Basic Geometric Constructions

Basic geometric constructions, including tubing, boundary twisting, pushing down intersections, and contraction followed by push-off are presented. These moves are used repeatedly later in the proof. New, detailed pictures illustrating these constructions are provided. The Clifford torus at an intersection point between two surfaces in 4-dimensional space is described. The chapter closes with an important application of some of these moves called the Geometric Casson Lemma. This lemma upgrades algebraically dual spheres to geometrically dual spheres, at the cost of introducing more self-intersections. It is also shown that an immersed Whitney move is a regular homotopy of the associated surfaces.

Polynomial Vector Fields on the Clifford Torus

First, we characterize all the polynomial vector fields in [Formula: see text] which have the Clifford torus as an invariant surface. Then we study the number of invariant meridians and parallels that such polynomial vector fields can have on the Clifford torus as a function of the degree of these vector fields.

Remarks on the self-shrinking Clifford torus

On the one hand, we prove that the Clifford torus in {\mathbb{C}^{2}} is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian F-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is rigid: it is locally unique as a self-shrinker for mean curvature flow, despite having infinitesimal deformations which do not arise from rigid motions. The proofs rely on analysing higher order phenomena: specifically, showing that the Clifford torus is not a local entropy minimiser even under Hamiltonian variations, and demonstrating that infinitesimal deformations which do not generate rigid motions are genuinely obstructed.

Characterization of Clifford Torus in Three-Spheres

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the three-dimensional sphere.

Uniqueness of Real Lagrangians up to Cobordism

We prove that a real Lagrangian submanifold in a closed symplectic manifold is unique up to cobordism. We then discuss the classification of real Lagrangians in ${\mathbb{C}} P^2$ and $S^2\times S^2$. In particular, we show that a real Lagrangian in ${\mathbb{C}} P^2$ is unique up to Hamiltonian isotopy and that a real Lagrangian in $S^2\times S^2$ is either Hamiltonian isotopic to the antidiagonal sphere or Lagrangian isotopic to the Clifford torus.

New Characterizations of the Clifford Torus and the Great Sphere

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we study ruled surfaces in a three-dimensional sphere with finite-type and pointwise 1-type spherical Gauss map. Concerning integrability and geometry, we set up new characterizations of the Clifford torus and the great sphere of 3-sphere and construct new examples of spherical ruled surfaces in a three-dimensional sphere.

Minimal surfaces in the three-dimensional sphere with high symmetry

Using the Lawson existence theorem of minimal surfaces and the symmetries of the Hopf fibration, we will construct symmetric embedded closed minimal surfaces in the three-dimensional sphere. These surfaces contain the Clifford torus, the Lawson’s minimal surfaces, and seven new minimal surfaces with genera [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We will also discuss the relation between such surfaces and the maximal extendable group actions on subsurfaces of the three-dimensional sphere.