We generalize the results of Adams–Schoenfeld [2], finding large classes of totally geodesic Seifert surfaces in hyperbolic knot and link complements, each covering a rigid 2-orbifold embedded in some hyperbolic 3-orbifold. In addition, we provide a uniqueness theorem and demonstrate that many knots cannot possess totally geodesic Seifert surfaces by giving bounds on the width invariant in the presence of such a surface. Finally, we utilize these examples to demonstrate that the Six Theorem is sharp for knot complements in the 3-sphere.

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A version of the Donaldson-Thomas invariants of a Calabi-Yau threefold is proposed as a conjectural mathematical definition of the Gopakumar-Vafa invariants. These invariants have a local version, which is verified to satisfy the required properties for contractible curves. This provides a new viewpoint on the computation of the local Gromov-Witten invariants of contractible curves by Bryan, Leung, and the author.

This note discusses the recent work of Huang-Schoen-Wang [13] on specifying total conserved quantities of a vacuum initial data set.

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in F. Almgren, and J. Pitts, corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$of positive Ricci curvature with 2 ≤ n ≤ 6. We characterize the Morse index, volume and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces.