arXiv research

New symmetric Willmore tori emerge from Clifford torus in Berger spheres.

We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also…

The Clifford torus minimizes Willmore energy closely for small perturbations.

In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus $\mathbb{S}^1(1)\times\mathbb{S}^1(1)$ is the unique compact orientable Lagrangian self-shrinker in $\mathbb{C}^2$ with $|A|^2\leq 2$, which gives an affirmative answer to Ca…

Square Clifford torus uniquely determined by isoperimetric ratio, rectangular torus not.

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 r…

We construct embedded closed minimal surfaces in the round three-sphere, resembling two parallel copies of the Clifford torus, joined by m^2 small catenoidal bridges symmetrically arranged along a square lattice of points on the torus.

We provide several rigidity results for the Clifford torus in the class of compact self-shrinkers for Lagrangian mean curvature flow.

The Clifford torus is a product surface in $\mathbb S^3$ and it is helicoidal. It will be shown that more minimal submanifolds of $\mathbb S^n$ have these properties.

The Clifford torus is unique when its isoperimetric ratio is prescribed.

New minimal surface doublings of Clifford Torus improve bounds on minimal surfaces in S^3.

Solves Neumann problem on CR manifold boundary.

Study of tori of revolution under Willmore flow converges to Clifford Torus.

A peculiarity of the geometry of the euclidean 3-sphere $\S3$ is that it allows for the existence of compact without boundary minimally immersed surfaces. Despite a wealthy of examples of such surfaces, the only known tori minimally embedded in $\S3$ are the ones congruent to the Clifford torus. In 1970 Lawson conjectu…

We construct the spectral curve and the Baker--Akhiezer function for the Dirac operator which corresponds to the Clifford torus via the Weierstrass representation. By constructing this Baker--Akhiezer function we demonstrate a general procedure for constructing Dirac operators and their Baker--Akhiezer functions corres…

We provide a characterization of the Clifford Torus in S3 via moving frames and contact structure equations. More precisely, we prove that minimal surfaces in S3 with constant contact angle must be the Clifford Torus. Some applications of this result are then given, and some examples are discussed.

Building on work of Kapouleas and Yang, we construct sequences of minimal surfaces embedded in the round 3-sphere which converge to the Clifford torus counted with multiplicity two and have second fundamental form blowing up at every point of the torus and genus tending to infinity. Each surface in a given sequence res…

We determine the Lagrangian monodromy group L(T) and the smooth monodromy group S(T) of a Clifford torus T in the symplectic 4-space. We show that L(T) is isomorphic to the infinite dihedral group, and S(T) is generated by three reflections. We give explicit formulas for both groups. We also show that if a Lagrangian t…

In 1965 Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in $R^3$ is at least $2π^2$ and attains this minimal value if and only if the torus is a Möbius transform of the Clifford torus. This was recently proved by Marques and Neves. In this paper, we show for tori there is …

The paper studies Willmore surfaces in 4D conformal manifolds and finds the Clifford torus is strictly Willmore-stable.

We define new Hamiltonian isotopy invariants for a monotone Lagrangian torus embedded in a symplectic 4-manifold. We show that, in the standard symplectic 4-space, these invariants distinguish a monotone Clifford torus from a Chekanov torus.

The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using the Hopf fibration. In this paper, all Hamiltonian stationary Lagrangian tori whic…

We use bifurcation theory to show the existence of infinite sequences isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.

The paper explores non-minimal solitons in the sphere with unique properties.

We show that any embedded minimal torus in S^3 is congruent to the Clifford torus. This answers a question posed by H.B. Lawson, Jr., in 1970.

The paper studies the biharmonic Clifford torus in 4D sphere, computing its index and nullity.

Study on minimal hypersurfaces in a unit sphere, proving specific isometries.

The paper identifies special Lagrangian shapes in 4D space.

Study minimizes CR surfaces in Heisenberg group with rotational symmetry.

Our goal is to generalize the Choe-Hoppe helicoid and Clifford cones in Euclidean space. By sweeping out $L$ indpendent Clifford cones in ${\mathbb{R}}^{2N+2}$ via the multi-screw motion, we construct minimal submanifolds in ${\mathbb{R}}^{L(2N+2)+1}$. Also, we sweep out the $L$-rays Clifford cone (introduced in Sectio…

For each integer $k \geq 2$, we apply gluing methods to construct sequences of minimal surfaces embedded in the round $3$-sphere. We produce two types of sequences, all desingularizing collections of intersecting Clifford tori. Sequences of the first type converge to a collection of $k$ Clifford tori intersecting with …

This note is motivated by Y.G. Oh's conjecture that the Clifford torus $L_n$ in $\mathbb{C}P^n$ minimizes volume in its Hamiltonian deformation class. We show that there exist explicit positive constants $a_n$ depending on the dimension with $a_2=3/π$ such that for any Lagrangian torus $L$ in the Hamiltonian class of $…

Ellipsoids host infinitely many minimal tori, bifurcating from a 2-torus orbit.

The study characterizes embedded minimal hypersurfaces in $S^{n+1}$ with symmetries.

We show that the surface energy introduced by Auckly and Sadun attains the minimum value at the Clifford torus among tori of revolution.

Paper finds conditions for minimal hypersurfaces in S^6 with constant scalar curvature.

The paper proves existence and behavior of Lagrangian tori in complex projective plane.

The paper proves conditions for closed minimally immersed hypersurfaces in a 5-sphere.

Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.

Optimizes the conformal capacity of linked curves in $S^3$.

New method for minimal submanifolds in spheres using eigenfunctions.

Paper proves minimal hypersurface index for specific cases.

We study solutions to the inverse mean curvature flow which evolve by homotheties of a given submanifold with arbitrary dimension and codimension. We first show that the closed ones are necessarily spherical minimal immersions and so we reveal the strong rigidity of the Clifford torus in this setting. Mainly we focus o…

We prove the Hamiltonian unknottedness of real Lagrangian tori in the monotone $S^2\times S^2$, namely any real Lagrangian torus in $S^2\times S^2$ is Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$. The proof is based on a neck-stretching argument, Gromov's foliation theorem, and the Cieliebak-Sc…

Defines algebraic structures in Lagrangian Floer cohomology using differential forms.

In this paper are given examples of tori T2 embedded in R3 with all their principal lines dense. These examples are obtained by stereographic projection of deformations of the Clifford torus in S3.

The Legendrian product of two Legendrian knots, as defined by Lambert-Cole, is a Legendrian torus. We show that this Legendrian torus is a twist spun whenever one of the Legendrian knot components is sufficiently large. We then study examples of Legendrian products which are not Legendrian isotopic to twist spuns. In o…

We construct biharmonic real hypersurfaces and Lagrangian submanifolds of Clifford torus type in $CP^n$ via the Hopf fibration; and get new examples of biharmonic submanifolds in $S^{2n+1}$ as byproducts .