Doubling Constant Mean Curvature Tori in the 3-Sphere
Adrian Butscher, Frank Pacard
备注：
22 pages. Final version improves the statement of the theorem, correct some errors and improves the presentation. Accepted for publication by Annali SNS Pisa
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 satisfies the CMC condition.
We construct closed embedded minimal surfaces in the round three-sphere, resembling two parallel copies of the equatorial two-sphere, joined by small catenoidal bridges symmetrically arranged either along two parallel circles of the equator, or along the equatorial circle and the poles. To carry out these constructions we refine and reorganize the doubling methodology in ways which we expect to apply also to further constructions. In particular we introduce what we call linearized doubling, which is an intermediate step where singular solutions to the linearized equation are constructed subjec…
We consider generalized Hodge-Laplace operators $αd δ+ βδd$ for $α, β> 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.
In earlier work of NK new closed embedded smooth minimal surfaces in the round three-sphere $\mathbb{S}^3(1)$ were constructed, each resembling two parallel copies of the equatorial two-sphere $\mathbb{S}^2_{eq}$ joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, or the equatorial circle and the poles. In this sequel we generalize those constructions so that the catenoidal bridges can concentrate along an arbitrary number of parallel circles, with the further option to include bridges at the poles. The current constructions follow the Line…
We consider proper-biharmonic flat tori with constant mean curvature (CMC) in spheres and find necessary and sufficient conditions for certain rectangular tori and square tori to admit full CMC proper-biharmonic immersions in $\mathbb{S}^n$, as well as the explicit expressions of some of these immersions.
We construct eternal mean curvature flows of tori in perturbations of the standard unit sphere $\Bbb{S}^3$. This has applications to the study of the Morse homologies of area functionals over the space of embedded tori in $\Bbb{S}^3$.
We show that the extrinsic diameter of immersed flat tori in the 3-sphere is $π$ under a certain topological condition for the projection of their asymptotic curves with respect to the Hopf fibration.
This paper is devoted to the study of minimal immersions of flat $n$-tori into spheres, especially those immersed by the first eigenfunctions (such immersion is called $λ_1$-minimal immersion), which also play important roles in spectral geometry. It is known that there are only two non-congruent $λ_1$-minimal $2$-tori in spheres, which are both flat. For higher dimensional case, the Clifford $n$-torus in $\mathbb{S}^{2n-1}$ might be the only known example in the literature. In this paper, by discussing the general construction of homogeneous minimal flat $n$-tori in spheres, we construct many…
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…
In this paper we give an explicit parametrisation of the moduli space of equivariant harmonic maps from a 2-torus to the 3-sphere. As Hitchin proved, a harmonic map of a 2-torus is described by its spectral data, which consists of a hyperelliptic curve together with a pair of differentials and a line bundle. The space of spectral data is naturally a fibre bundle over the space of spectral curves. For homogeneous tori the space of spectral curves is a disc and the bundle is trivial. For tori with a one-dimensional invariance group, we enumerate the path connected components of the space of spec…
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…
The study of minimal surfaces has a long history, due to the important applications. Given a fixed boundary, one wants to minimise the surface area: this can be used, for example, to minimise the area of the roof of a building. Similarly, looking for constant mean curvature (CMC) provides us with many interesting applications in physics: one of the easiest examples are soap bubbles. In this work however we occupy ourselves with minimal and constant mean curvature surfaces in the three-dimensional sphere $S^3$ and its dual space $Σ^3$. In Chapter 1 we give a brief overview of the tools of Riema…
There are not so many kinds of surface of revolution whose cut locus structure have been determined, although the cut locus structures of very familiar surfaces of revolution (in Euclidean space) such as ellipsoids, 2-sheeted hyperboloids, paraboloids, and tori are now known. Except for tori, the known cut locus structures are very simple, i.e., a single point or an arc. In this article, a new family {M_n}_n of 2-spheres of revolution with simple cut locus structure is introduced. This family is also new in the sense that the number of points on each meridian which assume a local minimum or ma…
A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below $8π$. Combining such a strict inequality with previous works by Keller-Mondino-Rivière and Mondino-Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution …
In this paper, we study biharmonic isometric immersions of a surface into and biharmonic Riemannian submersion from 3-dimensional Berger spheres. We obtain a classification of proper biharmonic isometric immersions of a surface with constant mean curvature into Berger 3-spheres. We also give a complete classification of proper biharmonic Hopf tori in Berger 3-sphere. For Riemannian submersions, we prove that a Riemannian submersion from Berger 3-spheres into a surface is biharmonic if and only if it is harmonic.
