Related papers: The Fu-Yau equation with negative slope parameter
We study an equation proposed by Fu and Yau as a natural $n$-dimensional generalization of a Strominger system that they solved in dimension $2$. It is a complex Hessian equation with right hand side depending on gradients. Building on the…
We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $\alpha'$. Actually we obtain at the same time a solution of the open case $\alpha'>0$, an improved solution of the known case $\alpha'<0$, and solutions for a family…
In this paper, we prove the existence of solutions to the Fu-Yau equation on compact K\"{a}hler manifolds. As an application, we give a class of non-trivial solutions of the modified Strominger system.
In this paper, we study the Fu-Yau equation on compact Hermitian manifolds and prove the existence of solutions of equation on astheno-K\"ahler manifolds. We also prove the uniqueness of solutions of Fu-Yau equation when the slope parameter…
In this note, we study the curvature flow to Nirenberg problem on $S^2$ with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed…
We describe how the Schoen-Yau proof of the positive mass theorem can be extended to arbitrary dimensions. To overcome the problem of singularities, we propose a new inductive scheme. To carry out the inductive step, we use a combination of…
We introduce an iterative scheme to solve the Yamabe equation $ - a\Delta_{g} u + S u = \lambda u^{p-1} $ on small domains $(\Omega,g)\subset {\mathbb R}^n$ equipped with a Riemannian metric $g$. Thus $g$ admits a conformal change to a…
In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set $\mathcal{S}$ of suitable weak solutions and the parameter $\alpha$ in the nonlinear term in the following…
Dealing with the generalized Calabi-Yau equation proposed by Gromov on closed almost-K\"ahler manifolds, we extend to arbitrary dimension a non-existence result proved in complex dimension 2.
This article surveys the development of the SYZ conjecture since it was proposed by Strominger, Yau and Zaslow in their famous 1996 paper, and discusses how it has been leading us to a thorough understanding of the geometry underlying…
We consider the following equations: \begin{equation*} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2}u(x)=f(v(x)), \\ (-\triangle)^{\beta/2}v(x)=g(u(x)), &x \in R^{n},\\ u,v\geq 0, &x \in R^{n}, \end{array} \right. \end{equation*} for…
The Wu--Yau theorem asserts that a compact K\"ahler manifold with negative holomorphic sectional curvature admits a cohomologous metric with negative Ricci curvature. We introduce a conjectural positive analog of the Wu--Yau theorem and…
We study the existence of non--trivial solutions to the Yamabe equation: $$-\Delta u+ a(x)= \mu u|u|^\frac4{n-2} \hbox{} \mu >0, x\in \Omega \subset {\mathbf R}^n \hbox{with} n\geq 4,$$ $$ u(x)=0 \hbox{on} \partial \Omega$$ under weak…
In this paper we investigated the point-wise time-space estimates of the fundamental solution for higher order Schr\"odinger equation. These estimates improve the result of Yao [2] and generalize the one of Cui [1].
The aim of this note is to clarify the structure of nontrivial global solutions with nonpositive ADM mass for the static, spherically symmetric Einstein-Yang-Mills equations with SU(2) gauge group. The presented numerical results…
By employing the higher dimensional version of the Wu-Yang Ansatz we obtain black hole solutions in the spherically symmetric Einstein-Yang-Mills (EYM) theory. Although these solutions were found recently by other means, our method provides…
We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n , [h])$ of a Poincar\'e-Einstein manifold $(X^{n+1} , g^+ )$ with either $n = 2$ or $n \geq 3$ and $(M^n , [h])$ is…
We show that a problem by Yau can not be true in general. The counterexamples are constructed based on the recent work of Wu and Zheng.
This article is devoted to the mathematical study of a new Navier-Stokes-alpha model with a nonlinear filter equation. For a given indicator function, this filter equation was first considered by W. Layton, G. Rebholz, and C. Trenchea to…
The Einstein-Yang-Mills equations are the source of many interesting solutions within general relativity, including families of particle-like and black hole solutions, and critical phenomena of more than one type. These solutions,…