François Labourie
A report on an article of Calegari, Marques and Neves about counting minimal surfaces. An idea of a proof is included using the concept of a laminar measure.
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Bourbaki Seminar 1081 : Min-max methods and the Willmore conjecture, after Fernando Codá Marques and André Arroja Neves
两年前，FC马克斯和A.A. Neves在三维球的封闭可求长二维流的框架内实现了几何测度论中的极小-极大方法，这是由于F。Almgren和J. Pitts。使用这种方法，他们成功地证明了著名的克利福德环面最小化所有非零亏格在$S^3$的封闭极小曲面之间的面积。他们的工作的另一个引人注目的结果是提供了Willmore猜想的证明。这次谈话的目的是讨论第一的一般框架，这两个定理的马克斯和内维斯。然后，我们将介绍其证明的结构和一些关键细节。然后，我们将解决这个显着的贡献微积分的范围内的表面上的变化在一个3维空间。
A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher dimensions and, more precisely, on any smooth closed riemannian manifod of arbitrary dimension between 3 and 7 there exists infinitely many closed embedded minimal surfaces. We are going to present the origins of this theorem as well as it's proof given recently by Antoine Song.
Let $M$ be a closed hyperbolic 3-manifold that admits no infinitesimal conformally-flat deformations. Examples of such manifolds were constructed by Kapovich. Then if $g$ is a Riemannian metric on $M$ with scalar curvature greater than or equal to $-6$, we find lower bounds for the areas of stable immersed minimal surfaces $Σ$ in $M$. Our bounds improve the closer $Σ$ is to being homotopic to a totally geodesic surface in the hyperbolic metric. We also consider a functional introduced by Calegari-Marques-Neves that is defined by an asymptotic count of minimal surfaces in $(M,g)$. We show this …
For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in $M$. This gives a quantitative version of the main result of \cite{irie-marques-neves}, by Irie and the first two authors, that established denseness of minimal hypersurfaces for generic metrics. As in \cite{irie-marques-neves}, the main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich and the first two authors in \cite{liokumovich-marques…
In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of the program (\cite{marques-icm}, \cite{marques-neves-cycles}, \cite{marques-neves-index}, \cite{neves-icm}) proposed by the authors with the goal of developing a Morse theory for the area functional.
We prove a Weyl Law for the phase transition spectrum based on the techniques of Liokumovich-Marques-Neves. As an application we give phase transition adaptations of the proofs of the density and equidistribution of minimal hypersufaces for generic metrics by Irie-Marques-Neves and Marques-Neves-Song, respectively. We also prove the density of separating limit interfaces for generic metrics in dimension 3, based on the recent work of Chodosh-Mantoulidis, and for generic metrics on manifolds containing only separating minimal hypersurfaces, e.g. $H_n(M,\mathbb{Z}_2) = 0$, for $4 \leq n+1 \leq 7…
Minimal Surface Entropy and Average Area Ratio
本文在闭可双曲化三维流形上建立了极小曲面熵与平均面积比的精确关系，证明在数量曲率≥-6的度量中双曲度量最大化前者，从而解决Gromov猜想，证明采用带手术的Ricci流与PSL(2,R)不变层流测度。
Scalar curvature rigidity with a volume constraint
受Brendle-Marques-Neves对Min-Oo猜想的反例的启发，我们证明了一个适用于半球的体积约束数量曲率刚性定理.
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 pr…
We show that in a closed 3-manifold with a generic metric of positive Ricci curvature, there are minimal surfaces of arbitrary large Morse index, which partially confirms a conjecture by F. Marques and A. Neves. We prove this by analyzing the lamination structure of the limit of minimal surfaces with bounded Morse index.
The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen-Cahn setting, a strong form of th…
We address the one-parameter minmax construction, via Allen--Cahn energy, that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold $N^{n+1}$ with $n\geq 2$ (see Guaraco's 2018 work). We obtain the following multiplicity-$1$ result: if the Ricci curvature of $N$ is positive then the minmax Allen--Cahn solutions concentrate around a multiplicity-$1$ hypersurface, that may have a singular set of dimension $\leq n-7$. This result is new for $n\geq 3$ (for $n=2$ it is also implied by the recent work by Chodosh--Mantoulidis).…
Width Stability of Rotationally Symmetric Metrics
本文证明了旋转对称条件下三维球面极小极大宽度稳定性的Marques-Neves猜想，并将其推广至任意维度$n\geqslant 3$。
Strong multiplicity one theorems and homological min-max theory
本文证明了单位3球面的第10至第13个极小极大宽度严格介于2π²和8π之间，并推广了X. Zhou的重数一定理。
Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds on the methods developed by F. C. Marques and A. Neves.
In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, we prove the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves shows that for any compact Riemannian manifolds with nonnegative R…
In this paper, we show that a closed manifold $M^{n+1} (n \geq 7)$ endowed with a $C^\infty$-generic (Baire sense) metric contains infinitely many singular minimal hypersurfaces with optimal regularity. Moreover, for $2 \leq n \leq 6$, our argument also implies the denseness of the minimal hypersurfaces realizing min-max widths for generic metrics. This partially supports equidistribution of the minimal hypersurfaces realizing min-max widths conjectured by F.C. Marques, A. Neves and A. Song.
Foliated Plateau problems and asymptotic counting of surface subgroups
本文研究三维流形中曲面子群依$k$-曲面面积的渐近计数，确定下界并证明刚性，通过解决Cartan--Hadamard流形中的一般叶状Plateau问题克服了现有证明的局限。
E. Calabi and J. Cao showed that a closed geodesic of least length in a two-sphere with nonnegative curvature is always simple. Using min-max theory, we prove that for some higher dimensions, this result holds without assumptions on the curvature. More precisely, in a closed $(n+1)$-manifold with $2 \leq n \leq 6$, a least area closed minimal hypersurface exists and any such hypersurface is embedded. As an application, we give a short proof of the fact that if a closed three-manifold $M$ has scalar curvature at least $6$ and is not isometric to the round three-sphere, then $M$ contains an embe…
Existence of minimal hypersurfaces with arbitrarily large area and possible obstructions
本文证明在3至7维闭黎曼流形中，要么存在任意大面积的极小超曲面，要么存在不可数个稳定极小超曲面，并应用于解析流形，证明中结合了Almgren-Pitts理论、Song的思想和Zhou的重数为一猜想解决。
While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Marques and Neves used the $p$-width to prove the existence of infinite many minimal hypersurfaces in compact manifolds with positive Ricci curvature. However, whether the $p$-width can be realized as the volume of minimal hypersurfaces is not known yet. We introduced the concept o…
On the Willmore energy of Möbius bands
本文证明以$\mathbb{S}^3$中大圆为边界的Möbius带中，最小Willmore能量由Morse指标2的嵌入极小带实现，并引入2参数典范族将Kusner猜想归结为Lawson带的刻画。
Let $(M,g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of S.-T. Yau's conjecture on the abundance of minimal surfaces and builds on a result of M. Gromov. Suppose that $(M,g)$ has bounded geometry, or more generally is thick at infinity. Then the following dichotomy holds for the space of closed hypersurfaces in $M$: either there are infinitely many saddle points of the $n$-volume functional, or there is none. Additionally, we give a new short proof of the existence of a finite volume minimal hypersurface in finite volu…
Width, Ricci curvature and minimal hypersurfaces
设$(M,g_0)$是一个闭黎曼流形，其维数为$n$，对于$3 \leq n \leq 7$，且具有非负的里奇曲率。令$g = ϕ^2 g_0$是$g_0$的共形类中的度量。证明了在体积为$C V^{\frac{n-1}{n}}$的$(M,g)$中存在光滑的闭嵌入极小超曲面，其中$V$是$(M,g)$的总体积，$C$是只依赖于$n$的常数.当$Ric(M,g_0) \geq -(n-1)$时，我们得到了一个类似的结合常数$C$仅取决于$n$和$(M,g_0)$的体积。我们的第二个结果涉及流形$(M,g)$的积极里奇曲率。我们得到了F的一个定理的有效形式。Coda Marques和A. Neves关于$(M,g)$上无穷多极小超曲面的存在性。我们证明了对于任何这样的流形，存在体积至多为$C_n V \left( sys_{n-1}(M)\right)^{-\frac{1}{n-1}} k ^ {\frac{1}{n-1}}$的$k$极小超曲面，其中$V$表示$(M,g_0)$的体积，$sys_{n-1}(M)$是非平凡极小超曲面的最小体积.
Free boundary minimal annuli in convex balls
该研究构造了黎曼3-球中含4参数的真嵌入零亏格曲面族，作为Marques-Neves典范曲面族的自由边界类比，还应用其得到Almgren-Pitts 4-宽度上界，并证明特定紧致3-流形中至少存在3个自由边界极小环面。
Min-max minimal surfaces, horizons and electrostatic systems
本文建立指标一极小曲面与广义相对论的联系，证明某类（电）静电系统的不稳定视界是面积泛函单参数极小极大问题的解且指标一，得到指标一极小曲面面积与电荷的不等式及全局版本与刚性结果。
Minimal surfaces in S^3: a survey of recent results
在这篇综述中，我们讨论了三球面S ^[3]中极小曲面方程的各个方面。在回顾了基本定义之后，我们描述了一族具有旋转对称性的浸入极小环面。然后，我们回顾了已知的嵌入S^3中的极小曲面的例子。除了赤道和克利福德环面，这些例子还包括劳森和Kapouleas-Yang例子，以及Choe和Soret最近发现的一个新的例子家族。接下来我们讨论S^3中极小曲面的唯一性定理，例如Almgren在亏格为0的情况下的工作，以及我们最近对亏格为1的嵌入极小曲面的劳森猜想的解决方案。更一般地说，我们表明，任何极小曲面的亏格1，这是亚历山德罗夫浸入必须是旋转对称的。我们还讨论了嵌入的极小曲面的Morse指数的Urbano估计，并给出了最近由Marques和Neves证明Willmore猜想的大纲。最后，我们描述的第一特征值的拉普拉斯算子的极小曲面的估计。
We study global variational properties of the space of solutions to $-\varepsilon^2Δu + W'(u)=0$ on any closed Riemannian manifold $M$. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min-max and have index 1. We show that if $\varepsilon$ is not small enough, in terms of the Cheeger constant of $M$, then there are no interesting solutions. However, we show that the number of mi…
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