Non-linear partial differential equations in conformal geometry

In this article, we first show that given a smooth function $ S $ either on closed manifolds $ (M, g) $ or compact manifolds $ (\bar{M}, g) $ with non-empty boundary, both for dimensions at least $ 3 $, the condition $ S \equiv 0 $, or $ S $ changes sign and $ \int_{M} S \dvol < 0 $ (with zero mean curvature if the boundary is not empty), is both the necessary and sufficient condition for prescribing scalar curvature problems within conformal class $ [g] $, provided that the ...

We lay out the foundations of the theory of second-order conformal superintegrable systems. Such systems are essentially Laplace equations on a manifold with an added potential: $(\Delta_n+V({\bf x}))\Psi=0$. Distinct families of second-order superintegrable Schr\"odinger (or Helmholtz) systems $(\Delta'_n+V'({\bf x}))\Psi=E\Psi$ can be incorporated into a single Laplace equation. There is a deep connection between most of the special functions of mathematical physics, these ...

This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bo...

Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 3$. In this paper, we give various properties of the eigenvalues of the Yamabe operator $L_g$. In particular, we show how the second eigenvalue of $L_g$ is related to the existence of nodal solutions of the equation $L_g u = \epsilon | u|^{N-2} u$, where $\epsilon = +1, 0,$ or -1.

In this paper we formulate new curvature functions on $\mathbb{S}^n$ via integral operators. For certain even orders, these curvature functions are equivalent to the classic curvature functions defined via differential operators, but not for all even orders. Existence result for antipodally symmetric prescribed curvature functions on $\mathbb{S}^n$ is obtained. As a corollary, the existence of a conformal metric for an antipodally symmetric prescribed $Q-$curvature functions ...

We study a fully nonlinear flow for conformal metrics. The long-time existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the $\sk$-Yamabe problem for locally conformal flat manifolds when $k \neq n/2$.

Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor $P$ to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and including 3 and n, this prescription equation is an Euler-Lagrange equation of some action if and only if the structure is locally conformally flat.

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 constant scalar curvature metric. The proof does not use the traditional functional minimization. Applications to the Yamabe problem on closed manifolds, manifolds with boundary, and noncompact manifolds are given in forthcoming papers.

We will prove the non-existence of positive radially symmetric solution of the nonlinear elliptic equation $\frac{n-1}{m}\Delta v^m+\alpha v+\beta x\cdot\nabla u=0$ in $R^n$ when $n\ge 3$, $0<m\le\frac{n-2}{n}$, $\alpha<0$ and $\beta\le 0$. Let $n\ge 3$ and $g=v^{\frac{4}{n+2}}dx^2$ be a metric on $\R^n$ where $v$ is a radially symmetric solution of the above elliptic equation in $R^n$ with $m=\frac{n-2}{n+2}$, $\alpha=\frac{2\beta+\rho}{1-m}$ and $\rho\in R$. For $n\ge 3$, $...

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.