Francisco Martin, Santiago Morales
In 1996, Nadirashvili used Runge's theorem to produce a complete minimal disc inside a ball in R^3. In this paper we generalize the techniques used by Nadirashvili to obtain new examples of complete minimal surfaces inside a ball in R^3, with the conformal structure of an annulus.
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We study minimal cylinders in the three-dimensional Heisenberg group ${\rm Nil}_3$ using the generalized Weierstrass type representation, the so-called loop group method. We characterize all non-vertical minimal cylinders in terms of pairs of two closed plane curves which have the same signed area. Moreover, as a byproduct of the construction, spacelike CMC cylinders can also be obtained.
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…
In this paper we consider the isoperimetric profile of convex cylinders $K\times\mathbb{R}^q$, where $K$ is an $m$-dimensional convex body, and of cylindrically bounded convex sets, i.e, those with a relatively compact orthogonal projection over some hyperplane of $\mathbb{R}^{n+1}$, asymptotic to a right convex cylinder of the form $K\times\mathbb{R}$, with $K\subset\mathbb{R}^n$. Results concerning the concavity of the isoperimetric profile, existence of isoperimetric regions, and geometric descriptions of isoperimetric regions for small and large volumes are obtained.
Minimal Lagrangian surfaces in $\mathbb{C}P^2$ via the loop group method Part II: The general case
本文将环路群方法从可缩黎曼曲面推广到任意黎曼曲面，构造 $\mathbb{C}P^2$ 中的极小拉格朗日浸入与柱面，并证明其逼近 Delaunay 柱面。
We analyse the asymptotic behaviour of solutions of the Teichmüller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition forces the domain to stretch out as a boundary bubble forms. Our main result then establishes that for prescribed boundary curves that satisfy Douglas' separation condition, these boundary bubbles will not only be harmonic but will themselves be branched minimal immersions. Toget…
Asymptotic analysis for approximate harmonic maps from degenerating cylinders and applications to minimal surfaces
本文研究了从退化环面或退化圆柱到紧致黎曼流形的映射序列的吹胀分析和定量行为，建立了广义能量恒等式，并证明了在无气泡区域，渐进行为要么是流形上的测地线，要么是子流形上的测地线状曲线。同时，研究了寻找具有自由边界条件的最小圆柱体的演化系统，得出了存在性结果，并揭示了颈部可能收敛到子流形上的测地线状曲线的新现象。
We investigate minimal helix submanifolds of any dimension and codimension immersed in Euclidean space. Our main result proves that a ruled minimal helix submanifold is a cylinder. As an application we classify complex helix submanifolds of $\mathbb{C}^n$: They are extrinsic products with a complex line as a factor. The key tool is Corollary 1.3 which allows us to classify Riemannian foliations of open subsets of the Euclidean space with minimal leaves. Finally, we consider the case of a helix hypersurface with constant mean curvature and prove that it is either a cylinder or an open part of a…
In this paper, we consider the problem of finding the hypersurface M^n in the Euclidean (n+1)-space R^{n+1} that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the hypersurfaces in the upper halfspace (R^{n+1})_{+} with lowest gravity center, for a fixed unit vector u in R^{n+1} . We first state that a singular minimal cylinder M^n in R^{n+1} is either a hyperplane or a α-catenary cylinder. It is also shown that this result remains true when M^n is a translation hypersurface and u a horizantal vector. As a …
Mean exit times from submanifolds with bounded mean curvature
该研究针对微分几何中子流形浸入问题，提出不依赖无穷远弱极大值原理的方法，证明无限平均退出时间的子流形无法等距极小浸入特定乘积空间的几类区域，并推广了随机完备子流形的不可浸入性结果，还给出小平均曲率子流形的矩塔估计。
We address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross section. For each cylinder of convex bounded cross-section, we show that there are only two ancient solutions which are asymptotic to this cylinder: the non-compact translating soliton and the compact oval solution obtained by gluing two translating solitons approaching each other from time $-\infty$ from two opposite ends.
In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real Kahler submanifold of codimension 2, if with complex dimension > 2, would be either holomorphic, or a cylinder, or complex ruled. In this article, we generalize their result to real analytic complete real Kahler submanifolds of codimension 4. The conclusion is that such the submanifold, if with complex dimension > 4, would be either partially holomorphic, or a cylinder, or a twisted cylinder in the sense that the complex relative nullity foliation is contained in a strictly larger holomorphic foliation, whose leaves are…
We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that also in this new setting and for arbitrary initial data, solutions of the Teichmüller harmonic map flow exist for all times. Furthermore, for solutions for which a three-point-condition does not degenerate as $t\to\infty$, we show convergence along a sequence $t_i\to\infty$ to a critical point of the area given either by a minimal cylinder or by two minimal dis…
Existence of Positive Scalar Curvature and Positive Yamabe constant on Hypersurfaces of Noncompact Cylinders
本文研究非紧圆柱超曲面上的正标量曲率与正 Yamabe 常数，利用 $g$-角度条件及度量的有界几何或有界曲率，将圆柱上的相应正性传递到截面流形。
We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabion complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness o…
We show that $H$-hypersurfaces of $\mathbb{H}^{n}\times \mathbb{R}$ contained in a vertical cylinder and with Ricci curvature with strong quadratic decay have mean curvature $| H| > (n-1)/n$.
Extending an example by Colding and Minicozzi, we construct a sequence of properly embedded minimal disks $Σ_i$ in an infinite Euclidean cylinder around the $x_3$-axis with curvature blow-up at a single point. The sequence converges to a non smooth and non proper minimal lamination in the cylinder. Moreover, we show that the disks $Σ_i$ are not properly embedded in a sequence of open subsets of $\mathbb{ R}^3$ that exhausts $\mathbb{ R}^3$.
In this note, we give natural extensions to cylinders and tori of a classical result due to T. Takahashi about minimal immersions into spheres. More precisely, we deal with Euclidean isometric immersions whose projections in R^N satisfy a spectral condition of their Laplacian.
We prove that if $M$ is a strictly stable complete minimal hypersurface in Euclidean space with finite density at infinity and which lies on one side of a minimal cylinder with cross-section a strictly stable area minimizing hypercone, then $M$ must be cylindrical. Applications will be given in the references [Sim20a], [Sim20b].
Minimal translation hypersurfaces in hyperbolic space
本文在双曲空间上半空间模型中，对两类极小平移超曲面进行了完整分类，发现I型为抛物柱面（推翻文献中的不存在性定理），II型为抛物柱面、垂直全测地超平面及Scherk曲面上的垂直柱面。
The fundamental properties of $J$-holomorphic maps depend on two inequalities: The gradient inequality gives a pointwise bound on the differential of a $J$-holomorphic map in terms of its energy. The cylinder inequality stipulates and quantifies the exponential decay of energy along cylinders of small total energy. We show these inequalities hold uniformly if the geometry of the target symplectic manifold and Lagrangian boundary condition is appropriately bounded.
We prove that there are no minimal hypersurfaces properly immersed in any region of the Euclidean space bounded by unstable minimal cones. We also prove the analogous result for $r$-minimal hypersurfaces.
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.
In this paper, we study complete oriented $f$-minimal hypersurfaces properly immersed in a cylinder shrinking soliton $(\mathbb{S}^n\times \mathbb{R}, \bar{g}, f)$. We prove that such hypersurface with $L_f$-index one must be either $\mathbb{S}^n\times\{0\}$ or $\mathbb{S}^{n-1}\times\mathbb{R}$, where $\mathbb{S}^{n-1}$ denotes the sphere in $\mathbb{S}^{n}$ of the same radius. Also we prove a pinching theorem for them.
We obtain a Bernstein type result for entire two dimensional minimal graphs in $\mathbb{R}^{4}$, which extends a previous one due to L. Ni. Moreover, we provide a characterization for complex analytic curves.
We construct open domains in Euclidean 3-space which do not admit complete properly immersed minimal surfaces with an annular end. These domains can not be smooth by a recent result of Martin and Morales
Minimal isometric immersions F in codimension two from a complete Kahler manifold into Euclidean space had been classified for dimension greater than or equal to 3. In this note we describe the non--minimal situation by showing that, if F is real analytic but not everywhere minimal, then F is a cylinder over a real Kahler surface G in Euclidean 6-space. In addition, G can be further described.
In this paper, we investigate minimal submanifolds in Euclidean space with positive index of relative nullity. Let $M^m$ be a complete Riemannian manifold and let $f\colon M^m\to\R^n$ be a minimal isometric immersion with index of relative nullity at least $m-2$ at any point. We show that if the Omori-Yau maximum principle for the Laplacian holds on $M^m$, for instance, if the scalar curvature of $M^m$ does not decrease to $-\infty$ too fast or if the immersion $f$ is proper, then the submanifold must be a cylinder over a minimal surface.
Consider a strictly convex bounded regular domain $C$ of $\R^3$. For any arbitrary finite topological type we find a compact Riemann surface $\mathcal{M}$, an open domain $M\subset \mathcal{M}$ with the fixed topological type, and a conformal complete proper minimal immersion $X:M\to C$ which can be extended to a continuous map $X:\overline{M}\to \overline{C}$.
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifolds which can be enumerated explicitely. One consequence of this result is the uniqueness of the area minimizing examples. Another is the asymptotic nonexistence of stable compact embedded minimal surfaces of positive genus bounded by two convex curves in parallel planes.
Compactness for minimal surfaces with injectivity radius bounded from below
证明了闭黎曼3-流形中面积有上界且单射半径有下界的闭嵌入极小曲面空间的紧致性定理，用单射半径下界替代了Choi-Schoen紧致性定理中的亏格界。
We give a fairly complete solution to the asymptotic Plateau Problem for minimal surfaces in H^2xR. In particular, we identify the collection of finite Jordan curves in the asymptotic cylinder which bounds a minimal surface in H^2xR.
Consider a sequence of minimal varieties M_i in a Riemannian manifold N such that the boundary measures are uniformly bounded on compact sets. Let Z be the set of points at which the areas of the M_i blow up. We prove that Z behaves in some ways like a minimal variety without boundary: in particular, it satisfies the same maximum and barrier principles that a smooth minimal submanifold satisfies. For suitable open subsets W of N, this allows one to show that if the areas of the M_i are uniformly bounded on compact subsets of W, then the areas are in fact uniformly bounded on all compact subset…
For any closed Riemannian three-manifold, we prove that for any sequence of closed embedded minimal surfaces with uniformly bounded index, the genus can only grow at most linearly with respect to the area.
We prove a compactness result for minimal hypersurfaces with bounded index and volume, which can be thought of as an extension of the compactness theorem of Choi-Schoen (Invent. Math. 1985) to higher dimensions.
We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold of dimension at most seven, can degenerate. Loosely speaking, our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces, up to controlled errors. Several compactness/finiteness theorems follows our local picture.
We show that immersed minimal surfaces of $\mathbb{R}^{3}$ with bounded curvature and proper self intersections are proper. We also show that the restriction of the immersing map to a wide component is always proper. When the immersing map is injective the whole surface is a wide component. Prior to these results it was only known that injectively immersed minimal surfaces with bounded curvature were proper.
We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces.
Calabi-Yau Conjecture for Minimal Hypersurfaces in $\mathbb{R}^4$ with bounded geometry
本研究解决了具有有界第二基本形式且第二贝蒂数有限的ℝ⁴中完全连通嵌入极小超曲面的卡拉比-丘猜想，明确其必为真。
On boundedness of singularities and minimal log discrepancies of Kollár components
本文研究klt奇点的有界性问题，证明在三维或Kollár分量极小对数差异有上界时，局部体积远离零的klt奇点有界至多相差特殊退化，并回答了关于对数典范阈值与局部体积关系的一个问题。
Renormalized volume of minimally bounded regions in asymptotically hyperbolic Einstein spaces
本文为渐近双曲Einstein流形中由Graham-Witten极小曲面与共形无穷远界定的区域定义了重整化体积，证明了其Gauss-Bonnet定理并计算了极小超曲面变分下的导数。
On boundedness of singularities and minimal log discrepancies of Kollár components, II
本文证明K-半稳定log Fano锥奇点的有界性由局部体积下界和Kollár分量极小对数差异上界决定，并据此确认了三维情形的有界性猜想及klt奇点局部体积的聚集性质。
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