## Type of Work

Article (10) Book (0) Theses (6) Multimedia (4)

## Peer Review

Peer-reviewed only (20)

## Supplemental Material

Video (0) Audio (0) Images (0) Zip (0) Other files (4)

## Publication Year

## Campus

UC Berkeley (0) UC Davis (0) UC Irvine (0) UCLA (0) UC Merced (0) UC Riverside (0) UC San Diego (0) UCSF (0) UC Santa Barbara (0) UC Santa Cruz (20) UC Office of the President (0) Lawrence Berkeley National Laboratory (0) UC Agriculture & Natural Resources (0)

## Department

## Journal

## Discipline

## Reuse License

BY - Attribution required (2)

## Scholarly Works (20 results)

In this dissertation we mainly study the geometric structure of vacuum static spaces and some related geometric problems.

In particular, we have made progress in solving the classification problem of vacuum static spaces and in proving the Besse conjecture which is about manifolds admitting solutions to the critical point equation in general dimensions. We obtain even stronger results in dimension 3.

We also extend the local scalar curvature rigidity result of Brendle-Marques-Neves on upper hemisphere to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the work of Corvino. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work of Hang-Wang.

As for generic Riemannian manifolds, we find a connection between Brown-York mass and the first Dirichlet Eigenvalue of a Schrödinger type operator. In particular, we prove a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigate compactly conformal deformations which either increase or decrease scalar curvature. We find local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist for closed or a type of non-compact manifolds with positive scalar curvature. These results together give an answer to a question arises naturally in the work of Corvino and Lohkamp.

In this thesis we study the stability of the Ricci flow. The stability problem of Ricci flow in different settings have been considered by Ye \cite{Ye}, Li-Yin \cite{LY}, Schn\"urer-Schulze-Simon \cite{SSS} and Bamler \cite{Bam} etc. We consider a more general case and extend the results to the general case, that is, in the setting of asymptotically hyperbolic Einstein (AHE) manifolds with rough initial data. First we introduce the background of the problem and results on the long time behavior of Ricci flow in detail. Then we compare the difference in methodology of theses results and extend to the AHE case. We consider the normalized Ricci flow on a AHE manifold with initial metrics which are perturbations of a non-degenerate AHE metric $h_0$. The key step is to obtain exponential decay of certain geometric quantities. Then we prove that the normalized Ricci flow converges exponentially fast to $h_0$, if the perturbation is $L^2$-bounded and $C^0$-small.

For a hypersurface in a conformal manifold, by following the idea of Fefferman and Graham's work, we use the conformal Gauss map and the conformal transform to construct the associate hypersurface in the ambient space. By evaluations of scalar Riemannian invariants of associate hypersurface, we find out a way to construct and collect scalar conformal invariants of the given hypersurface. This method provides chances for searching higher order partial differential equations which are similar like the Willmore equation.

- 1 supplemental PDF

This thesis is a study of the uniqueness of the higher stationary states of the Schr\"odinger--Newton system under the assumption of spherically symmetric solutions. We begin with a theory of dark matter put foward by Bray \cite{hubray} involving the Einstein--Klein--Gordon system of equations, and then pose the Schr\"odinger--Newton system as the low--field nonrelatavistic limit of the Einstein--Klein--Gordon system. From here, by imposing spherical symmetry, we show that the potential term in the Schr\"odinger--Newton system can be seen as a nonlinear perturbation from the Coulomb potential $\frac{1}{r}$ on the half--line $[0, \infty)$. After proving uniqueness of bound states for the Hydrogen atom on the half--line, we then proceed by defining weighted Banach spaces for which the Schr\"odinger operator representing the Hydrogen atom on the half--line is Fredholm of index 0. In the last chapter, we detail an iteration scheme involving the implicit function theorem to show a correspondence between bound state solutions of the Hydrogen atom on the half--line and bound state solutions of the full Schr\"odinger--Newton system to prove the uniqueness result.

The relationship between the geometry of a conformally compact manifold and the conformal geometry of its conformal infinity is of particular interest due to its association with the AdS/CFT correspondence of physics, a conjectured correlation between a string theory on a negatively curved Einstein manifold and a conformal field theory on its boundary at infinity. In the case of hyperbolic space $\mathbb{H}^{n+1}$ with conformal infinity the round sphere $\mathbb{S}^n$, a very precise relationship has been established between conformal invariants (the eigenvalues of the Schouten tensor) on $\mathbb{S}^n$ and Weingarten curvatures of immersed hypersurfaces \cite{MR2538508}. This same relationship has been extended to hyperbolic Poincar

- 1 supplemental file