Related papers: Gradient estimates via two-point functions for ell…
Let $(M^n,g)$ be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\Delta u+au\log u=0,$$ where $a$ is a nonzero…
In this paper, by slightly modifying Li-Yau's technique so that we can handle drifting Laplacians, we were able to find three different gradient estimates for the warping function, one for each sign of the Einstein constant of the fiber…
Let M be a smooth compact manifold without boundary. We consider two smooth Sub-Semi-Riemannian metrics on M. Under suitable conditions, we show that they are almost conformally isometric in an Lp sense. Assume also that M carries a…
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex…
We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci…
We generalize the Benjamini-Pemantle-Peres estimate relating hitting probability and Martin capacity to the setting of manifolds with Ricci curvature bounded below. As applications we obtain: (1) a sharp estimate for the probability that…
Consider a sample of $n$ points taken i.i.d from a submanifold $\Sigma$ of Euclidean space. We show that there is a way to estimate the Ricci curvature of $\Sigma$ with respect to the induced metric from the sample. Our method is grounded…
In this paper, we establish second order estimates for a general class of fully nonlinear equations with linear gradient terms on compact almost Hermitian manifolds. As an application, we first prove the existence of solutions for the…
We study the Ricci-Bourguignon flow on warped product manifolds with noncompact base. This setting leads naturally to a parabolic partial differential equation on the space of smooth warping functions, arising from the necessary and…
We review recent results on the study of the isoperimetric problem on Riemannian manifolds with Ricci lower bounds. We focus on the validity of sharp second order differential inequalities satisfied by the isoperimetric profile of possibly…
Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and…
In this paper, we prove sharp estimates and existence results for anisotropic nonlinear elliptic problems with lower order terms depending on the gradient. Our prototype is: $ \left\{ \begin{array}{ll} -\mathcal Q_{p}u =[H(Du)]^{q}+f(x)…
Let $M^n$ be an $n$-dimensional Riemannian manifold with boundary $\partial M$. Assume that Ricci curvature is bounded from below by $(n-1)k$, for $k\in \RR$, we give a sharp estimate of the upper bound of $\rho(x)=\dis(x, \partial M)$, in…
Conditions, related to Kulkarni's equivalence problem are considered for indefinite Riemannian and Kaehlerian manifolds. Corresponding theorems are obtained for the values of the Ricci tensor on isotropic vectors as well as for the values…
In this note, we extend the rigidity of Cheng-Yau gradient estimate in \cite{HXY} to surfaces with lower Ricci curvature bound. Motivated by these sharp Cheng-Yau gradient estimates, pointwise Cheng-Yau gradient estimates for higher…
We obtain sharp estimates for heat kernels and Green's functions on complete noncompact Riemannian manifolds with Euclidean volume growth and nonnegative Ricci curvature. We will then apply these estimates to obtain sharp Moser-Trudinger…
We are concerned with fully nonlinear elliptic equations on complex manifolds and search for technical tools to overcome difficulties in deriving a priori estimates which arise due to the nontrivial torsion and curvature, as well as the…
We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…
In this paper, we consider the following Dirichlet problem for the fully nonlinear elliptic equation of Grad-Mercier type under asymptotic convexity conditions \begin{equation*} \left\{ \begin{array}{ll} F(D^2u(x),Du(x),u(x),x)=g(|\{y\in…
We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully…