Surfaces In Rn Research Articles

Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions

We study complete minimal surfaces in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy W:=14∫|H→|2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}: =\\frac{1}{4} \\int |\\vec H|^2$$\\end{document}. In codimension one, we prove that the W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-Morse index for any inverted minimal sphere or real projective plane with m such ends is exactly m-3=W4π-3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m-3=\\frac{\\mathcal {W}}{4\\pi }-3$$\\end{document}. We also consider several geometric properties—for example, the property that all m asymptotic planes meet at a single point—of these minimal surfaces and explore their relation to the W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {W}$$\\end{document}-Morse index of their inverted surfaces.

Willmore flow of complete surfaces

We consider the Willmore flow equation for complete, properly immersed surfaces in Rn. Given bounded geometry on the initial surface, we extend the result in [5] and prove short time existence and uniqueness of the Willmore flow. We also show that a complete Willmore surface with low Willmore energy must be a plane, and that a Willmore flow with low initial energy and Euclidean volume growth must converge smoothly to a plane.

Spectral analysis on ruled surfaces with combined Dirichlet and Neumann boundary conditions

Let Ω be an unbounded two dimensional strip on a ruled surface in Rn+1, n > 1. Consider the Laplacian operator in Ω with Dirichlet and Neumann boundary conditions on opposite sides of Ω. We prove some results on the existence and absence of the discrete spectrum of the operator; which are influenced by the twisted and bent effects of Ω. Provided that Ω is thin enough, we show an asymptotic behavior of the eigenvalues. The interest in those considerations lies on the difference from the purely Dirichlet case. Finally, we perform an appropriate dilatation in Ω and we compare the results.

The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space

In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds with free boundary. We first generalize ideas of restricted normal cones introduced by Choe-Ghomi-Ritoré in [10] and obtain an optimal area estimate for generalized restricted normal cones. This area estimate, together with the ABP method of Cabré in [5], provides a new proof of the relative isoperimetric inequality obtained by Choe-Ghomi-Ritoré in [11]. Furthermore, we use this estimate and the idea of Brendle in his recent work [3] to obtain a relative isoperimetric inequality for minimal submanifolds with free boundary on a convex support surface in Rn+m, which is optimal and gives an affirmative answer to an open problem proposed by Choe in [9], Open Problem 12.6, when the codimension m≤2.

Classification of Complete Regular Minimal Surfaces in ℝn with Total Curvature −6π

In this paper, we classify the complete regular orientable minimal surfaces in Rn with total curvature −6π and give a method to construct a series of complete non-holomorphic minimal surfaces with total curvature −6π. Specially, we give a simplified classification in another method if the surfaces lie in R4.

The restricted content and the d-dimensional Analyst's Travelling Salesman Theorem for general sets

In his 1990 paper, Jones characterized subsets of rectifiable curves in R2 via a multiscale sum of β-numbers, which measure how far a given set deviates from a straight line at each scale and location. This characterization was extended by Okikiolu to subsets of Rn and by Schul to subsets of a Hilbert space.Recently, there has been some interest in characterizing subsets of higher dimensional surfaces in Rn. Using a variant of Jones' β-number introduced by Azzam and Schul, Villa gave a characterization of lower regular subsets of a certain class of topologically stable surfaces – introduced in a 2004 paper of David – via a multiscale sum of these new β-numbers.In this paper we remove the lower regularity condition and prove an analogous result for general d-dimensional subsets of Rn. To do this, we introduce the restricted content, which assigns ‘mass’ to any subset of Rn (even to sets with zero Hausdorff measure), and use it to define new d-dimensional variant of Jones' β-number that is defined for any set in Rn.

A Li–Yau inequality for the 1-dimensional Willmore energy

Abstract By the classical Li–Yau inequality, an immersion of a closed surface in ℝ n {\mathbb{R}^{n}} with Willmore energy below 8 ⁢ π {8\pi} has to be embedded. We discuss analogous results for curves in ℝ 2 {\mathbb{R}^{2}} , involving Euler’s elastic energy and other possible curvature functionals. Additionally, we provide applications to associated gradient flows.

Numerical roadmap of smooth bounded real algebraic surface

For a smooth bounded real algebraic surface in Rn, a roadmap of it is a one-dimensional semi-algebraic subset of the surface whose intersection with each connected component of the surface is nonempty and semi-algebraically connected. In this paper, we introduce the notion of a numerical roadmap of a surface, which is a set of disjoint polygonal chains such that there is a bijective map between the chains and the connected components of a given roadmap of the surface. Moreover, the chains are ϵ-close to the connected components. We present an algorithm to compute such a numerical roadmap through constructing a topological graph. The topological graph also enables us to compute an approximate graph and a more intrinsic connectivity graph to represent the roadmap and its connectivity property, which is important for applications such as determining if two points on the surface belong to the same connected component and if so, finding a connected path between them.

New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in ℝⁿ

The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immersions of a given bordered non-orientable surface to $\mathbb{R}^n$ is a real analytic Banach manifold, obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces, and show general position theorems for non-orientable conformal minimal surfaces in $\mathbb{R}^n$. We also give the first known example of a properly embedded non-orientable minimal surface in $\mathbb{R}^4$; a Mobius strip. All our new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables us to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, we construct proper non-orientable conformal minimal surfaces in $\mathbb{R}^n$ with any given conformal structure, complete non-orientable minimal surfaces in $\mathbb{R}^n$ with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits $n$ hyperplanes of $\mathbb{CP}^{n-1}$ in general position, complete non-orientable minimal surfaces bounded by Jordan curves, and complete proper non-orientable minimal surfaces normalized by bordered surfaces in $p$-convex domains of $\mathbb{R}^n$.

Jordan surface theorem for simple closed SST-surfaces

The present paper proposes a new type of Jordan surface theorem for simple closed SST-surfaces. To do this work, we use space set topological (referred to as SST-, for short) structures on compact or non-compact surfaces in Rn. Indeed, the SST-structure is a special kind of Alexandroff topological structure and further, it is indeed a locally finite (LF-, for brevity) topological one. Owing to the LF-topological structure, it can efficiently be used in studying objects from the viewpoints of pure and applied topology such as discrete geometry, combinatorial topology, computational topology, digital topology, digital geometry and so on. The present paper first establishes an SST-structure of a subdivided abstract cell (SAC-, for brevity) complex on X(⊂Rn) using a generalized face to face tessellation of X⊂Rn. Second, the paper introduces a strong SST-homeomorphism to classify SST-surfaces. Finally, unlike the typical Jordan surface theorem under Hausdorff topology or polyhedral geometry, the current theorem is formulated by using Alexandroff topological or SST-structure. Thus this theorem has strong advantages being compared with those associated with Euclidean topology, digital topology and polygonal geometry. Besides, based on the SST-structure on an SAC-complex, the paper suggests some remarks on digital manifolds, Khalimsky manifolds, (n−1)-dimensional polyhedral pseudomanifolds and so on.

Weak convergence of branched conformal immersions with uniformly bounded areas and Willmore energies

In this paper, we first extend the classical Helein’s convergence theorem to a sequence of rescaled branched conformal immersions. By virtue of this local convergence theorem, we study the blow-up behavior of a sequence of branched conformal immersions of a closed Riemann surface in ℝn with uniformly bounded areas and Willmore energies. Furthermore, we prove that the integral identity of Gauss curvature is true.

The Morse index of a triply periodic minimal surface

In the previous work, the first author established an algorithm to compute the Morse index and the nullity of an n-periodic minimal surface in Rn. In fact, the Morse index can be translated into the number of negative eigenvalues of a real symmetric matrix and the nullity can be translated into the number of zero-eigenvalue of a Hermitian matrix. The two key matrices consist of periods of the abelian differentials of the second kind on a minimal surface, and the signature of the Hermitian matrix gives a new invariant of a minimal surface. On the other hand, H family, rPD family, tP family, tD family, and tCLP family of triply periodic minimal surfaces in R3 have been studied in physics, chemistry, and crystallography. In this paper, we first determine the two key matrices for the five families explicitly. As its applications, by numerical arguments, we compute the Morse indices, nullities, and signatures for the five families.

A local regularity theorem for mean curvature flow with triple edges

Abstract Mean curvature flow of clusters of n-dimensional surfaces in ℝ n + k {\mathbb{R}^{n+k}} that meet in triples at equal angles along smooth edges and higher order junctions on lower-dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smooth mean curvature flow with triple edges is weakly close to a static union of three n-dimensional unit density half-planes, then it is smoothly close. Extending the regularity result to a class of integral Brakke flows, we show that this implies smooth short-time existence of the flow starting from an initial surface cluster that has triple edges, but no higher order junctions.

Convexity estimates for mean curvature flow with free boundary

We prove the estimates of [3] and [2] for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier S. Here S can be any embedded, oriented surface in Rn+1 of bounded geometry and positive inscribed radius. We also prove the estimate [4] in the case of convex flows and S=Sn, which gives an alternative proof to [6].

Rough singular integrals associated with surfaces of Van der Corput type

In this paper, the authors establish the Lp-mapping properties for a class of singular integrals along surfaces in ℝn of the form {ϕ(|u|)u′: u ∈ ℝn} as well as the related maximal operators provided that the function ϕ satisfies certain oscillatory integral estimates of Van der Corput type, and the integral kernels are given by the radial function {ie86-1} for γ > 1 and the sphere function {ie86-2} for some β > 0, which is distinct from H1(Sn−1).

Canonical embeddings of [formula omitted] into orientable n-dimensional closed PL manifolds for [formula omitted

For each embedded oriented circle c in an n-dimensional closed PL manifold M, we define a canonical embedding with respect to a triangulation of M, of S1×Δn−1 into a regular neighborhood of the embedded circle in M. For n>4, we give a necessary and sufficient condition for an embedding of a surface with boundary in M, such that the embedding together with the canonical embeddings on the boundary components of the surface, extends to an embedding of the regular neighborhood of the surface in Rn to a regular neighborhood of the embedded surface in M.

Using bigraphs to model topological graphs embedded in orientable surfaces

Natural and artificial environments, at scales ranging from cellular to geographic, have complex and changing spatial structures based on regions, as well as being inhabited by a multiplicity of dynamic entities. Milner’s theory of bigraphs provides a formal design tool for dynamic and complex systems. However, bigraphs have rather limited explicit capability to represent spatial properties and relationships, being only equipped with a place graph that can express the containment relation between locations. This paper develops constructions that provide explicit bigraph types for representing complex two-dimensional spatial configurations, and shows that such representations are unique up to topological equivalence. In particular, we show how bigraphs can uniquely represent topological graphs embedded in compact, orientable surfaces in ℜn, as well as in the Euclidean plane.

Closed surfaces with bounds on their Willmore energy

The Willmore energy of a closed surface in Rn is the integral of its squared mean curvature, and is invariant under Mobius transformations of Rn . We show that any torus in R3 with energy at most 8π − δ has a representative under the Mobius action for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on δ > 0. An analogous estimate is also obtained for closed, orientable surfaces of fixed genus p ≥ 1 in R3 or R4, assuming suitable energy bounds which are sharp for n = 3. Moreover, the conformal type is controlled in terms of the energy bounds. Mathematics Subject Classification (2010): 53A05 (primary); 53A30, 53C21, 49Q15 (secondary).

Soap Film Solutions to Plateau’s Problem

Plateau’s problem is to show the existence of an area-minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area-minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau’s problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator, a Poincare Lemma, and good compactness properties to solve Plateau’s problem for orientable, embedded surfaces. But integral currents cannot represent surfaces such as the Mobius strip or surfaces with triple junctions. In the class of varifolds, there are no existence theorems for a general Plateau problem. We use the chain complex of differential chains, a geometric Poincare Lemma, and good compactness properties of the complex to solve Plateau’s problem in such generality as to find the first solution which minimizes area taken from a collection of surfaces that includes all previous special cases, as well as all smoothly immersed surfaces of any genus type, orientable or nonorientable, and surfaces with multiple junctions. Our result holds for all dimensions and codimension-one surfaces in ℝ n .

The exterior Plateau problem in higher codimension

We prove existence theorems for two-dimensional, noncompact, complete minimal surfaces in ℝn of annular type, which span a given contour and have a finite total curvature end and prescribed asymptotical behavior. For arbitrary rectifiable Jordan curves, we show the existence of such surfaces with a flat end, i.e., within a bounded distance from a 2-plane. For more restricted classes of curves, we prove the existence of minimal surfaces with higher multiplicity flat ends as well as of surfaces with polynomial-type nonflat ends.