arXiv subjects
Publications and source records attributed to Xiaoxiang Chai.
We study hypersurfaces with prescribed null expansion in an initial data set. We propose a notion of stability and prove a topology theorem. Eichmair's Perron approach toward the existence of marginally outer trapped surface adapts to the settings of hypersurfaces with prescribed null expansion with only minor modifications.
We study a perturbation \begin{equation} Δu + P | \nabla u| = h | \nabla u|, \end{equation} of spacetime Laplacian equation in an initial data set $(M, g, p)$ where $P$ is the trace of the symmetric 2-tensor $p$ and $h$ is a smooth function.
We study harmonic maps from a 3-manifold with boundary to $\mathbb{S}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $π/ 2$. Furthermore we give some applications to mapping torus hyperbolic 3-manifolds.
Using the upper half space model, we evaluate a component of the hyperbolic mass functional evaluated on a special family of polyhedra extending a formula of Miao-Piubello.
We discuss asymptotically hyperbolic manifold with a noncompact boundary which is close to a horosphere in a certain sense. The model case is a horoball or the complement of a horoball in standard hyperbolic space. We show some geometric formulas.
We study the Minkowski formula of conformal Killing-Yano two-forms in a spacetime of constant curvature. We establish the spacetime Alexandrov theorem with a free boundary.
We show that the mass of an asymptotically hyperbolic manifold with a noncompact boundary can be evaluated via the Ricci tensor and the second fundamental form by using purely coordinates. The method is analog to Miao-Tam's approach to the asymptotically flat manifold.
Simon type monotonicity formulas for the Willmore functional $\int | \mathbf{H} |^2$ in the hyperbolic space $\mathbb{H}^n$ and $\mathbb{S}^n$ are obtained. The formula gives a lower bound of $\int_Σ | \mathbf{H} |^2$ where $Σ^2$ is any closed surface in $\mathbb{H}^n$.
We study Hawking mass and the Huisken's isoperimetric mass evaluated on surfaces with boundary. The convergence to an ADM mass defined on asymptotically flat manifold with a non-compact boundary are proved.
Built on a recent work of Almaraz, Barbosa, de Lima on positive mass theorems on asymptotically flat manifods with a noncompact boundary, we apply free boundary minimal surface techniques to prove their positive mass theorem and study the existence of positive scalar curvature metrics with mean convex boundary on a connected sum of the form $(\mathbb{T}^{n-1}\times[0,1])\#M_0$.