Related papers: A note on Kazdan-Warner type equations on compact …

Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic,…

Using the convex functions in Grassmannian manifolds we can carry out interior estimates for mean curvature flow of higher codimension. In this way some of the results of Ecker-Huisken can be generalized to higher codimension

We obtain generalized Wintgen inequalities for submanifolds in conformally flat manifolds. We give some applications for submanifolds in a Riemannian manifold of quasi-constant curvature. Equality cases are also considered.

We introduce a flow approach to the generalized Loewner-Nirenberg problem $(1.5)-(1.7)$ of the $\sigma_k$-Ricci equation on a compact manifold $(M^n,g)$ with boundary. We prove that for initial data $u_0\in C^{4,\alpha}(M)$ which is a…

We announce a new proof of the uniform estimate on the curvature of solutions to the Ricci flow on a compact K\"ahler manifold $M^n$ with positive bisectional curvature. In contrast to the recent work of X. Chen and G. Tian, our proof of…

We consider the functional of total variation of maps from an interval into a Riemannian submanifold of $\mathbb R^N$. We define a notion of strong solution to the system of equations corresponding to the $L^2$-gradient flow of this…

We construct a Cauchy type formula on open subdomains of Riemann surfaces

In this paper, we study the partial convexity of smooth solutions to the heat equation on a compact or complete non-compact Riemannian manifold M or Kahler-Ricci flow. We show that under a natural assumption, a new partial convexity…

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.

We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field…

We give some uniform estimates for constant mean curvature solutions of the conformal vacuum Einstein constraint equations on compact manifolds. Existence of those solutions was given in a paper by J. Isenberg.

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence…

A short proof of the convergence of the Kahler-Ricci flow on Fano manifolds admitting a Kahler-Einstein metric or a Kahler-Ricci soliton is given, using a variety of recent techniques

In this note, we prove two Kazdan-Warner type identities involving $v^{(2k)}$, the renormalized volume coefficients of a Riemannian manifold $(M^n,g)$, and $G_{2r}$, the so-called Gauss-Bonnet curvature, and a conformal Killing vector field…

We survey some recent developments on solutions of the K\"ahler-Ricci flow on compact K\"ahler manifolds which exist for all positive times.

We investigate the Riemann problem for the shallow water equations with variable and (possibly) discontinuous topography and provide a complete description of the properties of its solutions: existence; uniqueness in the non-resonant…

In this paper, we give a formulation of the variational iteration method that makes it suitable for the analysis of the solutions of Klein-Gordon equations with variable coefficients. We particularly study a Klein-Gordon problem which has…

We solve the modified Kazdan-Warner problem of finding metrics with prescribed scalar curvature and unit total volume.

We present compacton-like solution of the modified KdV equation and compare its properties with those of the compactons and solitons. We further show that, the nonlinear Schr{\"o}dinger equation with a source term and other higher order…

The self-similar solutions to the mean curvature flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone…