For an admissible class of smooth compact initial data sets with boundary, we prove a comparison theorem between theWang/Liu–Yau quasi-local mass of the boundary and the Hawking mass of strictly minimizing hulls in the Jang graphs of the domain. Using this, we prove a quasi-local Penrose inequality that involves these quasi-local masses of the boundary and the area of an outermost marginally outer trapped surface (MOTS) in the domain or the area of minimizing minimal surface within the Jang graphs. Moreover, we obtain sufficient conditions for the (non)existence of a MOTS within a domain, in the spirit of the folklore hoop conjecture.

We develop a Floer theoretical gluing technique and apply it to deal with the most generic singular fiber in the SYZ program, namely the product of a torus with the immersed two-sphere with a single nodal self-intersection. As an application, we construct immersed Lagrangians in $\operatorname{Gr}(2,\mathbb{C}^n)$ and $\operatorname{OG}(1,\mathbb{C}^5)$ and derive their SYZ mirrors. It recovers the Lie theoretical mirrors constructed by Rietsch. It also gives an effective way to compute stable disks (with non-trivial obstructions) bounded by immersed Lagrangians.

The authors derive a McShane identity for once-punctured super tori. Relying upon earlier work on super Teichmüller theory by second and third authors, they further develop the supergeometry of these surfaces and establish the asymptotic growth rate of the simple length spectra for once-punctured super tori.

We prove a perturbative result concerning the uniqueness of Kerr–Newman family of black holes: given an asymptotically flat space-time with bifurcate horizons, if it agrees with a non-extremal Kerr–Newman space-time asymptotically at infinity and it is sufficiently close to the Kerr–Newman family, then the space-time must be one of the Kerr–Newman solutions. The closeness to the Kerr–Newman family is measured by the smallness of a pair of Mars–Simon type tensors, which were introduced by Wong in $\href{https://doi.org/10.1007/s00023-009-0414-z}{[22]}$ to detect the Kerr–Newman family.

Meeks, Pérez and Ros conjectured that a closed Riemannian $3$-manifold which does not admit any closed embedded minimal surface whose two-sided covering is stable must be diffeomorphic to a quotient of the $3$-sphere. We give a counterexample to this conjecture.

Also, we show that if we consider immersed surfaces instead of embedded ones, then the corresponding statement is true.

In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger–Gromoll splitting theorem we give a new proof to a rigidity result for complete manifolds with nonnegative scalar curvature admitting a proper smooth map to $T^{n-1} \times \mathbf{R}$ with nonzero degree. Especially we introduce a new trick to obtain the compactness of limit hypersurface from locally graphical convergence. Based on the same idea we also show some new result—an optimal $2$-systole inequality for several classes of complete Riemannian manifolds with positive scalar curvature and the corresponding characterization in the equality case.