On shrinking Gradient Ricci Soliton With Nonnegative Sectional Curvature
In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.
math.DG↗
arXiv subjects
Mingliang Cai
Publications and source records attributed to Mingliang Cai.
Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds
The Witten spinorial argument has been adapted in several works over the years to prove positivity of mass in the asymptotically AdS and asymptotically hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar curvature rigidity result and a positive mass theorem for asymptotically hyperbolic manifolds that do not require a spin assumption. The positive mass theorem is reduced to the rigidity case by a deformation construction near the conformal boundary. The proof of the rigidity result is based on a study of minimizers of the BPS brane action.
math.DG↗
On the topology and area of higher dimensional black holes
Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higher dimensional analogues of some well known results for black holes in 3+1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking's black hole topology theorem for asymptotically flat ($Λ=0$) black hole spacetimes, and Gibbons' and Woolgar's genus dependent, lower entropy bound for topological black holes in asymptotically locally anti-de Sitter ($Λ<0$) spacetimes. In higher dimensions the genus is replaced by the so-called $σ$-constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.
hep-th↗
Boundaries of Zero Scalar Curvature in the AdS/CFT Correspondence
In hep-th/9910245, Witten and Yau consider the AdS/CFT correspondence in the context of a Riemannian Einstein manifold $M^{n+1}$ of negative Ricci curvature which admits a conformal compactification with conformal boundary $N^n$. They prove that if the conformal class of the boundary contains a metric of positive scalar curvature, then $M$ and $N$ have several desirable properties: (1) $N$ is connected, (2) the $n$th homology of the compactified $M$ vanishes, and (3) the fundamental group of $M$ is "bounded by" that of $N$. Here it is shown that all of these results extend to the case where the conformal class of the boundary contains a metric of nonnegative scalar curvature. (The case of zero scalar curvature is of interest as it is borderline for the stability of the theory.) The proof method used here is different from, and in some sense dual to, that used by Witten and Yau. While their method involves minimizing the co-dimension one brane action on $M$, and requires the machinery of geometric measure theory, the main arguments presented here use only geodesic geometry.
hep-th↗
Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar curvature
The following version of a conjecture of Fischer-Colbrie and Schoen is proved: If M is a complete Riemannian 3-manifold with nonnegative scalar curvature which contains a two-sided torus S which is of least area in its isotopy class then M is flat. This follows from a local version derived in the paper.
math.DG↗