Willmore Conjecture Research Articles

A short note about 1-width of Lens spaces

In this note, we explore the nature of Lens spaces to study the first width of those spaces, more precisely, we use the existence of a sharp sweep out associated to a Clifford torus to provide a simple and pretty application of the Willmore conjecture for the computation of the [Formula: see text]-width of Lens spaces.

The isoperimetric problem for lens spaces

We solve the isoperimetric problem in the lens spaces with large fundamental group. Namely, we prove that the isoperimetric surfaces are either geodesic spheres or quotients of Clifford tori. We also show that the isoperimetric problem in the lens spaces L(3, 1) and L(3, 2) follows from the proof of the Willmore conjecture by Marques and Neves.

Entropy of closed surfaces and min-max theory

Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding–Ilmanen–Minicozzi–White conjectured (since proved by Bernstein–Wang) that the entropy of any closed surface is at least that of the self-shrinking two-sphere. In this paper we give an alternative proof of their conjecture for closed embedded 2-spheres. Our results can be thought of as the parabolic analog to the Willmore conjecture and our argument is analogous in many ways to that of Marques–Neves on the Willmore problem. The main tool is the min-max theory applied to the Gaussian area functional in $\mathbb{R}^3$ which we also establish. To any closed surface in $\mathbb{R}^3$ we associate a four parameter canonical family of surfaces and run a min-max procedure. The key step is ruling out the min-max sequence approaching a self-shrinking plane, and we accomplish this with a degree argument. To establish the min-max theory for $\mathbb{R}^3$ with Gaussian weight, the crucial ingredient is a tightening map that decreases the mass of nonstationary varifolds (with respect to the Gaussian metric of $\mathbb{R}^3$) in a continuous manner.

Min–max hypersurface in manifold of positive Ricci curvature

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the Morse index, area and multiplicity of this singular min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and has Morse index one, or is a double cover of a non-orientable stable minimal hypersurface. As an essential technical tool, we prove a stronger version of the discretization theorem. The discretization theorem, first developed by Marques-Neves in their proof of the Willmore conjecture \cite{MN12}, is a bridge to connect sweepouts appearing naturally in geometry to sweepouts used in the min-max theory. Our result removes a critical assumption of \cite{MN12}, called the no mass concentration condition, and hence confirms a conjecture by Marques-Neves in \cite{MN12}.

Preface Issue 4-2014

The last few years have been a good time for solving long standing geometrical-topological conjectures. This issue reports on the solution of the Willmore conjecture –the “best” topological torus is a “real” torus with ratio of radii equal to √ 2– and one of Thurston’s conjectures –every hyperbolic 3-manifold can be fibered over a circle, up to passing to a finite cover. Starting in the 1960s, Thomas Willmore studied the integral of the squared mean curvature of surfaces in R3 as the simplest but most interesting frame invariant elastic bending energy. This energy had shown up already in the early 19th century – too early for a rigorous investigation. Willmore asked: What is the shape of a compact surface of fixed genus minimising the Willmore energy in this class? (In the 1990s, existence of minimisers was proved by Leon Simon, with a contribution by Matthias Bauer and Ernst Kuwert.) Willmore already knew that the genus-0minimiser is a sphere. Assuming that symmetric surfaces require less energy than asymmetric ones (which has not been proved, yet) he studied families of geometric tori with the smaller radius 1 fixed and found that the larger radius √ 2 would yield the minimum in this very special family. He conjectured that this particular torus would be the genus-1-minimiser. Almost 50 years later Fernando Marques and Andre Neves found and published a 100-page-proof. In the present survey article, disregarding technical details, they explain their approach and their fundamental ideas and provide also an excellent historical and mathematical background. After Thurston’s Geometrization Conjecture was finally solved about 10 years ago by Grigori Perelman, one of the most important questions in three-dimensional topology has been to understand the hyperbolic 3-manifolds. William Thurston’s 1982–paper “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry” (see the “Classics Revisited”–contribution of this year’s first issue) lists 24 at that time open problems in 3-manifolds-theory. Problem No. 18 reads as follows: “Does every finite volume hyperbolic 3-manifold have a finite cover which fibers over the circle?” This conjecture was indeed proved 2012 by Ian Agol and Dani Wise. Stefan Friedl explains their work and the requisite background in his very comprehensible survey article “Thurstons Vision for 3-Manifolds and the Virtual Fibering Theorem.” As Stefan Friedl explains, after Agol & Wise, only “Thurston’s last challenge” concerning ratios of volumes of hyperbolic 3-manifolds remains to be solved. In this issue the “Classics Revisited”-contribution (in collaboration with the Zentralblatt fur Mathematik) is written by Jean Mawhin. It concerns the “Cours d’Analyse Infinitesimale” of Charles-Jean de La Vallee Poussin, who is still famous for his proof (independent of Hadamard’s) of the prime number theorem. These textbooks were used in Belgium for an exceptionally long time. Reviewers praise the originality, clearness and elegance of the exposition and its merits as a handbook in analysis. In the “Book Reviews”-section Gunter M. Ziegler gives us a critical view of Alexander Soifer’s “The Mathematical Coloring Book”. Johannes Huebschmann evaluates the book on “Poisson Structures” written by Camille Laurent-Gengoux, Anne Picherau, and Pol Vanhaecke.

The Willmore Conjecture

The Willmore conjecture, proposed in 1965, concerns the quest to find the best torus of all. This problem has inspired a lot of mathematics over the years, helping bringing together ideas from subjects like conformal geometry, partial differential equations, algebraic geometry and geometric measure theory. In this article we survey the history of the conjecture and our recent solution through the min-max approach. We finish with a discussion of some of the many open questions that remain in the field.

What surface maximizes entanglement entropy?

For a given quantum field theory, provided the area of the entangling surface is fixed, what surface maximizes entanglement entropy? We analyze the answer to this question in four and higher dimensions. Surprisingly, in four dimensions the answer is related to a mathematical problem of finding surfaces which minimize the Willmore (bending) energy and eventually to the Willmore conjecture. We propose a generalization of the Willmore energy in higher dimensions and analyze its minimizers in a general class of topologies $S^m\times S^n$ and make certain observations and conjectures which may have some mathematical significance.

Mini-Workshop: The Willmore Functional and the Willmore Conjecture

The Willmore functional evaluated on a surface immersed into Euclidean space is given by the L^2 -norm of its mean curvature. The interest for studying this functional comes from various directions. First, it arises in applications from biology and physics, where it is used to model surface tension in the Helfrich model for bilipid layers, or in General Relativity where it appears in Hawking’s quasi-local mass. Second, the mathematical properties justify consideration of the Willmore functional in its own right. The Willmore functional is one of the most natural extrinsic curvature functionals for immersions. Its critical points solve a fourth order Euler-Lagrange equation, which has all minimal surfaces as solutions.

Progress in Surface Theory

Over the last 30 years global surface theory has become pivotal in the understanding of low dimensional global phenomena. At the same time surface geometry became a platform on which seemingly different areas of mathematics – such as geometric and topological analysis, integrable systems, algebraic geometry of curves, and mathematical physics – coalesce to produce far reaching ideas, conjectures, methods and results. The workshop hosted talks on the resolutions of famous conjectures in surface geometry, including the Willmore conjecture, and on exciting new progress in the understanding of moduli spaces of special surface classes.

The Min-Max theory and the Willmore conjecture

In 1965, T. J. Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in R 3 is at least 2 2 . We prove this conjecture using the min-max theory of minimal surfaces.

Min-max theory, Willmore conjecture and the energy of links

In this paper we give an overview of some aspects of the min-max theory of minimal surfaces, and discuss recent applications to conformally invariant problems in Geometry and Topology. The goal is to explain what the proofs of the Willmore conjecture for surfaces and the Freedman-He-Wang conjecture for links share in common. This is based on joint work of the authors [19] and on joint work of I. Agol and the authors [1].

Minimal surfaces in $$S^3$$ : a survey of recent results

In this survey, we discuss various aspects of the minimal surface equation in the three-sphere $$S^3$$ . After recalling the basic definitions, we describe a family of immersed minimal tori with rotational symmetry. We then review the known examples of embedded minimal surfaces in $$S^3$$ . Besides the equator and the Clifford torus, these include the Lawson and Kapouleas-Yang examples, as well as a new family of examples found recently by Choe and Soret. We next discuss uniqueness theorems for minimal surfaces in $$S^3$$ , such as the work of Almgren on the genus $$0$$ case, and our recent solution of Lawson’s conjecture for embedded minimal surfaces of genus $$1$$ . More generally, we show that any minimal surface of genus $$1$$ which is Alexandrov immersed must be rotationally symmetric. We also discuss Urbano’s estimate for the Morse index of an embedded minimal surface and give an outline of the recent proof of the Willmore conjecture by Marques and Neves. Finally, we describe estimates for the first eigenvalue of the Laplacian on a minimal surface.

Applications of Almgren-Pitts Min-max theory

We survey recent applications of Almgren-Pitts Min-max theory found by the authors. Namely, the proof of the Willmore conjecture, the proof of the Freedman–He–Wang conjecture on the Mobius energy of links (jointly with Ian Agol), and the proof that manifolds with positive Ricci curvature metrics admit an infinite number of minimal hypersurfaces.

On the Willmore Conjecture for Surfaces

On the Willmore Conjecture for Surfaces

Towards the Willmore conjecture

We develop a variety of approaches, mainly using integral geometry, to proving that the integral of the square of the mean curvature of a torus immersed in ${\mathbb R}^3$ must always take a value no less than $2\pi^2$ . Our partial results, phrased mainly within the $S^3$ -formulation of the problem, are typically strongest when the Gauss curvature can be controlled in terms of extrinsic curvatures or when the torus enjoys further properties related to its distribution within the ambient space (see Sect. 3). Corollaries include a recent result of Ros [20] confirming the Willmore conjecture for surfaces invariant under the antipodal map, and a strengthening of the expected results for flat tori. The value $2\pi^2$ arises in this work in a number of different ways – as the volume (or renormalised volume) of $S^3, SO(3)$ or $G_{2,4}$ , and in terms of the length of shortest nontrivial loops in subgroups of SO(4).

The Willmore conjecture for immersed tori with small curvature integral

The Willmore conjecture states that any immersion of a 2-torus into euclidean space satisfies . We prove it under the condition that the L p -norm of the Gaussian curvature is sufficiently small.

Finite-gap solutions of the modified novikov-veselov equations: Their spectral properties and applications

We construct finite-gap solutions to the modified Novikov-Veselov equations, describe their spectral properties and the reduction to the modified Korteweg--de Vries equation and explain its relation to soliton deformations of tori and the Willmore conjecture.

A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

Let M be a compact Riemannian manifold with a fixed conformal structure. Then we introduce the concept of conformal volume of M in the following manner. For each branched conformal immersion q9 of M into the unit sphere S n, we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of S". We let Vc(n, qg) be the max imum volume of these branched immersions. The conformal volume of M is defined to be the infimum of V.(n, q0) where qo ranges over all branched conformal immersions of M into the unit sphere S". In this paper, we study the case when M is a compact surface and we call the conformal volume of M to be the conformal area of M. We demonst ra te that this conformal invariant is non-trivial. In fact, we prove that if there exists a minimal immersion of M into S" where coordinate functions are first eigenfunctions, then the conformal area of M is given by the area of M with respect to the induced metric. This enables us to compute the conformal area for several surfaces. For example, the conformal area of R P z is 6n and the conformal area of the square torus is 27c 2. We believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces. We demonstrate this claim by applying the concept of conformal area to two different branches of surface theory. The first application is to study the total curvature of a compact surface in R". This problem has a long history. Fenchel and Fary [9] proved that for a closed curve o in R", Slk[>27c where k is its curvature. Then Milnor [12]

The Willmore Conjecture in the Real Projective Space

-We prove that for any torus M immersed in the real projective space RP3(1) with mean curvature H, we have that ∫ M (1 +H 2)dA ≥ π2 and that the equality holds only for the minimal Clifford torus. In terms of the three sphere, this result says that the Willmore conjecture is true for immersed tori in S3(1) invariant under the antipodal map. Mathematics Subject Classification: 53A10, 53A05, 53C42.