Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary

In this paper, we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds \(M^3\) with nonnegative Ricci curvature and strictly convex boundary \(\partial M\). Here we obtain a sharp upper bound for the length \(L(\partial \Sigma )\) of the boundary \(\partial \Sigma \) of a free boundary minimal surface \(\Sigma ^2\) in \(M^3\) in terms of the genus of \(\Sigma \) and the number of connected components of \(\partial \Sigma \), assuming \(\Sigma \) has index one. After, under a natural hypothesis on the geometry of M along \(\partial M\), we prove that if \(L(\partial \Sigma )\) saturates the respective upper bound, then \(M^3\) is isometric to the Euclidean 3-ball and \(\Sigma ^2\) is isometric to the Euclidean disk. In particular, we get a sharp upper bound for the area of \(\Sigma \), when \(M^3\) is a strictly convex body in \(\mathbb {R}^3\), which is saturated only on the Euclidean 3-balls (by the Euclidean disks). We also consider similar results for free boundary stable CMC surfaces.     

Rigidity of Hyperbolic Sets on Surfaces

Given a hyperbolic invariant set of a diffeomorphism on a surface,it is proved that, if the holonomies are sufficiently smooth,then the diffeomorphism on the hyperbolic invariant set is rigidin the sense that it is C¹⁺ conjugate to a hyperbolic affinemodel.     

Rigidity and Separation Indices of Graphs in Surfaces

Let Σ be a surface. We prove that rigidity indices of graphs which admit a polyhedral embedding in Σ and 5-connected graphs admitting an embedding in Σ are bounded by a constant depending on Σ. Moreover if the Euler characteristic of Σ is negative, then the separation index of graphs admitting a polyhedral embedding in Σ is also bounded. As a side result we show that distinguishing number of both Σ-polyhedral and 5-connected graphs which admit and embedding in Σ is also bounded.     

Classification of Flat Lagrangian Surfaces in Complex Lorentzian Plane

One of the most fundamental problems in the study of Lagrangian submanifolds from Riemannian geometric point of view is to classify Lagrangian immersions of real space forms into complex space forms. The main purpose of this paper is thus to classify flat Lagrangian surfaces in the Lorentzian complex plane C1^2. Our main result states that there are thirty-eight families of flat Lagrangian surfaces in C1^2. Conversely, every flat Lagrangian surface in C1^2 is locally congruent to one of the thirty-eight families.     

Rigidity of Area-Minimizing Hyperbolic Surfaces in Three-Manifolds

We prove that if M is a three-manifold with scalar curvature greater than or equal to ?2 and Σ?M is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of Σ is greater than or equal to 4π(g(Σ)?1), where g(Σ) denotes the genus of Σ. In the equality case, we prove that the induced metric on Σ has constant Gauss curvature equal to ?1 and locally M splits along Σ. We also obtain a rigidity result for cylinders (I×Σ,dt ²+g _Σ), where I=[a,b]?? and g _Σ is a Riemannian metric on Σ with constant Gauss curvature equal to ?1.     

Time-like Linear Weingarten Surfaces in Lorentzian Space Forms

In this paper we study Backlund transformations of time-like linear Weingarten surfaces with negative discriminant in Lorentzian space forms.     

On the Rigidity of a Class of Glued Surfaces

The method of integral formulas is applied to prove the rigidity of a class of closed nonconvex surfaces obtained by gluing together regular pieces of surfaces of positive Gaussian curvature with smooth boundaries.     

Irreducible Triangulations of Surfaces with Boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g ≥ 0 with b ≥ 0 boundary components is O(g + b). So far, the result was known only for surfaces without boundary (b = 0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.     

流动中边界面的识别

将流场的边界面定义为流动表面,在该表面上剪切率为0,并利用所建议的程序求解.该方法是基于速度向量场的计算,与坐标系的选择无关.     

Torsional Rigidity for Regions with a Brownian Boundary

Let ?? ^m be the m-dimensional unit torus, m ∈ ?. The torsional rigidity of an open set Ω ? ?? ^m is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = ?? ^m \β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in ??²\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of ??³\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage W _r(t)[0, t] of radius r(t) = o(t ^-1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ?³ and W ₁[0, t] in ? ^m , m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on ?? ^m , which has received a lot of attention in the literature in past years.     

Rigidity Theorems for Manifolds with Boundary and Nonnegative Ricci Curvature

In this paper, we obtain some rigidity theorems for compact Riemannian manifolds Ω with boundary M and nonnegative Ricci curvature; for instance, we prove that the existence of certain functions on M together with a lower bound c > 0 on the principal curvtures of M imply that Ω is an euclidean ball of radius $ {1\over c} $ .     

Minimal Surfaces with an Elastic Boundary

LetD:= { C ³ (

Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

In this article, we investigate the restrictions imposed by the dominant energy condition (DEC) on the topology and conformal type of possibly non-compact marginally outer trapped surfaces (thus extending Hawking’s classical theorem on the topology of black holes). We first prove that an unbounded, stable marginally outer trapped surface in an initial data set (M, g, k) obeying the dominant energy condition is conformally diffeomorphic to either the plane \({\mathbb{C}}\) or to the cylinder \({\mathbb{A}}\) and in the latter case infinitesimal rigidity holds. As a corollary, when the DEC holds strictly, this rules out the existence of trapped regions with cylindrical boundary. In the second part of the article, we restrict our attention to asymptotically flat data (M, g, k) and show that, in that setting, the existence of an unbounded, stable marginally outer trapped surface essentially never occurs unless in a very specific case, since it would force an isometric embedding of (M, g, k) into the Minkowski spacetime as a space-like slice.     

Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jordán, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in \({{\mathbb {R}}}^2\) . The analogous conditions are known to be insufficient to characterise generic global rigidity in higher dimensions. Recently Laman-type characterisations of rigidity have been obtained for generic frameworks in \({\mathbb {R}}^3\) when the vertices are constrained to lie on various surfaces, such as the cylinder and the cone. In this paper we obtain analogues of Hendrickson’s necessary conditions for the global rigidity of generic frameworks on the cylinder, cone and ellipsoid.     

Parabolicity and Hyperbolicity Conditions for Boundary Elements of Surfaces

Mathematical Notes -