Inverse Curvature Research Articles

Radio loudness and spindown of pulsars in Einstein-aether gravity

The exact analytic solutions of Maxwell equations for magnetic field exterior to a slowly rotating magnetized relativistic star in Einstein-aether gravity have been obtained. The obtained analytical expressions were used to derive Goldreich–Julian charge density in plasma magnetosphere and solutions of Poisson equation for the unscreened electric field being parallel to the magnetic field lines in the presence of the aether theory parameters were obtained. As astrophysical applications, we have explored the effects of Einstein-aether gravity on total energy losses of rotating magnetized relativistic star due to magnetodipolar and magnetospheric electromagnetic radiations. The performed study of deathline of radio pulsars through inverse Compton and curvature radiation shows that the position of deathline shifts up (down) in period and its derivative space, so-called P − P ̇ diagram in the presence of larger values of the Einstein-aether gravity parameter c 13 ( c 14 ). Finally, we have provided upper limits for Einstein-aether gravity parameters using observational data on precise measurements of periods of rotation from the different types of isolated neutron stars. In particular, we have shown that some combinations and range of the Einstein-aether are excluded from pulsar observations.

On an inverse curvature flow in two-dimensional space forms

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetric inequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $$n=2$$ case of a conjecture of Girão–Pinheiro.

On an area-preserving inverse curvature flow of convex closed plane curves

This paper deals with the 1/κα-type area-preserving nonlocal flow of smooth convex closed plane curves for all constant α>0. Under this flow, the convexity of the evolving curve is preserved. Due to the existence of finite time curvature blow-up examples, it is shown that, if the curvature κ will not blow up in finite time, the evolving curve will converge smoothly to a circle as t→∞.

Self-expanders to inverse curvature flows by homogeneous functions

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist complete rotationally symmetric self-expanders which are asymptotic to two round cylinders with different radii. These extend some earlier results of inverse mean curvature flow to a wider class of flows.

Minkowski inequalities and constrained inverse curvature flows in warped spaces

Abstract This paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows, we prove long-time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases, new Minkowski type inequalities are the consequence. In higher dimensions, we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.

Uniqueness theorems of self-conformal solutions to inverse curvature flows

It is known from the literature that round spheres are the only closed homothetic self-similar solutions to the inverse mean curvature flow and parabolic curvature flows by degree − 1 -1 homogeneous functions of principal curvatures in the Euclidean space. In this article, we prove that the round sphere is rigid in a stronger sense: under some natural conditions such as star-shapedness, round spheres are the only closed solutions to the above-mentioned flows which evolve by diffeomorphisms generated by conformal Killing fields.

On a length-preserving inverse curvature flow of convex closed plane curves

This paper deals with a 1/κα-type length-preserving nonlocal flow of convex closed plane curves for all α>0. Under this flow, the convexity of the evolving curve is preserved. For a global flow, it is shown that the evolving curve converges smoothly to a circle as t→∞. Some numerical blow-up examples and a sufficient condition leading to the global existence of the flow are also constructed.

Locally constrained inverse curvature flows

We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order—namely, the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to deduce a new Minkowski-type inequality in the anti–de Sitter Schwarzschild manifolds and a weighted isoperimetric-type inequality in hyperbolic space.

On a family of inverse curvature flows for closed convex plane curves

In this note we introduce a family of flows for closed convex curves in the plane. Along the flows the enclosed area of the curve is increasing, and the curve remains convex and converges to a circle. The flows include various flows studied by various authors as special cases.

Nonhomogeneous inverse Gauss curvature flow in ℍ³

In this paper, we consider nonhomogeneous inverse Gauss curvature flows in hyperbolic space H 3 \mathbb {H}^{3} . We prove that if the initial hypersurface Σ 0 \Sigma _0 has nonnegative scalar curvature, then the evolving surface Σ t \Sigma _{t} has nonnegative scalar curvature along the flow for t > 0 t>0 . The solution of the flow exists for all time and becomes more and more umbilic as t → ∞ t\rightarrow \infty .

A note on expansion of convex plane curves via inverse curvature flow

Andrews and Bryan (J Reine Angew Math 653:179–187, 2011) discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this comparison function they estimate the length of any chord from below in terms of the arc length between its endpoints and elapsed time. They apply this estimate to short segments and deduce directly that the maximum curvature decays exponentially to the curvature of a circle with the same length. We consider the expansion of convex curves under inverse (mean) curvature flow and show that the above comparison function also works in this case to obtain a new proof of the fact that the flow exists for all times and becomes round in shape, i.e. converges smoothly to the unit circle after an appropriate rescaling.

A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow

AbstractWe define a formal Riemannian metric on a given conformal class of metrics with signed curvature on a closed Riemann surface. As it turns out this metric is the well-known Mabuchi-Semmes-Donaldson metric of Kähler geometry in a different guise. The metric has many interesting properties, and in particular we show that the classical Liouville energy is geodesically convex. This suggests a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy with respect to this metric, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics by exploiting the metric space structure.

BAR domain proteins—a linkage between cellular membranes, signaling pathways, and the actin cytoskeleton

Actin filament assembly typically occurs in association with cellular membranes. A large number of proteins sit at the interface between actin networks and membranes, playing diverse roles such as initiation of actin polymerization, modulation of membrane curvature, and signaling. Bin/Amphiphysin/Rvs (BAR) domain proteins have been implicated in all of these functions. The BAR domain family of proteins comprises a diverse group of multi-functional effectors, characterized by their modular architecture. In addition to the membrane-curvature sensing/inducing BAR domain module, which also mediates antiparallel dimerization, most contain auxiliary domains implicated in protein-protein and/or protein-membrane interactions, including SH3, PX, PH, RhoGEF, and RhoGAP domains. The shape of the BAR domain itself varies, resulting in three major subfamilies: the classical crescent-shaped BAR, the more extended and less curved F-BAR, and the inverse curvature I-BAR subfamilies. Most members of this family have been implicated in cellular functions that require dynamic remodeling of the actin cytoskeleton, such as endocytosis, organelle trafficking, cell motility, and T-tubule biogenesis in muscle cells. Here, we review the structure and function of mammalian BAR domain proteins and the many ways in which they are interconnected with the actin cytoskeleton.

Inverse curvature flows in Riemannian warped products

The long-time existence and umbilicity estimates for compact, graphical solutions to expanding curvature flows are deduced in Riemannian warped products of a real interval with a compact fibre. Notably we do not assume the ambient manifold to be rotationally symmetric, nor the radial curvature to converge, nor a lower bound on the ambient sectional curvature. The inverse speeds are given by powers 0<p≤1 of a curvature function satisfying few common properties.

New Pinching Estimates for Inverse Curvature Flows in Space Forms

We prove new pinching estimate for the inverse curvature flow of strictly convex hypersurfaces in the space form $N$ of constant sectional curvature $K_N$ with speed given by $F^{-\alpha}$, where $\alpha\in (0,1]$ for $K_N=0,-1$ and $\alpha=1$ for $K_N=1$, $F$ is a smooth, symmetric homogeneous of degree one function which is inverse concave and has dual $F_*$ approaching zero on the boundary of the positive cone $\Gamma_+$. We show that the ratio of the largest principal curvature to the smallest principal curvature of the flow hypersurface is controlled by its initial value. This can be used to prove the smooth convergence of the flow.

Inverse curvature flow for hypersurfaces in Riemannian manifold and its application

This is a survey paper on the inverse curvature flow for hypersurfaces in Riemannian manifold.We first discuss the long time behavior of the inverse curvature flow in Euclidean space, and its application in proving the Alexandrov-Fenchel inequalities for star-shaped hypersurfaces.Then we discuss the related results in hyperbolic space and in sphere.Finally, we discuss the inverse mean curvature flow in Kottler space. Kottler space is an important example of warped product space,and is aymptotically locally hyperbolic at the infinity and satisfies the static equation. We will consider the convergence result of inverse mean curvature flow in such space and also discussits application in proving the Minkowski-type inequality for star-shaped and mean convex hypersurfaces. Inverse curvature flow is an active research area in recent years.We cannot include all results in this short article. For the convenience of the interested readers, we list a few related references on other topics that we do not mention.

Proper Curvature Collineations of Plane Symmetric Static Spacetime in F(R) Theory of Gravity

The purpose of this paper is to study the proper curvature collineations of plane symmetric static spacetime in f(R) theory of gravity. We have taken the metrics of both constant and non-constant curvature and formulated the curvature vector fields by using vanishing Lie derivative and direct integration technique. Here, the solutions of the Einstein’s field equations (EFEs) for both cases involve inverse curvature terms which may cause an increase in gravity that justifies the accelerated expansion of the universe. It turns out that curvature collineations become Killing vector fields and therefore, proper curvature collineations do not exists in both the cases.

Inverse curvature flows in asymptotically Robertson Walker spaces

In this paper we consider inverse curvature flows in a Lorentzian manifold N which is the topological product of the real numbers with a closed Riemannian manifold and equipped with a Lorentzian metric having a future singularity so that N is asymptotically Robertson Walker. The flow speeds are future directed and given by 1/F where F is a homogeneous degree one curvature function of class (K⁎) of the principal curvatures, i.e. the n-th root of the Gauss curvature. We prove longtime existence of these flows and that the flow hypersurfaces converge to smooth functions when they are rescaled with a proper factor which results from the asymptotics of the metric.

Numerical Investigation of Inverse Curvature Ogee Spillway

Design of water structures and their segments including spillways play an important role in water resources management and agricultural activities. In the the linear body part of an ogee spillway, for speeding up the flow rate, the flow should be transferred to the stilling basin by inverse curve so that the water energy can be reduced. This study aims to evaluate the effect of the inverse profile curvature on the pressure of spillway surface using Fluent software. For this purpose, five different curvatures of inverse profile were considered to be equal to no-curvature (zero), 1, 1.5, 2 and 2.5 of the spillway design head. The results indicated that by increasing the curvature radius, the maximum pressure dramatically reduced. And for this purpose, some relationships were given to predict the pressure reduction. Pressure increment in zero curvature is caused by sudden collision of flow lines and turbulence caused by it. By increase in inverse profile curvature, the turbulence is created in flow lines and the maximum pressure shows a lower value than before. In general, there was little change in the average absolute pressure.

Non-parametric Inverse Curvature Flows in the AdS-Schwarzschild Manifold

We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and the principle curvatures of the hypersurface converges to 1 exponentially fast.