Singularity Knots of Minimal Surfaces in $\mathbb{R}^4$

Some properties of simple minimal knots

A minimal knot is the intersection of a topologically embedded branched minimal disk in $\mathbb{R}^4$ $\mathbb{C}^2 $ with a small sphere centered at the branch point. When the lowest order terms in each coordinate component of the embedding of the disk in $\mathbb{C}^2$ are enough to determine the knot type, we talk of a simple minimal knot. Such a knot is given by three integers $N < p,q$; denoted by $K(N,p,q)$, it can be parametrized in the cylinder as $e^{iθ}\mapsto (e^{Niθ},\sin qθ,\cos pθ)$. From this expression stems a natural representation of $K(N,p,q)$ as an $N$-braid. In this paper…

极小曲面、纽结与神经网络

Minimal surfaces, Knots, and Neural Networks

基于物理信息神经网络求解双曲空间中的极小曲面方程，通过计算纽结边界的极小曲面及其自交数，为Fine猜想提供了实证支持。

Local removable singularity theorems for minimal laminations

In this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold. This removable singularity theorem is the key result used in our proof that a complete, embedded minimal surface in $\mathbb{R}^3$ with quadratic decay of curvature has finite total curvature.

Knots, minimal surfaces and J-holomorphic curves

Let $K$ be a knot in the 3-sphere, viewed as the ideal boundary of hyperbolic 4-space $\mathbb{H}^4$. We prove that the number of minimal discs in $\mathbb{H}^4$ with ideal boundary $K$ is a knot invariant. I.e.\ the number is finite and doesn't change under isotopies of $K$. In fact this gives a family of knot invariants, indexed by an integer describing the extrinsic topology of how the disc sits in $\mathbb{H}^4$. These invariants can be seen as Gromov--Witten invariants counting $J$-holomorphic discs in the twistor space $Z$ of $\mathbb{H}^4$. Whilst Gromov--Witten theory suggests the gene…

The Extended Bogomolny Equations and Generalized Nahm Pole Boundary Condition

In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $Σ\times \RP$ with Nahm pole singularity at $Σ\times \{0\}$ and the Hitchin component of the stable $SL(2,\mathbb{R})$ Higgs bundle; this verifies a conjecture of Gaiotto and Witten. We also develop a partial Kobayashi-Hitchin correspondence for solutions with a knot singularity in this program, corresponding to the non-Hitchin components in the moduli space of stable $SL(2,\mathbb{R})$ Higgs bundles. We also prove existence and uniqueness of solutions with knot singulariti…

Interaction Between Singularity Theory and the Minimal Model Program

We survey some recent topics on singularities, with a focus on their connection to the minimal model program. This includes the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds and the recent progress on the `local stability theory' of an arbitrary Kawamata log terminal singularity.