arXiv subjects
Publications and source records attributed to Jianguo Cao.
Let $M^{2n-1}$ be the smooth boundary of a bounded strongly pseudo-convex domain $Ω$ in a complete Stein manifold $V^{2n}$. Then (1) For $n \ge 3$, $M^{2n-1}$ admits a pseudo-Eistein metric; (2) For $n \ge 2$, $M^{2n-1}$ admits a Fefferman metric of zero CR Q-curvature; and (3) for a compact strictly pseudoconvex CR embeddable 3-manifold $M^3$, its CR Paneitz operator $P$ is a closed operator.
In this paper, we study certain compact 4-manifolds with non-negative sectional curvature $K$. If $s$ is the scalar curvature and $W_+$ is the self-dual part of Weyl tensor, then it will be shown that there is no metric $g$ on $S^2 \times S^2$ with both (i) $K > 0$ and (ii) $ {1/6} s - W_+ \ge 0$. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: ``If a simply-connected, closed 4-manifold $M^4$ admits a metric $g$ of non-negative curvature operator, then $M^4$ is one of $S^4$, $\Bbb CP^2$ and $S^2 \times S^2$". Our method is different from Hamilton's and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.
A Lipschitz hypersurface is a hypersurface which locally is the graph of a Lipschitz function. A Lipschitz (or C^1) hypersurface is said to be Levi-flat if it is locally foliated by complex manifolds of complex dimension (n-1). We shall prove that there exist no Lipschitz Levi-flat hypersurfaces in CP^n with n >= 3. Our new estimates on the d-bar-Cauchy problems are different from the earlier Siu's integral kernal method.
We will present a new proof of the Gromoll-Grove diameter rigidity theorem.
The Martin boundary of a Cartan-Hadamard manifold describes a fine geometric structure at infinity, which is a sub-space of positive harmonic functions. We describe conditions which ensure that some points of the sphere at infinity belong to the Martin boundary as well. In the case of the universal cover of a compact manifold with Ballmann rank one, we show that Martin points are generic and of full harmonic measure. The result of this paper provides a partial answer to an open problem of S. T. Yau.
Let $M^n$ be a complete, non-compact and $C^\infty$-smooth Riemannian manifold with nonnegative sectional curvature. Suppose $\Cal S$ is a soul of $M^n$. Then any distance non-increasing retraction $Ψ: M^n \to \Cal S$ must give rise to a $C^\infty$-smooth Riemannian submersion.
Let $Ω$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb CP^n$. We prove that the $\bar\partial-Neumann operator $N$ exists for $(p,q)$-forms on $Ω$. Furthermore, there exists a $t_0>0$ such that the operators $N$, $\bar\partial^*N$, $\bar\partial N$ and the Bergman projection are regular in the Sobolev space $W^t (\barΩ) $ for $t 0$.