We prove that the Calabi–Yau equation on the Kodaira–Thurston manifold has a unique solution for every $S^1$-invariant initial datum.

In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface $\Sigma$ in a compact $n$-manifold $M$, there is a Riemannian metric on $M$ such that the nodal set of its first nontrivial eigenfunction is $\Sigma$. We present a number of variations on this result, which enable us to show, in particular, that the first nontrivial eigenfunction can have as many non-degenerate critical points as one wishes.

Let $M$ be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to $[0,\infty) \times X$ for some compact connected Ricci-flat manifold $X$. We begin by proving general structure theorems for $M$; in particular we show that there is no loss of generality in assuming that $M$ is simply-connected and irreducible with $\mathrm{Hol}(M) = \mathrm{SU}(n)$, where $n$ is the complex dimension of $M$. If $n \gt 2$ we then show that there exists a projective orbifold $\overline{M}$ and a divisor $\overline{D} \in \lvert -K_{\overline{M}} \rvert$ with torsion normal bundle such that $\overline{M}$ is biholomorphic to $\overline{M} \setminus \overline{D}$, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting.We give examples where $\overline{M}$ is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair $\overline{M} \setminus \overline{D}$ we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi–Yau metrics on $\overline{M} \setminus \overline{D}$.

We study solutions of the mean curvature flow which are defined for all negative times, usually called ancient solutions. We give various conditions ensuring that a closed convex ancient solution is a shrinking sphere. Examples of such conditions are: a uniform pinching condition on the curvatures, a suitable growth bound on the diameter, or a reverse isoperimetric inequality. We also study the behaviour of uniformly $k$-convex solutions, and consider generalizations to ancient solutions immersed in a sphere.

Let $f : U \to \mathbb{A}^1$ be a regular function on a smooth scheme $U$ over a field $\mathbb{K}$. Pantev, Toën, Vaquié and Vezzosi [30, 37] define the ‘derived critical locus’ $\mathbf{Crit}(f)$, an example of a new class of spaces in derived algebraic geometry, which they call ‘$-1$-shifted symplectic derived schemes’.

They show intersections of algebraic Lagrangians in a smooth symplectic $\mathbb{K}$-scheme, and stable moduli schemes of coherent sheaves on a Calabi–Yau $3$-fold over $\mathbb{K}$, are also $-1$-shifted symplectic derived schemes. Thus, their theory may have applications in algebraic symplectic geometry, and in Donaldson–Thomas theory of Calabi–Yau $3$-folds.

This paper defines and studies a new class of spaces we call ‘algebraic $d$-critical loci’, which should be regarded as classical truncations of the $-1$-shifted symplectic derived schemes of [30]. They are simpler than their derived analogues. We also give a complex analytic version of the theory, and an extension to Artin stacks.

In the sequels [4–8] we will apply $d$-critical loci to motivic and categorified Donaldson–Thomas theory, and to intersections of complex Lagrangians in complex symplectic manifolds. We will show that the important structures one wants to associate to a derived critical locus—virtual cycles, perverse sheaves, $\mathfrak{D}$-modules, and mixed Hodge modules of vanishing cycles, and motivic Milnor fibres—can be defined for oriented $d$-critical loci.