $$ e^{f}div(e^{-f}|\nabla v|^{p-2}\nabla v)=-λ_{1,p}v^{p-1}.$$ in the distribution sense. We first give a local gradient estimate for $v$ provided the $m$-dimmensional Bakry-Émery curvature $Ric_f^{m}$ bounded from below. Consequently, we show that when $Ric_f^m\geq0$ then $v$ is constant if $v$ is of sublinear growth. At the same time, we prove a Harnack inequality for weighted $p$-harmonic functions. Moreover, we show global sharp gradient estimates for weighted $p$-eigenfunctions. Then we use these estimates to study geometric structures at infinity when the first eigenvalue $λ_{1,p}$ obtains its maximal value. Our achievements generalize several results proved ealier by Li-Wang, Munteanu-Wang,...(\cite{LW1, LW2, MW1, MW2})

math.DG↗

arXiv2015

Gradient estimates of Hamilton - Souplet - Zhang type for a general heat equation on Riemannian manifolds

The purpose of this paper is to study gradient estimate of Hamilton - Souplet - Zhang type for the general heat equation $$ u_t=Δ_V u + au\log u+bu $$ on noncompact Riemannian manifolds. As its application, we show a Harnak inequality for the heat solution and a Liouville type theorem for a nonlinear elliptic equation. Our results are an extention and improvement of the work of Souplet - Zhang (\cite{SZ}), Ruan (\cite{Ruan}), Yi Li (\cite{Yili}) and Huang-Ma (\cite{HM}).

math.DG↗

arXiv2015

A note on a Sung-Wang's paper

The purpose of this note is to study the connectedness at infinity of manifold by using the theory of $p$-harmonic functions. We show that if the first eigenvalue $λ_{1,p}$ for the $p$-Laplacian achievies its maximal value on a Kähler manifold or a quaternionic Kähler manifold then such a manifold must be connected at infinity unless it is a topological cylinder with an explicit warped product metric.

math.DG↗

arXiv2014

A generalization of almost Schur lemma on CR manifolds

In this paper, we study a general almost Schur Lemma on pseudo-Hermitian (2n+1)-manifolds $(M,J,θ)$ for $n\geq2$. When the equality of almost Schur inequality holds, we derive the contact form $θ$ is pseudo-Einstein and the pseudo-Hermitian scalar curvature is constant.

math.DG↗

arXiv2011

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface $M$ has sufficiently small total scalar curvature then $M$ has only one end. We also obtain a vanishing theorem for $L^2$ harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.

math.DG↗