Keita Kunikawa, Yohei Sakurai
In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation.
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In this paper, by employ the cutoff function and the maximum principle, some Hamilton-Souplet-Zhang type gradient estimates for porous medium type equation are deduced. As a special case, an Hamilton-Souplet-Zhang type gradient estimates of the heat equation is derived which is different from the result of Souplet-Zhang. Furthermore, our results generalize those of Zhu. As application, some Livillous theorems for ancient solution are derived.
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}).
In this paper, we consider gradient estimates for two type of nonlinear parabolic equations under the Ricci flow: one is the equation $$u_t=Δu+au\log u+bu$$ with $a,b$ two real constants, the other is $$u_t=Δu+λu^α$$ with $λ,α$ two real constants. By a suitable scaling for the above two equations, we obtain Hamilton-Souplet-Zhang type gradient estimates.
In this article we derive both Hamilton type and Souplet-Zhang type gradient estimations for a system of semilinear equations along a geometric flow on a weighted Riemannian manifold.
In this paper, we study the nonlinear parabolic equation with two exponents on complete noncompact Riemannian maniflods. The special types of such equation include the Fisher-KPP equation, the parabolic Allen-Cahn equation and the Newell-Whitehead equation. We get the Souplet-Zhang's gradeint estimates for the positive solutions to such equation. We also obtain the Liouville theorem for positive ancient solutions. Our results extend those of Souplet-Zhang (Bull. London. Math. Soc. 38:1045-1053, 2006) and Zhu (Acta Mathematica Scientia 36B(2): 514-526, 2016).
In this paper, we study elliptic gradient estimates for a nonlinear $f$-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton's and Souplet-Zhang's gradient estimates for positive solutions to the nonlinear $f$-heat equation only assuming the Bakry-Émery Ricci tensor is bounded below. As applications, we prove parabolic Liouville properties for some kind of ancient solutions to the nonlinear $f$-heat equation. Some special cases are also discussed.
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.
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.
In this paper, motivated by the work of Qi S. Zhang in [28], we derive Li-Yau gradient bounds for positive solutions of the f-heat equation on closed manifolds with Bakry-Emery Ricci curvature bounded below.
In this paper, we prove Souplet-Zhang type gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with the compact boundary under the Dirichlet boundary condition when the Bakry-Emery Ricci tensor and the weighted mean curvature are both bounded below. As an application, we obtain a new Liouville type result for some space-time functions on such smooth metric measure spaces. These results generalize previous linear equations to a nonlinear case.
One of the main results of the paper arXiv:1108.0967 by Gross-Tosatti-Zhang establishes estimates on the collapsing of Ricci-flat Kahler metrics on holomorphic torus fibrations. We remove a projectivity assumption from these estimates and simplify some of the underlying analysis.
Let $(M,g)$ be a complete non-compact Riemannian manifold with the $m$-dimensional Bakry-Émery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation \[ u_t=Δu-\nablaϕ\cdot\nabla u-au\log u-bu, \] where $ϕ$ is a $C^2$ function, and $a\neq0$ and $b$ are two real constants. This work generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).
A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound
本文针对负Ricci下界闭流形上的热方程正解，提出用经典极大值原理直接推导尖锐Li-Yau估计的方法，简化了已有证明，并将思路推广至Ricci流下的热方程与共轭热方程，得到带最优系数的Li-Yau型估计。
We give an elementary proof to the asymptotic expansion formula of Rochon-Zhang for the unique complete Kähler-Einstein metric of Cheng-Yau, Kobayashi, Tian-Yau and Bando on quasi-projective manifolds. The main tools are the solution formula for second order ODE's with constant coefficients and spectral theory for Laplacian operator on a closed manifold.
Gradient estimates for the porous medium type equations and fast diffusion type equations on complete noncompact metric measure space with compact boundary
该研究在带紧边界的完备非紧度量测度空间及流形上，分别推导了两类含漂移拉普拉斯算子的非线性扩散方程的Li-Yau梯度估计、Souplet-Zhang型估计与局部梯度估计。
On the Miyaoka-Yau inequality for manifolds with nef anti-canonical line bundle
本文在 K. Zhang 工作的基础上，结合 Fujita–Odaka 的 delta 不变量下界条件，研究具有 nef 反典范线丛的射影流形上的 Miyaoka–Yau 型不等式。
We study the fast diffusion equation (FDE) with a linear forcing term under the Ricci flow on complete manifolds with bounded curvature and nonnegative curvature operator. We prove Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of the FDE. In addition, we use similar method to prove certain Li-Yau-Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained by X. Cao and R. Hamilton, X. Cao, and S. Kuang and Q. Zhang to noncompact setting.
$(\varepsilon, δ)$--Quasi-Negative Curvature and Positivity of the Canonical Bundle
本研究推广拟负曲率概念提出$(\u03b5,\u03b4)$--拟负性，结合实双截曲率与加权正交Ricci曲率得到典范丛第一陈类n次积分正性的间隙型定理，拓展了已有相关研究结果。
Let $(X,J)$ be an almost-complex manifold. In \cite{li-zhang} Li and Zhang introduce $H^{(p,q),(q,p)}_J(X)_{\rr}$ as the cohomology subgroups of the $(p+q)$-th de Rham cohomology group formed by classes represented by real pure-type forms. Given a proper, surjective, pseudo-holomorphic map between two almost-complex manifolds we study the relationship among such cohomology groups. Similar results are proven in the symplectic setting for the cohomology groups introduced in \cite{tsengyauI} by Tseng and Yau and a new characterization of the Hard Lefschetz condition in dimension $4$ is provided.
A transcendental non-Archimedean Calabi--Yau Theorem with applications to the cscK problem
本文证明紧致Kähler流形上模型一致K-稳定性蕴含唯一cscK度量，通过发展超越非阿基米德多重位势理论，推广了Calabi-Yau定理。
Families of singular Kähler-Einstein metrics
本文通过改进Yau和Kolodziej的技术，建立了退化复Monge-Ampère方程的精确估计，用于研究奇异凯勒-爱因斯坦度量族的性质及模空间扩展。
Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons
本文证明了非紧复流形上在无穷远处Ricci曲率趋于零的收缩梯度Kähler-Ricci孤立子在双全纯同构下的唯一性，且该方法可广泛应用于锥形Calabi-Yau度量唯一性及极化Fano纤维化同构的证明。
Deformations of Fano manifolds with weighted solitons
本文研究Fano流形上加权孤立子的形变问题，给出了加权孤立子可沿Kuranishi族小形变延拓的等价条件，推广了Cao-Sun-Yau-Zhang关于Kaehler-Einstein度量的相关结果。
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