The Experts below are selected from a list of 33984 Experts worldwide ranked by ideXlab platform
Hwajeong Kim - One of the best experts on this subject based on the ideXlab platform.
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Morse theory for Minimal Surfaces in manifolds
Annals of Global Analysis and Geometry, 2018Co-Authors: Hwajeong KimAbstract:A Morse theory of a given function gives information of the numbers of critical points of some topological type. A Minimal surface, bounded by a given curve in a manifold, is characterized as a harmonic extension of a critical point of the functional \({\mathcal E}\) which corresponds to the Dirichlet integral. We want to obtain Morse theories for Minimal Surfaces in Riemannian manifolds. We first investigate the higher differentiabilities of \({\mathcal E}\). We then develop a Morse inequality for Minimal Surfaces of annulus type in a Riemannian manifold. Furthermore, we also construct body handle theories for Minimal Surfaces of annulus type as well as of disc type. Here we give a setting where the functional \({\mathcal E}\) is non-degenerated.
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Unstable Minimal Surfaces of annulus type in manifolds
Advances in Geometry, 2009Co-Authors: Hwajeong KimAbstract:Unstable Minimal Surfaces are the unstable stationary points of the Dirichlet integral. In order to obtain unstable solutions, the method of the gradient flow together with the minimax- principle is generally used, an application of which was presented in (19) for Minimal Surfaces in Euclidean space. We extend this theory to obtain unstable Minimal Surfaces in Riemannian manifolds. In particular, we consider Minimal Surfaces of annulus type.
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Unstable Minimal Surfaces of annulus type in manifolds
Advances in Geometry, 2009Co-Authors: Hwajeong KimAbstract:Unstable Minimal Surfaces are the unstable stationary points of the Dirichlet integral. In order to obtain unstable solutions, the method of the gradient flow together with the minimax-principle is generally used, an application of which was presented in [Struwe, J. Reine Angew. Math. 349: 1–23, 1984] for Minimal Surfaces in Euclidean space. We extend this theory to obtain unstable Minimal Surfaces in Riemannian manifolds. In particular, we consider Minimal Surfaces of annulus type.Peer Reviewe
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Unstable Minimal Surfaces of annulus type in manifolds
arXiv: Differential Geometry, 2006Co-Authors: Hwajeong KimAbstract:Unstable Minimal Surfaces are the unstable stationary points of the Dirichlet-Integral. In order to obtain unstable solutions, the method of the gradient flow together with the minimax-principle is generally used. The application of this method for Minimal Surfaces in the Euclidean spacce was presented in \cite{s3}. We extend this theory for obtaining unstable Minimal Surfaces in Riemannian manifolds. In particular, we handle Minimal Surfaces of annulus type, i.e. we prescribe two Jordan curves of class $C^3$ in a Riemannian manifold and prove the existence of unstable Minimal Surfaces of annulus type bounded by these curves.
Guozhao Wang - One of the best experts on this subject based on the ideXlab platform.
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quintic parametric polynomial Minimal Surfaces and their properties
Differential Geometry and Its Applications, 2010Co-Authors: Guozhao WangAbstract:Abstract In this paper, quintic parametric polynomial Minimal surface and their properties are discussed. We first propose the sufficient condition of quintic harmonic polynomial parametric surface being a Minimal surface. Then several new models of Minimal Surfaces with shape parameters are derived from this condition. We also study the properties of new Minimal Surfaces, such as symmetry, self-intersection on symmetric planes and containing straight lines. Two one-parameter families of isometric Minimal Surfaces are also constructed by specifying some proper shape parameters.
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Parametric polynomial Minimal Surfaces of arbitrary degree
arXiv: Graphics, 2010Co-Authors: Guozhao WangAbstract:Weierstrass representation is a classical parameterization of Minimal Surfaces. However, two functions should be specied to construct the parametric form in Weierestrass representation. In this paper, we propose an explicit parametric form for a class of parametric polynomial Minimal Surfaces of arbitrary degree. It includes the classical Enneper surface for cubic case. The proposed Minimal Surfaces also have some interesting properties such as symmetry, containing straight lines and self-intersections. According to the shape properties, the proposed Minimal surface can be classied into four categories with respect to n = 4k 1 n = 4k + 1, n = 4k and n = 4k + 2. The explicit parametric form of corresponding conjugate Minimal Surfaces is given and the isometric deformation is also implemented.
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parametric polynomial Minimal Surfaces of degree six with isothermal parameter
Geometric Modeling and Processing, 2008Co-Authors: Gang Xu, Guozhao WangAbstract:In this paper, parametric polynomial Minimal Surfaces of degree six with isothermal parameter are discussed. We firstly propose the sufficient and necessary condition of a harmonic polynomial parametric surface of degree six being a Minimal surface. Then we obtain two kinds of new Minimal Surfaces from the condition. The new Minimal Surfaces have similar properties as Enneper's Minimal surface, such as symmetry, self-intersection and containing straight lines. A new pair of conjugate Minimal Surfaces is also discovered in this paper. The new Minimal Surfaces can be represented by tensor product Bezier surface and triangular Bezier surface, and have several shape parameters. We also employ the new Minimal Surfaces for form-finding problem in membrane structure and present several modeling examples.
Enrico Valdinoci - One of the best experts on this subject based on the ideXlab platform.
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Nonlocal Minimal Surfaces
Lecture Notes of the Unione Matematica Italiana, 2016Co-Authors: Claudia Bucur, Enrico ValdinociAbstract:In this chapter, we introduce nonlocal Minimal Surfaces. We first discuss a Bernstein type result in any dimension, namely the property that an s-Minimal graph in \(\mathbb{R}^{n+1}\) is flat (if no singular cones exist in dimension n); we will then prove that an s-Minimal surface whose prescribed data is a subgraph, is itself a subgraph. The non-existence of nontrivial s-Minimal cones in dimension 2 is then proved. Moreover, some boundary regularity properties will be discussed at the end of this chapter: quite surprisingly, and differently from the classical case, nonlocal Minimal Surfaces do not always attain boundary data in a continuous way (not even in low dimension). A possible boundary behavior is, on the contrary, a combination of stickiness to the boundary and smooth separation from the adjacent portions.
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boundary behavior of nonlocal Minimal Surfaces
arXiv: Analysis of PDEs, 2015Co-Authors: Serena Dipierro, O Savin, Enrico ValdinociAbstract:We consider the behavior of the nonlocal Minimal Surfaces in the vicinity of the boundary. By a series of detailed examples, we show that nonlocal Minimal Surfaces may stick at the boundary of the domain, even when the domain is smooth and convex. This is a purely nonlocal phenomenon, and it is in sharp contrast with the boundary properties of the classical Minimal Surfaces. In particular, we show stickiness phenomena to half-balls when the datum outside the ball is a small half-ring and to the side of a two-dimensional box when the oscillation between the datum on the right and on the left is large enough. When the fractional parameter is small, the sticking effects may become more and more evident. Moreover, we show that lines in the plane are unstable at the boundary: namely, small compactly supported perturbations of lines cause the minimizers in a slab to stick at the boundary, by a quantity that is proportional to a power of the perturbation. In all the examples, we present concrete estimates on the stickiness phenomena. Also, we construct a family of compactly supported barriers which can have independent interest.
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regularity properties of nonlocal Minimal Surfaces via limiting arguments
arXiv: Analysis of PDEs, 2011Co-Authors: Luis A Caffarelli, Enrico ValdinociAbstract:We prove an improvement of flatness result for nonlocal Minimal Surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal Minimal cones are flat and that all the nonlocal Minimal Surfaces are smooth when the dimension of the ambient space is less or equal than 7 and $s$ is close to 1.
Metin Gürses - One of the best experts on this subject based on the ideXlab platform.
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Sigma Models and Minimal Surfaces
Letters in Mathematical Physics, 1998Co-Authors: Metin GürsesAbstract:Correspondence is established between sigma models, Minimal Surfaces and the Monge–Ampére equation. The Lax pairs of the Minimality condition of the Minimal Surfaces and the Monge–Ampére equations are given. Existence of infinitely many nonlocal conservation laws is shown and some Bäcklund transformations are also given.
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Sigma Models and Minimal Surfaces
Letters in Mathematical Physics, 1998Co-Authors: Metin GürsesAbstract:Correspondence is established between sigma models, Minimal Surfaces and the Monge–Ampere equation. The Lax pairs of the Minimality condition of the Minimal Surfaces and the Monge–Ampere equations are given. Existence of infinitely many nonlocal conservation laws is shown and some Backlund transformations are also given.
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Sigma Models and Minimal Surfaces
arXiv: Exactly Solvable and Integrable Systems, 1997Co-Authors: Metin GürsesAbstract:The correspondance is established between the sigma models, the Minimal Surfaces and the Monge-Ampere equation. The Lax -Pairs of the Minimality condition of the Minimal Surfaces and the Monge-Ampere equations are given. Existance of infinitely many nonlocal conservation laws is shown and some Backlund transformations are also given.
Luis A Caffarelli - One of the best experts on this subject based on the ideXlab platform.
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obstacle type problems for Minimal Surfaces
arXiv: Analysis of PDEs, 2016Co-Authors: Luis A Caffarelli, Daniela De Silva, O SavinAbstract:We study certain obstacle type problems involving standard and nonlocal Minimal Surfaces. We obtain optimal regularity of the solution and a characterization of the free boundary.
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regularity properties of nonlocal Minimal Surfaces via limiting arguments
arXiv: Analysis of PDEs, 2011Co-Authors: Luis A Caffarelli, Enrico ValdinociAbstract:We prove an improvement of flatness result for nonlocal Minimal Surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal Minimal cones are flat and that all the nonlocal Minimal Surfaces are smooth when the dimension of the ambient space is less or equal than 7 and $s$ is close to 1.
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nonlocal Minimal Surfaces
arXiv: Analysis of PDEs, 2009Co-Authors: Luis A Caffarelli, Jeanmichel Roquejoffre, O SavinAbstract:The de Giorgi theory for Minimal Surfaces consists in studying sets whose indicator function is a (local) minimum of the BV norm. In this paper we replace the BV norm by the $H^\sigma$ norm, with $\sigma<1/2$, and try to understand what the minimisers look like. Parallel to the de Giorgi theory we prove that, if the boundary of a minimiser is sufficiently flat in the unit ball, then it is a smooth piece of hypersurface.
