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Register a new userYau Type Gradient Estimates For $Delta u + au(log u)^{p}+bu=0$ On Riemannian Manifolds
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In this paper, we consider the gradient estimates of the positive solutions to the following equation defined on a complete Riemannian manifold $(M, g)$ $$Delta u + au(log u)^{p}+bu=0,$$ where $a, bin mathbb{R}$ and $p$ is a rational number with $p=frac{k_1}{2k_2+1}geq2$ where $k_1$ and $k_2$ are positive integer numbers. we obtain the gradient bound of a positive solution to the equation which does not depend on the bounds of the solution and the Laplacian of the distance function on $(M, g)$. Our results can be viewed as a natural extension of Yaus estimates on positive harmonic function.
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In this short note, we use a unified method to consider the gradient estimates of the positive solution to the following nonlinear elliptic equation $Delta u + au^{p+1}=0$ defined on a complete noncompact Riemannian manifold $(M, g)$ where $a > 0$ and $ p <frac{4}{n}$ or $a < 0$ and $p >0$ are two constants. For the case $a>0$, this improves considerably the previous known results except for the cases $dim(M)=4$ and supplements the results for the case $dim(M)leq 2$. For the case $a<0$ and $p>0$, we also improve considerably the previous related results. When the Ricci curvature of $(M,g)$ is nonnegative, we also obtain a Liouville-type theorem for the above equation.
In this paper, we consider a class of important nonlinear elliptic equations $$Delta u + a(x)ulog u + b(x)u = 0$$ on a collapsed complete Riemannian manifold and its parabolic counterpart under integral curvature conditions, where $a(x)$ and $b(x)$ are two $C^1$-smooth real functions. Some new local gradient estimates for positive solutions to these equations are derived by Mosers iteration provided that the integral Ricci curvature is small enough. Especially, some classical results are extended by our estimates and a few interesting corollaries are given. Furthermore, some global gradient estimates are also established under certain geometric conditions. Some estimates obtained in this paper play an important role in a recent paper by Y. Ma and B. Wang [17], which extended and improved the main results due to B. Wang [29] to the case of integral Ricci curvature bounds.
We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear isotropic elliptic equations on compact Riemannian manifolds, depending only on dimension and a lower bound for the Ricci curvature. These estimates imply sharp gradient bounds relating the gradient of an arbitrary solution at given height to that of a symmetric solution on a warped product model space. We also discuss the problem on Finsler manifolds with nonnegative weighted Ricci curvature, and on complete manifolds with bounded geometry, including solutions on manifolds with boundary with Dirichlet boundary condition. Particular cases of our results include gradient estimates of Modica type.
An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of $(-1,1)$. Such a 3-manifold $M$ admits a foliation of parallel surfaces, whose locus in Teichm{u}ller space is represented as a path $gamma$, we show that $gamma$ joins the conformal structures of the two components of the conformal boundary of $M$. Moreover, we obtain an upper bound for the Teichm{u}ller distance between any two points on $gamma$, in particular, the Teichm{u}ller distance between the two components of the conformal boundary of $M$, in terms of the principal curvatures of the minimal surface in $M$. We also establish a new potential for the Weil-Petersson metric on Teichm{u}ller space.
The purpose of this paper is to study a complete orientable minimal hypersurface with finite index in an $(n+1)$-dimensional Riemannian manifold $N$. We generalize Theorems 1.5-1.6 (cite{Seo14}). In 1976, Schoen and Yau proved the Liouville type theorem on stable minimal hypersurface, i.e., Theorem 1.7 (cite{SchoenYau1976}). Recently, Seo (cite{Seo14}) generalized Theorem 1.7 (cite{SchoenYau1976}). Finally, we generalize Theorems 1.7 (cite{SchoenYau1976}) and 1.8 (cite{Seo14})
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