Let $X$ be a complex manifold, and let $S\hookrightarrow X$ be an embedding of a complex submanifold. Assuming that the embedding is $(k-1)$-linearizable or $(k-1)$-comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism $F$ from a small neighborhood of the zero section in the normal bundle $N_S$ to a small neighborhood of $S$ in $X$ such that $F$ is in a precise sense holomorphic up to the $(k-1)$th order. Using this $F$, we obtain optimal estimates on asymptotic rates for asymptotically conical (AC) Calabi–Yau (CY) metrics constructed by Tian and Yau. Furthermore, when $S$ is an ample divisor satisfying an appropriate cohomological condition, we relate the order of comfortable embedding to the weight of the deformation of the normal isolated cone singularity arising from the deformation to the normal cone. We also give an example showing that the condition of comfortable embedding depends on the splitting liftings. We then prove an analytic compactification result for the deformation of the complex structure on an affine cone that decays to any positive order at infinity. This can be seen as an analytic counterpart of Pinkham’s result on deformations of cone singularities with negative weights.

Let $G$ be a semisimple real algebraic Lie group of real rank at least $2$, and let $U$ be the unipotent radical of a nontrivial parabolic subgroup. We prove that a discrete Zariski-dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous works of Selberg and Oh.

We prove the uniqueness of blowups and $C^{1,log}$-regularity for the free-boundary of minimizers of the Alt–Caffarelli functional at points where one blowup has an isolated singularity. We do this by establishing a (log-)epiperimetric inequality for the Weiss energy for traces close to that of a cone with isolated singularity, whose free boundary is graphical and smooth over that of the cone in the sphere. With additional assumptions on the cone, we can prove a classical epiperimetric inequality which can be applied to deduce a $C^{1,\alpha }$-regularity result. We also show that these additional assumptions are satisfied by the De Silva–Jerison-type cones, which are the only known examples of minimizing cones with isolated singularity. Our approach draws a connection between epiperimetric inequalities and the Łojasiewicz inequality, and, to our knowledge, provides the first regularity result at singular points in the one-phase Bernoulli problem.