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 $L_n$ we have $vol(L) \geq a_n vol (L_n)$. The proof uses the recent work of C.H. Cho on Floer homology of the Clifford tori. A formula from integral geometry enables us to derive the estimate. We wish to point out that a general lower bound on the volume of $L$ exists from the work of C. Viterbo. Our lower bound $a_2= 3/π$ is the best one we know.
The tori $T_r = r S^1 \times s S^1 \subset S^3$, where $r^2 + s^2 = 1$, are constrained Willmore surfaces, i.e. critical points of the Willmore functional among tori of the same conformal type. We compute which of the $T_r$ are stable critical points.
For each integer m>1 and l>0 we construct a pair of compact embedded minimal surfaces of genus 1+4m(m-1)l. These surfaces desingularize the m Clifford tori meeting each other along a great circle at the angle of π/m. They are invariant under a finite group of screw motions and have no reflection symmetry across a great sphere.
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 maximal symmetry along these two circles. Near each of the circles, after rescaling, the sequences converge smoothly on compact subsets to a Karcher-Scherk tower of order $k$. Sequences of the second type desingularize a collection of the same $k$ Clifford tori supplemented by an…
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.
Extending work of Kapouleas and Yang, for any integers $N \geq 2$, $k, \ell \geq 1$, and $m$ sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus $k\ell m^2(N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of $O(4)$, where $D_n$ is the dihedral group of order $2n$. Each such surface resembles the union of $N$ nested topological tori, all small perturbations of a single Clifford torus $\mathbb{T}$, that have been connected by $k\ell m^2 (N-1)$ small catenoidal tunnels, with $k \ell m^2$ tunnels join…
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 which is contained in a hypersphere in the complex Euclidean plane are constructed explicitly. Then it is shown that they are homogeneous tori. For the construction, flat quaternionic connections of Hamiltonian stationary Lagrangian tori are considered and a spectral curve of an asso…
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 conjectured that the Clifford torus is, up to congruences, the only torus minimally embedded in $\S3$. We prove here Lawson conjecture to be true. Two results are instrumental to this work, namely, a characterization of the Clifford torus in terms of its first eingenfunctions (\cite{MR})…
We formulate a class of minimal tori in S^3 in terms of classical mechanics, reveal a curious property of the Clifford torus, and note that the question of periodicity can be made more explicit in a simple way.
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 a gap to the next critical point of the Willmore energy and we discuss an application to the Willmore flow. We also prove an energy gap from the Clifford torus to surfaces of higher genus.
The Clifford tori in the 3-sphere are a one-parameter family of flat, two-dimensional, constant mean curvature (CMC) surfaces. This paper demonstrates that new, topologically non-trivial CMC surfaces resembling a pair of neighbouring Clifford tori connected at a sub-lattice consisting of at least two points by small catenoidal bridges can be constructed by perturbative PDE methods. That is, one can create an approximate solution by gluing a rescaled catenoid into the neighbourhood of each sub-lattice point; and then one can show that a perturbation of this approximate submanifold exists which …
We study the energy functional on the set of Lagrangian tori in $\mathbb{C}P^2$ . We prove that the value of the energy functional on a certain family of Hamiltonian minimal Lagrangian tori in $\mathbb{C}P^2$ is strictly larger than energy of the Clifford torus.
We give a classification of irreducible four-dimensional symmetric spaces $G/H$ which admit compact Clifford-Klein forms, where $G$ is the transvection group of $G/H$. For this, we develop a method that applies to particular 1-connected solvable symmetric spaces. We also examine a `solvable analogue' of Kobayashi's conjecture for reductive groups and find an evidence that the reductive assumption in Kobayashi's conjecture is crucial.
The Marques-Neves theorem asserts that among all the torodial (i.e. genus 1) closed surfaces, the Clifford torus has the minimal Willmore energy $\int H^2 \, dA$. % It is a natural conjecture that if one prescribes the isoperimetric Since the Willmore energy is invariant M{ö}bius transformations, it can be shown that there is a one-parameter family, up to homotheties, of genus 1 Willmore minimizers. It is then a natural conjecture that such a minimizer is unique if one prescribes its isoperimetric ratio. In this article, we show that this conjecture can be reduced to the positivity question of…
Using Takahashi theorem we propose an approach to extend known families of minimal tori in spheres. As an example, the well-known two-parametric family of Lawson tau-surfaces including tori and Klein bottles is extended to a three-parametric family of tori and Klein bottles minimally immersed in spheres. Extremal spectral properties of the metrics on these surfaces are investigated. These metrics include i) both metrics extremal for the first non-trivial eigenvalue on the torus, i.e. the metric on the Clifford torus and the metric on the equilateral torus and ii) the metric maximal for the fir…
The Willmore conjecture states that any immersion F:T^2 -> R^n of a 2-torus into flat euclidean space satisfies $\int_{T^2} H^2\geq 2π^2$. We prove it under the condition that the L^p-norm of the Gaussian curvature is sufficiently small.
We study the quaternionic Calabi-Yau problem in HKT geometry introduced by Alesker and Verbitsky on 8-dimensional 2-step nilmanifolds with an abelian hypercomplex structure. We show that the quaternionic Monge-Ampère equation on these manifolds can always be solved for every data which is invariant by the action of a 3-dimensional torus.
We prove the metric version of the SYZ conjecture for a class of Calabi-Yau hypersurfaces inside toric Fano manifolds, by solving a variational problem whose minimizer may be interpreted as a global solution of the real Monge-Ampère equation on certain polytopes. This does not rely on discrete symmetry.
In this paper we study Lagrangian tori in ${\mathbb C}P^2$. A two-dimensional periodic Schrödinger operator is associated with every Lagrangian torus in ${\mathbb C}P^2$. We introduce an energy functional for tori as an integral of the potential of the Schrödinger operators, which has a natural geometrical meaning. We study the energy functional on two families of Lagrangian tori and propose a conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori. In particular we show that if the deformation preserves a conformal…
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 Castro-Lerma's conjecture. We also prove that the Clifford torus is the unique compact orientable embedded Lagrangian self-shrinker with nonnegative or nonpositive Gauss curvature in $\mathbb{C}^2$.
In this note, we prove the magnetic spectral gap-labelling conjecture as stated in [arXiv:1508.01064], in all dimensions, for principal solenoidal tori.
We consider the combinatorial Laplacian on a sequence of discrete tori which approximate the m-dimensional torus. In the special case m=1, Friedli and Karlsson derived an asymptotic expansion of the corresponding spectral zeta function in the critical strip, as the approximation parameter goes to infinity. There, the authors have also formulated a conjecture on this asymptotics, that is equivalent to the Riemann conjecture. In this paper, inspired by the work of Friedli and Karlsson, we prove that a similar asymptotic expansion holds for m=2. Similar argument applies to higher dimensions as we…
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 resembles a pair of tori close to the limit torus and joined by many catenoidal bridges arranged over a rectangular lattice on the limit. The collection of sequences is indexed by the ratio of the lengths of the lattice edges, which may be any prescribed positive rational. Unlike th…
In this article we reduce the geometric stability conjecture for the scalar torus rigidity theorem to the conformal case via the Yamabe problem. Then we are able to prove the case where a sequence of Riemannian manifolds is conformal to a uniformly controlled sequence of flat tori and satisfies the geometric stability conjecture. We are also able to handle the case where a sequence of Riemannian manifolds is conformal to a sequence of constant negative scalar curvature Riemannian manifolds which converge to a flat torus in $C^1$. The full conjecture from the conformal perspective is also discu…
