Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. In “Symplectic embeddings into four-dimensional concave toric domains”, the author, Choi, Frenkel, Hutchings and Ramos computed the ECH capacities of all “concave toric domains”, and showed that these give sharp obstructions in several interesting cases. We show that these obstructions are sharp for all symplectic embeddings of concave toric domains into “convex” ones. In an appendix with Choi, we prove a new formula for the ECH capacities of convex toric domains, which shows that they are determined by the ECH capacities of a corresponding collection of balls.

We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N \leq -a^2 \leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by a and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface $\Sigma$ must be proper in $N$ and its total curvature must be equal to $2 \pi \chi (\Sigma)$. If $N$ is a hyperbolic $3$-manifold of finite volume and $\Sigma$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than $1$, then we prove that each end of $\Sigma$ is asymptotic (with finite positive integer multiplicity) to a totally umbilic annulus, properly embedded in $N$.

Building on the geometric theory of uniruled projective manifolds by Hwang–Mok, which relies on the study of varieties of minimal rational tangents (VMRTs) from both the algebro-geometric and the differential-geometric perspectives, Mok, Hong–Mok and Hong–Park have studied standard embeddings between rational homogeneous spaces $X = G/P$ of Picard number $1$. Denoting by $S \subset X$ an arbitrary germ of complex submanifold which inherits from $X$ a geometric structure defined by taking intersections of VMRTs with tangent subspaces and modeled on some rational homogeneous space $X_0 = G_0 / P_0$ of Picard number $1$ embedded in $X = G/P$ as a linear section through a standard embedding, we say that $(X_0, X)$ is rigid if there always exists some $\gamma \in \mathrm{Aut}(X)$ such that $S$ is an open subset of $\gamma (X_0)$. We prove that a pair $(X_0, X)$ of sub-diagram type is rigid whenever $X_0$ is nonlinear, which in the Hermitian symmetric case recovers Schubert rigidity for nonlinear smooth Schubert cycles, and which in the general rational homogeneous case goes beyond earlier works dealing with images of holomorphic maps. Our methods apply to uniruled projective manifolds $(X, \mathcal{K})$, for which we introduce a general notion of sub-VMRT structures $\varpi : \mathscr{C} (S) \to S$, proving that they are rationally saturated under an auxiliary condition on the intersection $\mathscr{C} (S) := \mathscr{C} (X) \cap \mathbb{P} T (S)$ and a nondegeneracy condition for substructures expressed in terms of second fundamental forms on VMRTs. Under the additional hypothesis that minimal rational curves are of degree $1$ and that distributions spanned by sub-VMRTs are bracket generating, we prove that $S$ extends to a subvariety $Z \subset X$. For its proof, starting with a “Thickening Lemma” which yields smooth collars around certain standard rational curves, we show that the germ of submanifold $(S; x_0)$ and, hence, the associated germ of sub-VMRT structure on $(S; x_0)$ can be propagated along chains of “thickening” curves issuing from $x_0$, and construct by analytic continuation a projective family of chains of rational curves compactifying the latter family, thereby constructing the projective completion $Z$ of $S$ as its image under the evaluation map.

In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap (the difference between the first two eigenvalues) conjecture for convex domains in the Euclidean space [3] and conjectured similar results hold for spaces with constant sectional curvature. We prove the conjecture for the sphere. Namely when $D$, the diameter of a convex domain in the unit $\mathbb{S}^n$ sphere, is $\leq \frac{\pi}{2}$, the gap is greater than the gap of the corresponding 1-dim sphere model. We also prove the gap is $\geq 3 \frac{\pi^2}{D^2}$ when $n \geq 3$, giving a sharp bound. As in [3], the key is to prove a super log-concavity of the first eigenfunction.