In this paper we will show how to classify Legendrian and transverse knots in the knot type of “sufficiently positive” cables of a knot in terms of the classification of the underlying knot. We will also completely explain the phenomena of “Legendrian large” cables. These are Legendrian representatives of cables that have Thurston–Bennequin invariant larger than the framing coming from the cabling torus. Such examples have only recently, and unexpectedly, been found. We will also give criteria that determines the classification of Legendrian and transverse knots the knot type of negative cables.

$\def\Volume{\operatorname{Volume}}$Let $(M^n, g)$ be a compact $n$-dimensional Riemannian manifold without boundary, where $g = (g_{ij})$ is $C^1$-smooth. Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on $M$. Let $B$ be a ball on $M$. We prove that the number of nodal domains of $u_k$ that intersect $B$ is not greater than\[C_1 \dfrac{\Volume_g (B)}{\Volume_g(M)} k + C_2 k^\frac{n-1}{n} \: ,\]where $C_1, C_2$ depend on $M$. The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains.

For a sequence of immersed connected closed Hamiltonian stationary Lagrangian submaniolds in $\mathbb{C}^n$ with uniform bounds on their volumes and the total extrinsic curvatures, we prove that a subsequence converges either to a point or to a Hamiltonian stationary Lagrangian $n$-varifold locally uniformly in $C^k$ for any nonnegative integer $k$ away from a finite set of points, and the limit is Hamiltonian stationary in $\mathbb{C}^n$. We also obtain a theorem on extending Hamiltonian stationary Lagrangian submanifolds $L$ across a compact set $N$ of Hausdorff codimension at least $2$ that is locally noncollapsing in volumes matching its Hausdorff dimension, provided the mean curvature of $L$ is in $L^n$ and a condition on local volume of $L$ near $N$ is satisfied.

We consider critical points of a class of functionals on compact four-dimensional manifolds arising from Regularized Determinants for conformally covariant operators, whose explicit form was derived in $\href{https://doi.org/10.1090/S0002-9939-1991-1050018-8}{[10]}$, extending Polyakov’s formula. These correspond to solutions of elliptic equations of Liouville type that are quasilinear, of mixed orders and of critical type. After studying existence, asymptotic behaviour and uniqueness of fundamental solutions, we prove a quantization property under blow-up, and then derive existence results via critical point theory.

We study the jumps in the archimedean height of the Ceresa cycle, as introduced by R. Hain in his work on normal functions on moduli spaces of curves, and as further analyzed by P. Brosnan and G. Pearlstein in terms of asymptotic Hodge theory. Our work is based on a study of the asymptotic behavior of the Hain–Reed beta-invariant in degenerating families of curves. We show that the height jump of the Ceresa cycle at a given stable curve is equal to the so-called “slope” of the dual graph of the curve, and we characterize those stable curves for which the height jump vanishes. We also obtain an analytic formula for the height of the Ceresa cycle for a curve over a function field over the complex numbers, and characterize in analytic terms when the height of the Ceresa cycle vanishes.

We prove that a shrinking gradient Ricci soliton which agrees to infinite order at spatial infinity with one of the standard cylinders $\mathbb{S}^k \times \mathbb{R}^{n-k}$ for $k \geq 2$ along some end must be isometric to the cylinder on that end. When the underlying manifold is complete, it must be globally isometric either to the cylinder or (when $k =n-1$) to its $\mathbb{Z}_2$-quotient.

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at infinity. The 3d flying wings are collapsed. For dimension $n \geq 4$, we find a family of $\mathbb{Z}_2 \times O(n-1)$-symmetric but non-rotationally symmetric $n$-dimensional steady gradient solitons with positive curvature operator. We show that these solitons are non-collapsed.

We prove the following comparison theorem for metrics with nonnegative scalar curvature, also known as the dihedral rigidity conjecture by Gromov: for $n \leq 7$, if an $n$-dimensional prism has nonnegative scalar curvature and weakly mean convex faces, then its dihedral angle cannot be everywhere not larger than its Euclidean model, unless it is isometric to an Euclidean prism. The proof relies on constructing certain free boundary minimal hypersurface in a Riemannian polyhedron, and extending a dimension descent idea of Schoen–Yau. Our result is a localization of the positive mass theorem.

In this paper, we prove that in any compact Riemannian manifold with smooth boundary, of dimension at least $3$ and at most $7$, there exist infinitely many almost properly embedded free boundary minimal hypersurfaces. This settles the free boundary version of Yau’s conjecture. The proof uses adaptions of A. Song’s work and the early works by Marques–Neves in their resolution to Yau’s conjecture, together with Li–Zhou’s regularity theorem for free boundary min-max minimal hypersurfaces.