William P. Minicozzi
The study of embedded minimal surfaces in $\RR^3$ is a classical problem, dating to the mid 1700's, and many people have made key contributions. We will survey a few recent advances, focusing on joint work with Tobias H. Colding of MIT and Courant, and taking the opportunity to focus on results that have not been highlighted elsewhere.
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In this paper we give two examples of sequences of embedded minimal planar domains in $\mathbb{R}^3$ which converge to singular laminations of $\mathbb{R}^3$. In contrast with the situation for embedded minimal disks, these examples do not arise from complete embedded minimal planar domains and highlight some of the subtleties inherent in understanding refined properties of embedded minimal planar domains.
We show that any embedded minimal torus in S^3 is congruent to the Clifford torus. This answers a question posed by H.B. Lawson, Jr., in 1970.
given two minimal surfaces embedded in $\S3$ of genus $g$ we prove the existence of a sequence of non-congruent compact minimal surfaces embedded in $\S3$ of genus $g$ that converges in $C^{2,α}$ to a compact embedded minimal surface provided some conditions are satisfied. These conditions also imply that, if any of these two surfaces is embedded by the first eigenvalue, so is the other.
We present a new construction of embedded minimal surfaces in hyperbolic space with $3$ asymptotically totally geodesic ends and arbitrary finite genus.
We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded minimal submanifolds in simply connected noncompact globally symmetric spaces.
We show that the image of a nonconstant conformal harmonic map $\mathbb C\to \mathbb R^3$, not necessarily proper and possibly with branch points, intersects every properly embedded nonflat minimal surface of bounded curvature in $\mathbb R^3$. The same holds if $\mathbb C$ is replaced by any open conformal surface without nonconstant bounded subharmonic functions.
We show that a compact embedded minimal or constant mean curvature annulus with non-vanishing Gaussian curvature which is tangent to two spheres of same radius or tangent to a sphere and meeting a plane in constant contact angle is rotational.
In this paper, we discuss complete minimal immersions in $\mathbb{R}^N$($N\geq4$) with finite total curvature and embedded planar ends. First, we prove nonexistence for the following cases: (1) genus 1 with 2 embedded planar ends, (2) genus $\neq4$, hyperelliptic with 2 embedded planar ends like the Lagrangian catenoid. Then we show the existence of embedded minimal spheres in $\mathbb{R}^4$ with 3 embedded planar ends. Moreover, we construct genus $g$ examples in $\mathbb{R}^4$ with $d$ embedded planar ends such that $g\geq 1$ and $g+2\leq d\leq 2g+1$. These examples include a family of embed…
Examples of complete minimal surfaces properly embedded in H^2 x R have been extensively studied and the literature contains a plethora of nontrivial ones. In this paper we construct a large class of examples of complete minimal surfaces embedded in H^2 x R, not necessarily proper, which are invariant by a vertical translation or by a hyperbolic or parabolic screw motion. In particular, we construct a large family of non-proper complete minimal disks embedded in H^2 x R invariant by a vertical translation and a hyperbolic screw motion and whose importance is twofold. They have finite total cur…
This paper proves that classical minimal surfaces of arbitrary topological type with total boundary curvature at most 4πmust be smoothly embedded. Related results are proved for varifolds and for soap film surfaces.
Genus three embedded doubly periodic minimal surfaces with parallel ends
本文构造了一族亏格为三的嵌入双周期极小曲面，具有四个平行端口，并解决了二维周期问题。
We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil_3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil_3 is entire. Also, the sister surface of an entire minimal graph in Nil_3 is an entire constant mean curvature 1/2 graph in H^2 x R, and conversely. This gives a classification of all entire constant mean curvature 1/2 graphs in H^2 x R. Finally we construct properly embedded constant mean curvature 1/2 …
We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies apriori area and curvature estimates for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness result for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.
Uniqueness of embedded minimal Lagrangian tori in $\mathbb{CP}^2$
本文证明复射影平面中任意嵌入极小拉格朗日环面必全等于克利福德环面，方法为利用五维球面归一化水平提升构造两点函数。
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The ends of a complete embedded minimal surface of {\em finite total curvature} are well understood (every such end is asymptotic to a catenoid or to a plane). We give a similar characterization for a large class of ends of {\em infinite total curvature}, showing that each such end is asymptotic to a helicoid. The result applies, in particular, to the genus one helicoid and implies that it is embedded outside of a compact set in ${\mathbb R}^3$.
In 1997, Collin proved that any properly embedded minimal surface in $\mathbb{R}^3$ with finite topology and more than one end has finite total Gaussian curvature. Hence, by an earlier result of Lopez and Ros, catenoids are the only non-planar, non-simply connected, properly embedded, minimal planar domains in $\mathbb{R}^3$ of finite topology. In 2005, Meeks and Rosenberg proved that the only simply connected, properly embedded minimal surfaces in $\mathbb{R}^3$ are planes and helicoids. Around 1860, Riemann defined a one-parameter family of periodic, infinite topology, properly embedded, min…
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 consider two natural classes of minimal laminations in three-manifolds. Both classes may be thought of as limits - in different senses - of embedded minimal disks. In both cases, we prove that, under a natural geometric assumption on the three-manifold, the leaves of these laminations are topologically either disks, annuli or Moebius bands. This answers a question posed by Hoffman and White.
We consider the question of existence of embedded doubly periodic minimal surfaces in Euclidean 3-space with Scherk-type ends, surfaces that topologically are Scherk's doubly periodic surface with handles added in various ways. We extend the existence results of H. Karcher and F. Wei to more cases, and we find other cases where existence does not hold.
Any sequence of properly embedded minimal disks in an open subset U of Euclidean 3-space has a subsequence such that the curvatures blow up on a relatively closed subset K of U and such that the disks converge in the complement of K to a minimal lamination of U\K. Assuming results of Colding-Minicozzi and an extension due to Meeks, we prove that such a blow-up set K must be contained in a C^1 embedded curve.
On the First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere
针对单位球面中闭嵌入极小超曲面，改进了其Laplace-Beltrami算子第一非零特征值的下界，结果略优于Duncan-Sire-Spruck的界。
On the Compactness Theorem for Embedded Minimal Surfaces in 3-manifolds with Locally Bounded Area and Genus
本文研究面积与亏格局部有界的真嵌入极小曲面序列的紧性，证明其在离散集外光滑收敛并分析奇异点附近的放大行为。
For any prescribed closed subset of a line segment in Euclidean 3-space, we construct a sequence of minimal disks that are properly embedded in an open solid cylinder around the segment and that have curvatures blowing up precisely at the points of the closed set.
Existence of embedded minimal tori in three-spheres with positive Ricci curvature
该研究通过证明三维球面嵌入环面与球面空间面积泛函的强Morse不等式，得出正Ricci曲率三维球面至少含4个嵌入极小环面、崎岖度量下至少9个的结论，证明基于Simon-Smith min-max理论的重数一性定理。
For a compact connected Lie group $G$ acting as isometries on a compact orientable Riemannian manifold $M^{n+1},$ and cohomogeneity not equal to 0 or 2, we prove the existence of a nontrivial embedded $G$-invariant minimal hypersurface, that is smooth outside a set of Hausdorff dimension at most $n-7.
We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We give an explicit expression for these surfaces. Among band types there is a class of properly embedded p-minimal surfaces of so called helicoid type. We classify all the helicoid type p-minimal surfaces. This class of p-minimal surfaces includes all the entire p-minimal graphs…
We construct three kinds of complete embedded minimal surfaces in $\Bbb H^2\times \Bbb R$. The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These two are conjugate surfaces just as the helicoid and the catenoid are in $\mathbb R^3$. The third one is a finite total curvature surface which is conformal to $\mathbb S^2\setminus\{p_1,...,p_k\}, k\geq3.$
In this paper we prove that a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface $\overline{M}$ with boundary punctured in a finite number of interior points and that $M$ can be represented in terms of meromorphic data on its conformal completion $\overline{M}$. In particular, we demonstrate that $M$ is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of $M$.
We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some periodic minimal surfaces in $\mathbb{S}^2\times\mathbb{R}$ via conjugate constructions. The resulting surfaces can be seen as the analogy to the Schwarz P-surface in these homogeneous 3-manifolds.
In this paper we study the mean curvature flow of embedded disks with free boundary on an embedded cylinder or generalised cone of revolution, called the support hypersurface. We determine regions of the interior of the support hypersurface such that initial data is driven to a curvature singularity in finite time or exists for all time and converges to a minimal disk. We further classify the type of the singularity. We additionally present applications of these results to the uniqueness problem for minimal hypersurfaces with free boundary on such suppport hypersurfaces; the results obtained t…
We prove that the area of each nonflat genus zero free boundary minimal surface embedded in the unit $3$-ball is less than the area of its radial projection to $\mathbb{S}^2$. The inequality is asymptotically sharp, and we prove any sequence of surfaces saturating it converges weakly to $\mathbb{S}^2$, as currents and as varifolds.
In this paper we prove two theorems. The first one is a structure result that describes the extrinsic geometry of an embedded surface with constant mean curvature (possibly zero) in a homogeneously regular Riemannian three-manifold, in any small neighborhood of a point of large almost-maximal curvature. We next apply this theorem and the Quadratic Curvature Decay Theorem (previously proven by the same authors in [14]) to deduce compactness, descriptive and dynamics-type results concerning the space $D(M)$ of non-flat limits under dilations of any given properly embedded minimal surface $M$ in …
Limiting behavior of sequences of properly embedded minimal disks
本文发展极小 θ-图与极限层叠理论，分析极限叶及曲率爆破集，证明无曲率爆破时悬链面族仍可作为嵌入极小圆盘序列的极限叶，并据此构造双曲空间中非真嵌入的完备单连通极小曲面。
Let $α$ be a polygonal Jordan curve in $\bfR^3$. We show that if $α$ satisfies certain conditions, then the least-area Douglas-Radó disk in $\bfR^3$ with boundary $α$ is unique and is a smooth graph. As our conditions on $α$ are not included amongst previously known conditions for embeddedness, we are enlarging the set of Jordan curves in $\bfR^3$ which are known to be spanned by an embedded least-area disk. As an application, we consider the conjugate surface construction method for minimal surfaces. With our result we can apply this method to a wider range of complete catenoid-ended minimal …
We survey what is known about minimal surfaces in $\bold R^3 $ that are complete, embedded, and have finite total curvature. The only classically known examples of such surfaces were the plane and the catenoid. The discovery by Costa, early in the last decade, of a new example that proved to be embedded sparked a great deal of research in this area. Many new examples have been found, even families of them, as will be described below. The central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the spac…
In the recent paper \cite{DGNP} we have proved that the only stable $C^2$ minimal surfaces in the first Heisenberg group $\Hn$ which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this paper we extend the result in \cite{DGNP} to $C^2$ complete embedded minimal surfaces in $\mathbb H^1$ with empty characteristic locus. We prove that every such a surface without boundary must be a vertical plane.
We add two new 1-parameter families to the short list of known embedded triply periodic minimal surfaces of genus 4 in $\mathbb{R}^3$. Both surfaces can be tiled by minimal pentagons with two straight segments and three planar symmetry curves as boundary. In one case (which has the appearance of the CLP surface of Schwarz with an added handle) the two straight segments are parallel, while they are orthogonal in the second case. The second family has as one limit the Costa surface, showing that this limit can occur for triply periodic minimal surfaces. For the existence proof we solve the 1-dim…
In $\mathbb{S}^2 \times \mathbb{R}$ there is a two-parameter family of properly embedded minimal annuli foliated by circles. In this paper we show that this family contains all properly embedded minimal annuli. We use the description of minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$ by periodic harmonic maps $G : \mathbb{C} \to \mathbb{S}^2$ of finite type. Due to the algebraic geometric correspondence of Hitchin [14], these harmonic maps are parametrized by hyperelliptic algebraic curves together with Abelian differentials with prescribed poles. We deform annuli by deforming spectral data…
First eigenvalue of embedded minimal surfaces in $S^3$
该研究针对$S^3$中嵌入极小曲面的拉普拉斯算子第一特征值问题，证明其满足$λ_1≥1+ε_g$（$ε_g$为仅依赖曲面亏格的正常数），改进了Choi-Wang的已有结果。
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 construct embedded minimal surfaces which are $n$-periodic in $\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz reflection extends to a complete minimal surface. Studying the group of Schwarz reflections, we can characterize those Jordan curves for which the complete surface is embedded. For example, for $n=4$ exactly five such Jordan curves generate embedded surfaces. Our results apply to surface classes other than minimal as well, for instance polygonal surfaces.
We prove the three embeddedness results as follows. $({\rm i})$ Let $Γ_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of $Γ_{2m+1}<2mπ$. In particular, the total curvature of $Γ_5<4π$ and thus any minimal surface $Σ\subset \mathbb{R}^n$ bounded by $Γ_5$ is embedded. Let $Γ_5$ be a piecewise geodesic Jordan curve with $5$ vertices in $\mathbb{H}^n$. Then any minimal surface $Σ\subset \mathbb{H}^n$ bounded by $Γ_5$ is embedded. If $Γ_5$ is in a geodesic ball of radius $\fracπ{4}$ in $\mathbb{S}^n_+$, …
We establish curvature estimates and a convexity result for mean convex properly embedded $[φ,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^3$, i.e., $φ$-minimal surfaces when $φ$ depends only on the third coordinate of $\mathbb{R}^3$. Led by the works on curvature estimates for surfaces in 3-manifolds, due to White for minimal surfaces, to Rosenberg, Souam and Toubiana, for stable CMC surfaces, and to Spruck and Xiao for stable translating solitons in $\mathbb{R}^3$, we use a compactness argument to provide curvature estimates for a family of mean convex $[φ,\vec{e}_{3}]$-minimal surfaces in …
These notes outline recent developments in classical minimal surface theory that are essential in classifying the properly embedded minimal planar domains M in R^3 with infinite topology (equivalently, with an infinite number of ends). This final classification result by Meeks, Perez, and Ros states that such an M must be congruent to a homothetic scaling of one of the classical examples found by Riemann in 1860. These examples {\cal R}_s, 0<s<\infty, are defined in terms of the Weierstrass {\cal P}-functions {\cal P}_t on the rectangular elliptic curve {\C} / {< 1, t\sqrt{-1}>}, are singly-pe…
We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This determines its asymptotic behavior. Some rigidity theorems are obtained.
We construct a sequence of compact embedded minimal disks in a ball in Euclidean 3-space, whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the x_3-axis. This extends a result of Colding and Minicozzi, who constructed a sequence for which the curvature blows up only at the center of the ball, and is a partial affirmative answer to the larger question of the existence of a sequence for which the curvature blows up precisely on a prescribed closed set on the x_3-axis.
The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture
本文证明单位球面中嵌入极小超曲面的第一非零特征值等于其维数，从而肯定解决 Yau 猜想，并由此推导出多个刚性定理。
In this paper we study the moduli space of properly Alexandrov-embedded, minimal annuli in $\mathbb{H}^2 \times \mathbb{R}$ with horizontal ends. We say that the ends are horizontal when they are graphs of $\mathcal{C}^{2, α}$ functions over $\partial_\infty \mathbb{H}^2$. Contrary to expectation, we show that one can not fully prescribe the two boundary curves at infinity, but rather, one can prescribe the bottom curve, but the top curve only up to a translation and a tilt, along with the position of the neck and the vertical flux of the annulus. We also prove general existence theorems for m…
In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with nonpositive curvature. We show this result by constructing a non-properly embedded minimal plane in hyperbolic 3-space. Hence, this gives a counterexample to Calabi-Yau conjecture for embedded minimal surfaces in the negative curvature case.
This is a survey of our work on embedded minimal disks.
Let $(M,g)$ be an asymptotically flat $3$-manifold containing no closed embedded minimal surfaces. We prove that for every point $p\in M$ there exists a complete properly embedded minimal plane in $M$ containing $p$.
Minimal surfaces in $\mathbb{H}^2\times\mathbb{R}$ with finite total curvature and embedded ends
本文研究 $\mathbb{H}^2\times\mathbb{R}$ 中具有嵌入端和有限全曲率的完备极小曲面，分类了全曲率为 $-4\pi$ 和 $-6\pi$ 的渐近边界，并通过渐近 Plateau 问题构造了新的极小曲面族。
Existence of rotationally symmetric embedded f-minimal tori
该研究推广了Angenent的收缩环面，证明了特定欧氏空间中满足条件的旋转对称嵌入f-极小环面的存在性。
We give a generalization of Meeks-Yau's celebrated embeddedness result for the solutions of the Plateau problem for extreme curves.
We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's result concerning asymmetric complete minimal submanifolds of Euclidean space with finite total curvature.
We first consider a uniqueness problem for embedded free boundary minimal annuli in the three-dimensional Euclidean unit half-ball. Then, we obtain symmetry properties for compact embedded free boundary minimal surfaces in the unit ball. Finally, we obtain several uniqueness results for the critical catenoid under symmetry conditions on the boundary. For example, we show that if an embedded free boundary minimal annulus whose boundary consists of two congruent components and has a reflection symmetry by a plane, then it is congruent to the critical catenoid.
This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key is to understand the structure of an embedded minimal disk in a ball in $\RR^3$. This was undertaken in [CM3], [CM4] and the global version of it will be completed here; see [CM15] for discussion of the local case and [CM13], [CM14] where we have surveyed our results about embedded minimal disks.
The Weierstrass representation for minimal surfaces in $\mathbb{R}^3$ provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmonic surfaces, which in general don't have the same topological limitations of minimal surfaces and can have complicated embedded ends. In this paper we demonstrate the flexibility of using the Weierstrass representation for harmonic surfaces to combine embedded harmonic ends togeth…
We prove the existence of a one parameter family of minimal embedded hypersurfaces in $R^{n+1}$, for $n \geq 3$, which generalize the well known 2 dimensional "Riemann minimal surfaces". The hypersurfaces we obtain are complete, embedded, simply periodic hypersurfaces which have infinitely many parallel hyperplanar ends. By opposition with the 2-dimensional case, they are not foliated by spheres.
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