Evolution of the Torsional Rigidity Under Geometric Flows

This work consists an introduction to the classical and quantum information theory of geometric flows of (relativistic) Lagrange–Hamilton mechanical systems. Basic geometric and physical properties of the canonical nonholonomic deformations of G. Perelman entropy functionals and geometric flows evolution equations of classical mechanical systems are described. There are studied projections of such F- and W-functionals on Lorentz spacetime manifolds and three-dimensional spacelike hypersurfaces. These functionals are used for elaborating relativistic thermodynamic models for Lagrange–Hamilton geometric evolution and respective generalized Hamilton geometric flow and nonholonomic Ricci flow equations. The concept of nonholonomic W-entropy is developed as a complementary one for the classical Shannon entropy and the quantum von Neumann entropy. There are considered geometric flow generalizations of the approaches based on classical and quantum relative entropy, conditional entropy, mutual information, and related thermodynamic models. Such basic ingredients and topics of quantum geometric flow information theory are elaborated using the formalism of density matrices and measurements with quantum channels for the evolution of quantum mechanical systems.

In this survey paper we discuss ancient solutions to different geometric flows, such as the Ricci flow, the mean curvature flow and the Yamabe flow. We survey the classification results of ancient solutions in the Ricci flow and the mean curvature flow. We also discuss methods for constructing new ancient solutions to the Yamabe flow, indicating that the classification results for this flow are impossible to expect.

Since its discovery, Hamilton’s Li–Yau–Hamilton (LYH) estimate has proven its importance in many different works (for example [7], [9], [10], [6]), and as a result, similar inequalities have subsequently appeared in the study of numerous other geometric flows – the mean curvature flow [11], the Kahler– Ricci flow [1], and the Gauss curvature flow [2], among others. The importance of LYH-type estimates is underscored by the fact that the discovery of an LYH estimate without curvature assumption is a large step in Hamilton’s program for Geometrization. Traditionally, positivity of curvature in some form has always been needed for the existence of an LYH estimate. However, work of Hamilton [10] and Ivey [13] in three dimensions indicate that because the curvature operator becomes (in a sense) close to positive near singularities, and because there is an LYH estimate for positive curvature operator, there should be an LYH estimate without any curvature assumptions. Because of this, one approach towards finding an inequality on spaces of arbitrary curvature is to perturb the LYH estimates that are found when there is positive curvature. In fact, using this point of view, an LYH estimate was discovered on surfaces where some negative curvature was allowed [12]. This provides great motivation to find and understand the LYH estimates that do exist when positivity is assumed, and to discover the deeper reasons why such inequalities exist. One very interesting approach towards understanding LYH estimates attempts to view these inequalities in a geometric setting. This was first accomplished in the work of Chow and Chu [3], where a degenerate metric and a space-time connection is place on the flow. In this setting, Hamilton’s original LYH quantity appears very naturally and geometrically, being closely related to the curvature of this conenction. In subsequent papers, Chow and Chu [4], as well as Chow and Knopf [5] have since gained more understanding of this point of view. They have been able to refine it, clarify-

We present a new isogeometric analysis (IGA) approach based on extended Loop subdivision scheme for solving various geometric flows defined on subdivision surfaces. The studied flows include the second‐order, fourth‐order, and sixth‐order geometric flows, such as averaged mean curvature flow, constant mean curvature flow, and minimal mean‐curvature‐variation flow, which are generally derived by minimizing the associate energy functionals with ‐gradient flow respectively. The geometric flows are discretized by means of subdivision based IGA, where the finite element space is formulated by the limit form of the extended Loop subdivision for different initial control meshes. The basis functions, consisting of quartic box‐splines corresponding to each subdivided control mesh, are utilized to represent the geometry exactly. For the cases of the evolution of open surfaces with any shape boundary, high‐order continuous boundary conditions derived from the mixed variational forms of the geometric flows should be implemented to be consistent with the isogeometric concept. For time discretization, we adopt an adaptive semi‐implicit Euler scheme. By several numerical experiments, we study the convergence behaviors of the proposed approach for solving the geometric flows with high‐order boundary conditions. Moreover, the numerical results also show the accuracy and efficiency of the proposed method.

Simulating the behavior of soap films and foams is a challenging task. A direct numerical simulation of films and foams via the Navier-Stokes equations is still computationally too expensive. We propose an alternative formulation inspired by geometric flow. Our model exploits the fact, according to Plateau's laws, that the steady state of a film is a union of constant mean curvature surfaces and minimal surfaces. Such surfaces are also well known as the steady state solutions of certain curvature flows. We show a link between the Navier-Stokes equations and a recent variant of mean curvature flow, called hyperbolic mean curvature flow , under the assumption of constant air pressure per enclosed region. Instead of using hyperbolic mean curvature flow as is, we propose to replace curvature by the gradient of the surface area functional. This formulation enables us to robustly handle non-manifold configurations; such junctions connecting multiple films are intractable with the traditional formulation using curvature. We also add explicit volume preservation to hyperbolic mean curvature flow, which in fact corresponds to the pressure term of the Navier-Stokes equations. Our method is simple, fast, robust, and consistent with Plateau's laws, which are all due to our reformulation of film dynamics as a geometric flow.

We will give a survey of recent research progress on ancient or eternal solutions to geometric flows such as the Ricci flow, the Mean Curvature flow and the Yamabe flow. We will address the classification of ancient solutions to parabolic equations as well as the construction of new ancient solutions from the gluing of two or more solitons.

Exact solutions for the torsion of a bar, flow, and heat transfer in a duct, all having rounded triangular cross sections, are presented. The family of shapes has the circle and the equilateral triangle as extreme cases. Geometrical properties, torsional rigidity, flow rates, and Nusselt numbers are determined.

In this work, we investigate the properties of the Abelian gauge vector field in the background of a string-cigar braneworld. Both the thin and thick brane limits are considered. The string-cigar scenario can be regarded as an interior and exterior string-like solution. The source undergoes a geometric Ricci flow which is related to a variation of the bulk cosmological constant. The Ricci flow changes the width and amplitude of the massless mode at the brane core and recovers the usual string-like behavior at large distances. By means of suitable numerical methods, we attain the Kaluza–Klein (KK) spectrum for the string-like and the string-cigar models. For the string-cigar model, the KK modes are smooth near the brane and their amplitude are enhanced by the brane core. Furthermore, the analogue Schrödinger potential is also regulated by the geometric flow.

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$ ($n\geqslant2$) is mean convex and star-shaped. Several interesting examples and some hyperbolic evolution equations for geometric quantities of the evolving hypersurfaces have been shown. Besides, under different assumptions for the initial velocity, we can get the expansion and the convergence results of a hyperbolic inverse mean curvature flow in the plane $\mathbb{R}^2$, whose evolving curves move normally.

A smooth solution {Γ(t)}t∈[0,T]⊂\Rd of a parabolic geometric flow is characterized as the reachability set of a stochastic target problem. In this control problem the controller tries to steer the state process into a given deterministic set \Tc with probability one. The reachability set, V(t), for the target problem is the set of all initial data x from which the state process \xx(t)∈\Tc for some control process ν. This representation is proved by studying the squared distance function to Γ(t). For the codimension k mean curvature flow, the state process is dX(t)=2√PdW(t), where W(t) is a d-dimensional Brownian motion, and the control P is any projection matrix onto a (d−k)-dimensional plane. Smooth solutions of the inverse mean curvature flow and a discussion of non smooth solutions are also given.

We develop an approach to the theory of relativistic geometric flows and emergent gravity defined by entropy functionals and related statistical thermodynamics models. Nonholonomic deformations of G. Perelman’s functionals and related entropic values used for deriving relativistic geometric evolution flow equations. For self-similar configurations, such equations describe generalized Ricci solitons defining modified Einstein equations. We analyse possible connections between relativistic models of nonholonomic Ricci flows and emergent modified gravity theories. We prove that corresponding systems of nonlinear partial differential equations, PDEs, for entropic flows and modified gravity posses certain general decoupling and integration properties. There are constructed new classes of exact and parametric solutions for nonstationary configurations and locally anisotropic cosmological metrics in modified gravity theories and general relativity. Such solutions describe scenarios of nonlinear geometric evolution and gravitational and matter field dynamics with pattern-forming and quasiperiodic structure and various space quasicrystal and deformed spacetime crystal models. We analyse new classes of generic off-diagonal solutions for entropic gravity theories and show how such solutions can be used for explaining structure formation in modern cosmology. Finally, we speculate why the approaches with Perelman–Lyapunov type functionals are more general or complementary to the constructions elaborated using the concept of Bekenstein–Hawking entropy.

Brownian motion with respect to a metric depending on time; definition, existence and applications to Ricci flow

The meeting continued the biannual conference series Differentialgeometrie im Großen at the MFO which was established in the 60's by Klingenberg and Chern. Traditionally, the conference series covers a wide scope of different aspects of global differential geometry and its connections with topology, geometric group theory and geometric analysis. The Riemannian aspect is emphasized, but the interactions with the developments in complex geometry, symplectic/contact geometry/topology and mathematical physics play also an important role. Within this spectrum each particular conference gives special attention to two or three topics of particular current relevance. The scientific program consisted of 22 (almost) one hour talks leaving ample time for informal discussions. This time, a main focus of the workshop were geometric evolution equations . 6 talks discussed the Ricci flow and its applications to the geometrization of 3-manifolds, the Ricci flow in arbitrary dimension and applications to Riemannian manifolds of positive curvature, the Kähler–Ricci flow and the (inverse) mean curvature flow. A second focus was the geometry of singular spaces (5 talks), that is, metric spaces with sectional curvature bounds (in the sense of Aleksandrov), Gromov-hyperbolic spaces and Carnot spaces with connections to geometric group theory. One of the talks discussed the theory of collapse with lower curvature bound which is another ingredient (independent of Ricci flow) in the argument for geometrization in dimension 3 . Other talks covered results about geometric structures on manifolds (hyperbolic geometry and representation varieties), from geometric analysis (Dirac operators, metrics of positive scalar curvature), symplectic and contact geometry (open book decompositions, confoliations, construction of special Lagrangian submanifolds with isolated conical singularities), and complex geometry (extremal metrics on Kähler manifolds, Kähler–Ricci flow). There were 52 participants from 8 countries, more specifically, 23 participants from Germany, 12 from France, 6 from other European countries, and 11 from North-America. 6 were young researchers (less than 10 years after diploma or B.A.), both on doctoral and postdoctoral level. The organizers would like to thank the institute staff for their great hospitality and support before and during the conference. The financial support of the European Union (in particular for young participants) is gratefully acknowledged.

We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assume the conformal invariant $Y(M,\partial M)<0$. Given any negative smooth functions $f$ in $M$ and $h$ on $\partial M$, there exists a unique conformal metric of $g_0$ such that its scalar curvature equals $f$ and mean curvature curvature equals $h$. The first two methods are sub-super-solution method and subcritical approximation, and the third method is a geometric flow. In the flow approach, assume another conformal invariant $Q(M,\pa M)$ is a negative real number, for some class of initial data, we prove the short time and long time existences of the so-called prescribed scalar curvature plus mean curvature flows, as well as their asymptotic convergence. Via a family of such flows together with some additional variational arguments, under the flow assumptions we prove existence and uniqueness of positive minimizers of the associated energy functional and also the above result by analyzing asymptotic limits of the flows and the relations among some conformal invariants.

We develop geometrical models of vision consistent with the characteristics of the visual cortex and study geometric flows in the relevant model geometries. We provide a novel sub-Riemannian model of the primary visual cortex, which models orientation-frequency selective phase shifted cortex cell behavior and the associated horizontal connectivity. We develop an image enhancement algorithm using sub-Riemannian diffusion and Laplace-Beltrami flow in the model framework. We provide two geometric models for multi-scale orientation map and orientation-frequency preference map construction which employ Bargmann transform in high dimensional cortical spaces. We prove the uniqueness of the solution to sub-Riemannian mean curvature flow equation in the Heisenberg group geometry. An iterative diffusion process followed by a maximum selection mechanism was proposed by Citti and Sarti in the sub-Riemannian setting of the roto-translation group. They conjectured that this two-fold procedure is equivalent to a mean curvature flow. However a complete proof was missing, even in the Euclidean setting. We prove in the Euclidean setting that this two fold procedure is equivalent to mean curvature flow.