Minimal tori that are linearly full in the 3-sphere possess a natural invariant g called their spectral genus, which was introduced by Hitchin. We show that for each g>0, there are countably many real g-dimensional families of minimally immersed tori with spectral genus g (two of these dimensions are just reparametrisations of the torus). Each family maps from a fixed torus of rectangular conformal type.
We use Whitham deformations to give a complete account of spectral data of real solutions of the sinh--Gordon equation of spectral genus 2. We parameterise the closure of spectral data of constant mean curvature tori in $\mathbb{R}^3$ by an isosceles right triangle and analyse its boundary. We prove that the Wente family, which is described by spectral data with real coefficients, is parameterised by the bisector of the right angle. Our methods combine blowups of Whitham deformations and spectral data in an innovative way that changes the underlying integrable system.
We prove a formula for the determinant of Laplacian on an arbitrary compact polyhedral surface of genus one. This formula generalizes the well-known Ray-Singer result for a flat torus. A special case of flat conical metrics given by the modulus of a meromorphic quadratic differential on an elliptic surface is also considered. We study the determinant of Laplacian as a functional on the moduli space of meromorphic quadratic differentials with L simple poles and L simple zeros and derive formulas for variations of this functional w. r. t. natural coordinates on this moduli space. A new proof of …
In this paper we study equivariant constrained Willmore tori in the 3-sphere. These tori admit a 1-parameter group of Möbius symmetries and are critical points of the Willmore energy under conformal variations. We show that the associated spectral curve of an equivariant torus is given by a double covering of $\mathbb C$ and classify equivariant constrained Willmore tori by the genus g of their spectral curve. In this case the spectral genus satisfies $g \leq 3.$
We prove existence results that give information about the space of minimal immersions of 2-tori into $ S ^ 3 $. More specifically, we show that \begin{enumerate} \item For every positive integer $ n $, there are countably many real $n $-dimensional families of minimally immersed 2-tori in $ S ^ 3 $. Every linearly full minimal immersion $ T ^ 2\to S ^ 3 $ belongs to exactly one of these families. \item Let $ \mathcal A $ be the space of rectangular 2-tori. There is a countable dense subset $\mathcal B $ of $\mathcal A $ such that every torus in $\mathcal B$ can be minimally immersed into $ S …
We extend the techniques in a previous paper to calculate the Heegaard Floer homology groups for fibered 3-manifolds M whose monodromy is a power of a Dehn twist about a genus-1 separating circle on a surface of genus g > 1. We only consider non-torsion Spin^c-structures on M.
In this paper, we show that an embedded Weingarten surface in S^3 of genus 1 must be rotationally symmetric, provided that certain structure conditions are satisfied. The argument involves an adaptation of our proof of Lawson's Conjecture for minimal tori.
Constant mean curvature (CMC) tori in Euclidean 3-space are described by an algebraic curve, called the spectral curve, together with a line bundle on this curve and a point on $ S ^ 1 $, called the Sym point. For a given spectral curve the possible choices of line bundle and Sym point are easily described. The space of spectral curves of tori is totally disconnected. Hence to characterise the "moduli space" of CMC tori one should, for each genus $g$, determine the closure $\overline{\mathcal{P}^g}$ of spectral curves of CMC tori within the spectral curves of CMC planes having spectral genus $…
We show the existence of several new families of non-compact constant mean curvature surfaces: (i) singly-punctured surfaces of arbitrary genus $g \geq 1$, (ii) doubly-punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.
We show that all superconformal harmonic immersions from genus one surfaces into de Sitter spaces $ S ^ {2n}_1 $ with globally defined harmonic sequence are of finite-type and hence result merely from solving a pair of ordinary differential equations. As an application, we prove that all Willmore tori in $ S ^ 3 $ without umbilic points can be constructed in this simple way.
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.
We show that any minimal torus in $S^3$ which is Alexandrov immersed must be rotationally symmetric. An analogous result holds for surfaces of constant mean curvature.
Special class of surfaces in five-dimensional sphere in $C^3$ is considered. Immersion equations for minimal tori of that class are shown to be reducible to the equation $u_{z\bar z}=e^u-e^{-2u}$ which is integrable by means of inverse scattering method. Finite-gap minimal tori are constructed.
In this paper, we study trigonal minimal surfaces in flat tori. First, we show a topological obstruction similar to that of hyperelliptic minimal surfaces. Actually, the genus of trigonal minimal surface in 3-dimensional flat torus must be 1 (mod 3). Next, we construct an explicit example in the higher codimensional case. This surface satisfies good properties and is theoretically distinct from traditional examples.
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.
In this paper, we consider new components of the key space of the Moduli space of minimal surfaces in flat 4-tori and calculate their dimensions. Moreover, we construct an example of minimal surfaces in 4-tori and obtain an element of the Moduli. In the process of the construction, we give an example of minimal surfaces with good property in a 3-torus distinct from classical examples.
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.
In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 1$ and $m\geq 1$ ideal boundary components. We show that the width for the area functional can be achieved by a bubble tree limit consisting of branched genus $g$ free boundary minimal surfaces with nodes, and possibly finitely many branched minimal spheres and free boundary minimal disks.
In this paper by reduction we construct a family of conformally flat Hamiltonian-minimal Lagrangian tori in $\mathbb{CP}^3$ as the image of the composition of the Hopf map $\mathcal{H}: \mathbb{S}^7\to \mathbb{CP}^3$ and a map $ψ:\mathbb{R}^3 \to \mathbb{S}^7$ with certain conditions.
We prove that for each positive integer g, there exists a complete minimal surface of genus g that is properly embedded in three-dimensional euclidean space and that is asymptotic to the helicoid.
We investigate which orbits of an $n$-dimensional torus action on a $2n$-dimensional toric Kähler manifold $M$ are minimal. In other words, we study minimal submanifolds appearing as the fibres of the moment map on a toric Kähler manifold. Amongst other questions we investigate and give partial answers to the following: (1) How many such minimal Lagrangian tori exist? (2) Can their stability, as critical points of the area functional, be characterised just from the ambient geometry? (3) Given a toric symplectic manifold, for which sets of orbits $S$, is there a compatible toric Kähler metric w…
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…
We show that for every non-negative integer n there is a real n-dimensional family of minimal Lagrangian tori in CP^2, and hence of special Lagrangian cones in C^3 whose link is a torus. The proof utilises the fact that such tori arise from integrable systems, and can be described using algebro-geometric (spectral curve) data.
Constant mean curvature surfaces in $S^3$ can be studied via their associated family of flat connections. In the case of tori this approach has led to a deep understanding of the moduli space of all CMC tori. For compact CMC surfaces of higher genus the theory is far more involved due to the non abelian nature of their fundamental group. In this paper we extend the spectral curve theory for tori developed in \cite{Hi, PiSt} and for genus $2$ surfaces \cite{He3} to CMC surfaces in $S^3$ of genus $g=k\cdot l$ with commuting $\mathbb Z_{k+1}$ and $\mathbb Z_{l+1}$ symmetries. We determine their a…
A Euclidean minimal torus with planar ends gives rise to an immersed Willmore torus in the conformal 3--sphere $S^3=\R^3\cup \{\infty\}$. The class of Willmore tori obtained this way is given a spectral theoretic characterization as the class of Willmore tori with reducible spectral curve. A spectral curve of this type is necessarily the double of the spectral curve of an elliptic KP soliton. The simplest possible examples of minimal tori with planar ends are related to 1--gap Lamé potentials, the simplest non--trivial algebro geometric KdV potentials. If one allows for translational periods, …
We employ min-max techniques to show that the unit ball in $\mathbb{R}^3$ contains embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
H-minimal Lagrangian submanifolds in general Kähler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with negative first Chern class.
Generalizing earlier work by Ros in ambient dimension three, we prove an affine lower bound for the Morse index of closed minimal hypersurfaces inside a flat torus in terms of their first Betti number (with purely dimensional coefficients).
We construct higher genus Riemann's minimal surfaces properly embedded in the Euclidean space. To do that we glue end by end a Costa-Hoffman-Meeks examples to two halves genus zero Riemann's minimal surfaces. In first we need to perform a deformation of a Costa-Hoffman-Meeks example to prescribe the flux vector along the catenoidal ends. Then we study the mapping property of the Jacobi operator on the half Riemann example as a perturbation analysis of a CMC-Delaunay half cylinder.
We construct Weierstrass data for higher genus embedded doubly periodic minimal surfaces and present numerical evidence that the associated period problem can be solved. In the orthogonal ends case, there previously was only one known surface for each genus. We illustrate multiple new examples for each genus g>2. In the parallel ends case, the known examples limit as a foliation of parallel planes with nodes. We construct a new example for each genus g>2 that limit as g-1 singly periodic Scherk surfaces glued between two doubly periodic Scherk surfaces and also as a singly periodic surface wit…
Let $\mathcal{K}$ be the space of properly embedded minimal tori in quotients of $\R^3$ by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that $\mathcal{K}$ is a 3-dimensional real analytic manifold that reduces to the finite coverings of the examples defined by Karcher, Meeks and Rosenberg in \cite{ka4,ka6,mr3}. The degenerate limits of surfaces in $\mathcal{K}$ are the catenoid, the helicoid and three 1-parameter families of surfaces: the simply and doubly periodic Scherk minimal surfaces and the Riemann minimal examp…
Let M be a 3-manifold (possibly with boundary). We show that, for any positive integer g, there exists an open nonempty set of metrics on M for each of which there are stable compact embedded minimal surfaces of genus g with arbitrarily large area. This extends the result of Colding and Minicozzi for g=1.
We prove optimal genus bounds for minimal surfaces arising from the min-max construction of Simon-Smith. This confirms a conjecture made by Pitts-Rubinstein in 1986.
We use Colding--Minicozzi lamination theory to study the systole of large genus minimal surfaces in an ambient three-manifold of positive Ricci curvature.
In this paper we show that all totally real superconformal minimal tori in $CP^2$ correspond with doubly-periodic finite gap solutions of the Tzitzeica equation $$ω_{z{\overline z}}=e^{-2ω}-e^ω$$ Using the results on the Tzitzeica equation in integrable system theory, we describe explicitly all these tori by Prym-theta functions.
For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cover and that the Busemann functions with these directions are unique up to adding constants. Moreover, using a kind of weak KAM theory, we show that for almost all types of minimal geodesics in the sense of Morse, there is precisely one minimal geodesic of this type.
