Convex Boundary Research Articles

Comparison theorems for manifolds with mean convex boundary

Let Mn be an n-dimensional Riemannian manifold with boundary ∂M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k ∈ ℝ, we give a sharp estimate of the upper bound of ρ(x) = d (x, ∂M), in terms of the mean curvature bound of the boundary. When ∂M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kähler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kähler manifold and also estimate the first eigenvalue of the real Laplacian.

Connectivity by geodesics in open subsets of globally hyperbolic spacetimes

We discuss the geodesic connectedness problem in open subsets with convex boundary of globally hyperbolic spacetimes endowed with a complete, timelike or lightlike, Killing vector field. Furthermore, we furnish applications to generalized plane waves.

Properties of squeezing functions and global transformations of bounded domains

The central purpose of the present paper is to study boundary behaviors of squeezing functions on some bounded domains. We prove that the squeezing function of any strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate is proved for the squeezing function on any bounded domain near its globally strongly convex boundary points. We also study the stability properties of squeezing functions on a sequence of bounded domains, and give some comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions. As applications, we give new proofs of several well-known results about geometry of strongly pseudoconvex domains, and prove that all Cartan-Hartogs domains are homogenous regular. Finally, some related problems for further study are proposed.

Hamiltonian Floer homology for compact convex symplectic manifolds

We construct absolute and relative versions of Hamiltonian Floer homology algebras for strongly semi-positive compact symplectic manifolds with convex boundary, where the ring structures are given by the appropriate versions of the pair-of-pants products. We establish the absolute and relative Piunikhin-Salamon-Schwarz isomorphisms between these Floer homology algebras and the corresponding absolute and relative quantum homology algebras. As a result, the absolute and relative analogues of the spectral invariants on the group of compactly supported Hamiltonian diffeomorphisms are defined.

A sharp lower bound for the first eigenvalue on Finsler manifolds with nonnegative weighted Ricci curvature

Let (M,F) be an n-dimensional compact Finsler manifold without boundary or with a convex boundary and λ1 be the first (nonzero) closed or Neumann eigenvalue of the Finsler Laplacian on M with nonnegative weighted Ricci curvature. In this paper, we prove that λ1≥π2d2, where d is the diameter of M, and that the equality holds if and only if M is a 1-dimensional circle or a 1-dimensional segment, which generalize the well-known Zhong–Yang’s sharp estimate in Riemannian geometry (Zhong and Yang, 1984).

High speed observation of HIFU-induced cavitation cloud near curved rigid boundaries

This paper focuses on the experimental study of the influence of surface curvature to the behaviour of HIFU-induced cavitation cloud. A Q-switched ruby pulse laser is used to induce cavitation nuclei in deionized water. A piezoelectric ultrasonic transducer (1.7 MHz) provides a focused ultrasound field to inspire the nucleus to cavitation cloud. A PZT probe type hydrophone is applied for measuring the HIFU sound field. It was observed that the motion of cavitation cloud located near the boundary is significantly influenced by the distance between cloud and boundary, as well as the curvature of the boundary. The curvature was defined by parameters λ and ξ. Convex boundary, concave boundary, and flat boundary correspond to ξ <1, ξ >1 and ξ = 1, respectively. Different behaviours of the cloud, including the migration of the cloud, the characteristics of oscillation, etc., were observed under different boundary curvatures by high-speed photography. Sonoluminescence of the acoustic cavitation bubble clouds were also studied to illustrate the characteristics of acoustic streaming.

Manipulation of cell mechanotaxis by designing curvature of the elasticity boundary on hydrogel matrix

Directional cell migration induced by the stiffness gradient of cell culture substrates is known as a subset of the mechanical-cue-induced taxis, so-called mechanotaxis, typically durotaxis toward hard region. To establish the general conditions of biomaterials to manipulate the mechanotaxis, the effect of the shape of the elasticity transition boundary between hard and soft regions of a substrate on mechanotaxis should be systematically determined as well as the conditions of elasticity gradient strength. Here, as a simplified factor of expressing variations in the shape of the elasticity boundary in living tissues, we focus on the curvature of the elasticity boundary. Mask-free photolithographic microelasticity patterning of photocurable gelatin gel was employed to systematically prepare elasticity boundaries with various curvatures, and the efficiency of mechanotaxis of fibroblast cells around each curved boundary was examined. Highly efficient usual durotaxis was induced on a convex boundary with 100 μm in radius and on a concave boundary with 750 μm in radius of curvature. Interestingly, biased migration toward soft regions of the gel, i.e., inverse durotaxis, was first observed for concave boundaries with 50 μm or 100 μm in radius of curvature, which was named as “negative mechanotaxis”. The curvature of the elasticity boundary was found to markedly affect the efficiency of induction and the direction of mechanotaxis. The mechanism responsible for this phenomenon and the implication for the curvature effect in in vivo systems are discussed.

Complex manifolds in $$q$$ q -convex boundaries

We consider a \(C^\infty \) boundary \(b\Omega \subset {\mathbb {C}}^n\) which is \(q\)-convex in the sense that its Levi-form has positive trace on every complex \(q\)-plane. We prove that \(b\Omega \) is tangent of infinite order to the complexification of each of its submanifolds which is complex tangential and of finite bracket type. This generalizes Diederich and Fornaess (Ann Math 107:371–384, 1978) from pseudoconvex to \(q\)-convex domains. We also readily prove that the rows of the Levi-form are \(\frac{1}{2}\)-subelliptic multipliers for the \(\bar{\partial }\)-Neumann problem on \(q\)-forms (cf. Ho in Math Ann 290:3–18, 1991). This allows to run the Kohn algorithm of Acta Math 142:79–122 (1979) in the chain of ideals of subelliptic multipliers for \(q\)-forms. If \(b\Omega \) is real analytic and the algorithm gets stuck on \(q\)-forms, then it produces a variety of holomorphic dimension \(q\), and in fact, by our result above, a complex \(q\)-manifold which is not only tangent but indeed contained in \(b\Omega \). Altogether, the absence of complex \(q\)-manifolds in \(b\Omega \) produces a subelliptic estimate on \(q\)-forms.

Eigenvalue estimate and compactness for closedf-minimal surfaces

Let be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-´ Emery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted Laplacian for closed em- bedded f-minimal hypersurfaces contained in . Using this estimate, we prove a compactness theorem for the space of closed embedded f- minimal surfaces with the uniform upper bounds of genus and diameter in a complete 3-manifold with Bakry-´ Emery Ricci curvature bounded below by a positive constant and admitting an exhaustion by bounded domains with convex boundary.

Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies an apriori curvature estimate 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 theorem for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.

Bypass attachments and homotopy classes of 2-plane fields in contact topology

We use the generalized Pontryagin-Thom construction to analyze the effect of attaching a bypass on the homotopy class of the contact structure. In particular, given a 3-dimensional contact manifold with convex boundary, we show that the bypass triangle attachment changes the homotopy class of the contact structure relative to the boundary, and the difference is measured by the Hopf invariant.

A review of non local continuum damage: Modelling of failure?

Failure of quasi-brittle materials such as concrete needs a proper description of strain softening due to progressive micro-cracking and the introduction of an internal length in the constitutive model in order to achieve non zero energy dissipation. This paper reviews the main results obtained with the non local damage model, which has been among the precursors of such models. In most cases up to now, the internal length has been considered as a constant. There is today a consensus that it should not be the case as models possess severe shortcomings such as incorrect averaging near the boundaries of the solid considered and non local transmission across non convex boundaries. An interaction-based model in which the weight function is constructed from the analysis of interaction has been proposed. It avoids empirical descriptions of the evolution of the internal length. This model is also recalled and further documented. Additional results dealing with spalling failure are discussed. Finally, it is pointed out that this model provides an asymptotic description of complete failure, which is consistent with fracture mechanics.

Interaction of panel flutter with inviscid boundary layer instability in supersonic flow

AbstractWe investigate the stability of an elastic plate in supersonic gas flow. This problem has been studied in many papers regarding panel flutter, where uniform flow is usually considered. In this paper, we take the boundary layer on the plate into account and investigate its influence on plate stability. Three problem formulations are studied. First, we investigate the stability of travelling waves in an infinite-length plate. Second, the nature of the instability (absolute or convective instability) is examined. Finally, by using solutions of the first two problems, instability of a long finite-length plate is studied by using Kulikovskii’s global instability criterion. The following results are obtained. All the eigenmodes of a finite-length plate are split into two types, which we call subsonic and supersonic. The influence of the boundary layer on these eigenmodes can be of two kinds. First, for a generalized convex boundary layer profile (typical for accelerating flow), supersonic eigenmodes are stabilized by the boundary layer, whereas subsonic disturbances are destabilized. Second, for a profile with a generalized inflection point (typical for constant and decelerating flows), supersonic eigenmodes are destabilized in a thin boundary layer and stabilized in a thick layer; subsonic eigenmodes are damped. The correspondence between the influence of the boundary layer on panel flutter and the stability of the boundary layer over a rigid wall is established. Examples of stable boundary layer profiles of both types are given.

Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru (Splitting of 3-manifolds and rigidity of area-minimizing surfaces, arXiv:1107.5346, 2011). We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.

F-Minimal Surface and Manifold with Positive m-Bakry–Émery Ricci Curvature

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the $n$-shpere, or the $n$-dimensional hemisphere. Finally, for compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature and $f$-mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if only if the manifold is isometric to an Euclidean ball.

APPLICATIONS OF TOPONOGOV’S COMPARISON THEOREMS FOR OPEN TRIANGLES

Recently we generalized Toponogov's comparison theorem to a complete Riemannian manifold with smooth convex boundary, where a geodesic triangle was replaced by an open (geodesic) triangle standing on the boundary of the manifold, and a model surface was replaced by the universal covering surface of a cylinder of revolution with totally geodesic boundary. The aim of this article is to prove splitting theorems of two types as an application. Moreover, we establish a weaker version of our Toponogov comparison theorem for open triangles, because the weaker version is quite enough to prove one of the splitting theorems.

Level curves of constant mean curvature graphs over convex domains

We consider a compact graph of constant mean curvature H≠0 and with planar convex boundary. We prove that a level curve is convex provided its height with respect to the boundary plane is bigger than 1/(2|H|).

A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary

Let $M$ be a compact $n$-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary $\partial M$. Assume that the mean curvature $H$ of the boundary $\partial M$ satisfies $H \geq (n-1) k >0$ for some positive constant $k$. In this paper, we prove that the distance function $d$ to the boundary $\partial M$ is bounded from above by $\frac{1}{k}$ and the upper bound is achieved if and only if $M$ is isometric to an $n$-dimensional Euclidean ball of radius $\frac{1}{k}$.

Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximisation for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimisation problems with the help of convex semidefinite programming (optimisation over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach.

Analytic functions with polar and logarithmic singularities and locally convex boundary values

We consider a family of functions which have a pole of order m and a logarithmic term at the infinity and generalize univalent convex functions defined in the exterior of the unit disc. We prove sharp estimates for the derivative and the Schwarzian and describe some geometric properties of such functions.