For $k \geq 2$, let $M^{4k-1}$ be a closed $(2k-2)$-connected manifold. If $k \equiv 1 \mod 4$ assume further that $M$ is $(2k-1)$-parallelisable. Then there is a homotopy sphere $\Sigma^{4k-1}$ such that $M \sharp \Sigma$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.

The classification of Willmore two-spheres in the $n$-dimensional sphere $S^n$ is a long-standing problem, solved only when $n = 3, 4$ by Bryant, Ejiri, Musso and Montiel independently. In this paper we give a classification when $n = 5$. There are three types of such surfaces up to Möbius transformations: (1) superconformal surfaces in $S^4$; (2) minimal surfaces in $R^5$; (3) adjoint transforms of superconformal minimal surfaces in $R^5$. In particular, Willmore surfaces in the third class are not $S$-Willmore (i.e., without a dual Willmore surface).

To show the existence of Willmore two-spheres in $S^5$ of type (3), we describe all adjoint transforms of a superconformal minimal surface in $R^n$ and provide some explicit criterions on the immersion property. As an application, we obtain new immersed Willmore two-spheres in $S^5$ and $S^6$, which are not $S$-Willmore.

In this paper, we study closed embedded minimal hypersurfaces in a Riemannian $(n+1)$-manifold $(2 \leq n \leq 6)$ that minimize area among such hypersurfaces. We show they exist and arise either by minimization techniques or by min–max methods: they have index at most $1$. We apply this to obtain a lower area bound for such minimal surfaces in some hyperbolic $3$-manifolds.

We prove a compactness result for minimal hypersurfaces with bounded index and volume, which can be thought of as an extension of the compactness theorem of Choi–Schoen [4] to higher dimensions.

We show that the generating series of generalized Donaldson–Thomas invariants on the local projective plane with any positive rank is described in terms of modular forms and theta type series for indefinite lattices. In particular, it absolutely converges to give a holomorphic function on the upper half plane.