Laplacian Flow Research Articles

Laplacian solitons on nilpotent Lie groups

We investigate the existence of closed $G_2$-structures which are solitons for the Laplacian flow on nilpotent Lie groups. We obtain that seven of the twelve Lie algebras admitting a closed $G_2$-structure do admit a Laplacian soliton. Moreover, one of them admits a continuous family of Laplacian solitons which are pairwise non-homothetic and the Laplacian flow evolution on four of the Lie groups is not diagonal.

Cohomogeneity‐one G2‐Laplacian flow on the 7‐torus

We prove the hypersymplectic flow of simple type on standard torus $\mathbb{T}^4$ exists for all time and converges to the standard flat structure modulo diffeomorphisms. This result in particular gives the first example of a cohomogeneity-one $G_2$-Laplacian flow on a compact $7$-manifold which exists for all time and converges to a torsion-free $G_2$ structure modulo diffeomorphisms.

Shi-type estimates and finite time singularities of flows of G2 structures

In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to more general flows of G$_2$ structures including the modified Laplacian co-flow. Then we prove a version of $\kappa$-non-collapsing theorem. We will use both of them to study finite time singularities of general flows of G$_2$ structures.

Compact Solvmanifolds with a Closed G<sub>2</sub>-Structure

We consider a parametrized family of compact G2-calibrated solvmanifolds, and construct associative (so volume-minimizing submanifolds) 3-tori with respect to the closed G2-structure. We also study the Laplacian flow of this closed G2 form on the solvable Lie group underlying to each of these solvmanifolds, and show long time existence of the solution.

On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing I: Analysis

Variable exponent spaces have found interesting applications in real world problems. In imaging models, the variable exponent can approach the critical value 1 and this poses unique challenges in proving existence of solutions for the corresponding PDEs. In this work, we develop some new functional framework to study time-dependent parabolic variable exponent flows. Specifically, we consider bounded vectorial partial variation (BVPV) space and its variable exponent counterpart. We then prove the existence of weak solutions to the critical vectorial p(t,x)-Laplacian flow in the variable exponent BVPV space. For time-independent critical vectorial p(x)-Laplacian flow we obtain a unique semigroup solution. Our results are in particular valid in the scalar case and solve a long standing open problem.

Differential inequalities in multi-agent coordination and opinion dynamics modeling

Many distributed algorithms for multi-agent coordination employ the simple averaging dynamics, referred to as the Laplacian flow. Besides the standard consensus protocols, examples include, but are not limited to, algorithms for aggregation and containment control, target surrounding, distributed optimization and models of opinion formation in social groups. In spite of their similarities, each of these algorithms has been studied using separate mathematical techniques. In this paper, we show that stability and convergence of many coordination algorithms involving the Laplacian flow dynamics follow from the general consensus dichotomy property of a special differential inequality. The consensus dichotomy implies that any solution to the differential inequality is either unbounded or converges to a consensus equilibrium. In this paper, we establish the dichotomy criteria for differential inequalities and illustrate their applications to multi-agent coordination and opinion dynamics modeling.

Laplacian solitons: Questions and homogeneous examples

We give the first examples of closed Laplacian solitons which are shrinking, and in particular produce closed Laplacian flow solutions with a finite-time singularity. Extremally Ricci pinched G2-structures (introduced by Bryant) which are steady Laplacian solitons have also been found. All the examples are left-invariant G2-structures on solvable Lie groups.

Laplacian flow of homogeneous G2-structures and its solitons

We use the bracket flow/algebraic soliton approach to study the Laplacian flow of $G_2$-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a $G$-invariant $G_2$-structure on a homogeneous space $G/K$ that flows by pull-back of automorphisms of $G$ up to scaling). Algebraic solitons are geometrically characterized among Laplacian solitons as those with a `diagonal' evolution. Unlike the Ricci flow case, where any homogeneous Ricci soliton is isometric to an algebraic soliton, we have found, as an application of the above characterization, an example of a left-invariant closed semi-algebraic soliton on a nilpotent Lie group which is not equivalent to any algebraic soliton. The (normalized) bracket flow evolution of such a soliton is periodic. In the context of solvable Lie groups with a codimension-one abelian normal subgroup, we obtain long time existence for any closed Laplacian flow solution; furthermore, the norm of the torsion is strictly decreasing and converges to zero. We also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding Laplacian solitons.

Laplacian flow for closed G2 structures: Shi-type estimates, uniqueness and compactness

We develop foundational theory for the Laplacian flow for closed G2 structures which will be essential for future study. (1). We prove Shi-type derivative estimates for the Riemann curvature tensor Rm and torsion tensor T along the flow, i.e. that a bound onΛ(x,t)=|∇T(x,t)|g(t)2+|Rm(x,t)|g(t)212\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\Lambda(x,t)=\\left(|\\nabla T(x,t)|_{g(t)}^2+|Rm(x,t)|_{g(t)}^2\\right)^{\\frac 12}$$\\end{document}will imply bounds on all covariant derivatives of Rm and T. (2). We show that {Lambda(x,t)} will blow up at a finite-time singularity, so the flow will exist as long as {Lambda(x,t)} remains bounded. (3). We give a new proof of forward uniqueness and prove backward uniqueness of the flow, and give some applications. (4). We prove a compactness theorem for the flow and use it to strengthen our long time existence result from (2) to show that the flow will exist as long as the velocity of the flow remains bounded. (5). Finally, we study soliton solutions of the Laplacian flow.

Laplacian Flow of Closed G $$_2$$ 2 -Structures Inducing Nilsolitons

We study the existence of left invariant closed \(G_2\)-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these \(G_2\)-structures, we show long time existence and uniqueness of solution for the Laplacian flow on the noncompact manifold. Moreover, considering the Laplacian flow on the associated Lie algebra as a bracket flow on \({\mathbb {R}}^7\) in a similar way as in Lauret (Commun Anal Geom 19(5):831–854, 2011) we prove that the underlying metrics \(g(t)\) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in the nilpotent Lie group, as \(t\) goes to infinity.

A parabolic flow of balanced metrics

Abstract We prove a general criterion to establish existence and uniqueness of a short-time solution to an evolution equation involving “closed” sections of a vector bundle, generalizing a method used by Bryant and Xu [8] for studying the Laplacian flow in G 2 $G_{2}$ -geometry. We apply this theorem in balanced geometry introducing a natural extension of the Calabi flow to the balanced case. We show that this flow has always a unique short-time solution belonging to the same Bott–Chern cohomology class of the initial balanced structure and that it preserves the Kähler condition. Finally, we study explicitly the flow on the Iwasawa manifold.

Laplacian solitons and symmetry in [formula omitted]-geometry

In this paper, it is shown that (with no additional assumptions) on a compact 7-dimensional manifold which admits a G2-structure, soliton solutions to the Laplacian flow of R. Bryant can only be shrinking or steady. We also show that the space of symmetries (vector fields that annihilate via the Lie derivative) of a torsion-free G2-structure on a compact 7-manifold is canonically isomorphic to H1(M,R) by a direct method. Some comparisons with Ricci solitons are also discussed and emphasized, along with some future directions of exploration.

Anomalous large-time behaviour of the $p$-Laplacian flow in an exterior domain in low dimension

We study the large time behaviour of weak nonnegative solutions of the p -Laplace equation posed in an exterior domain in space dimension N<p with boundary conditions u = 0 . The description is done in terms of matched asymptotics: the outer asymptotic profile is a dipole-like self-similar solution with a singularity at x = 0 and anomalous similarity exponents. The inner asymptotic behaviour is given by a separate-variable profile. We gather both estimates in a global approximant and we also study the behaviour of the free boundary for compactly supported solutions. We complete in this way the analysis made in a previous work for high space dimensions N ≥ p , a range in which the large-time influence of the holes is less dramatic.

Distributed Coverage Verification in Sensor Networks Without Location Information

In this paper, we present three distributed algorithms for coverage verification in sensor networks with no location information. We demonstrate how, in the absence of localization devices, simplicial complexes and tools from algebraic topology can be used in providing valuable information about the properties of the cover. Our approach is based on computation of homologies of the Rips complex corresponding to the sensor network. First, we present a decentralized scheme based on Laplacian flows to compute a generator of the first homology, which represents coverage holes. Then, we formulate the problem of localizing coverage holes as an optimization problem for computing a sparse generator of the first homology. Furthermore, we show that one can detect redundancies in the sensor network by finding a sparse generator of the second homology of the cover relative to its boundary. We demonstrate how subgradient methods can be used in solving these optimization problems in a distributed manner. Finally, we provide simulations that illustrate the performance of our algorithms.

On the enriched RBF method for singular potential problems

The performance of Kansa's method in the solution of elliptic partial differential equations (PDEs) with singular boundary conditions is addressed. Like in all global numerical schemes, low-order singularities bring about Gibbs’ oscillations that deteriorate the accuracy and the convergence rate of Kansa's method to a great extent. Moreover, they may render it uncapable of handling common problems of incompressible flow. Focussing on a problem of Laplacian flow which is linked to the benchmark Motz problem, it is shown how all these difficulties can be overcome by enriching the radial basis function (RBF) interpolant with the proper singularity-capturing terms. This simple modification even enables Kansa's method to outperform the finite element method (FEM) in the conservation of Laplacian flow through an irregular channel.

Laplacian Equivalents to Subsonic Potential Flows

This paper presents a mapping technique for the solution of Laplacian equivalents to steady subsonic potential flows in the plane. The method employs a coordinate transformation to map a given closed profile and domain in a compressible flow to a related closed profile and domain in a Laplacian flow, with differences in the geometries stemming only from compressibility effects. Unlike certain past methods, the mapping is valid for nonlinear flows with arbitrary density–velocity relations. The theory is implemented in an iterative numerical method in which the Laplacian flow is solved with a boundary element technique, and the compressible flow is then found from a finite difference solution to the mapping. Validation is provided by comparing the resulting subsonic flow solutions to knownresultsintheliteratureforbothacircularcylinderandaNACA0012profile.Themethodisexactinthesense that the inverse mapping of the equivalent Laplacian field recovers the subsonic full-potential solution over the originalprofile,uptonumericalaccuracy.Becauseincompressiblepotential flowsareLaplacian,themappingyields insights into the phenomenon of compressibility itself in the subsonic setting, and it may have utility in certain aerodynamic test applications.

A Fast, Two-Dimensional Panel Method

A new two-dimensional panel technique has been developed to solve Laplacian flows, which eliminates the edge effect present in traditional panel methods. Such a method is very useful for applications where the velocity induced by the panels is required at arbitrary locations. Particle based flow solvers are a prime example. The method, however, requires considerably more computational effort. In this paper the method is modified to improve computational efficiency by adapting the fast multipole algorithm for the panel method. Significant improvement in computational efficiency is obtained while ensuring that the edge effects are eliminated.

Mesh regularization and adaptive smoothing

The paper presents a set of mesh smoothing tools developed to increase mesh regularity, reduce oversmoothing, and enhance crease lines. Mesh smoothing with simultaneous increasing mesh regularity and reducing oversmoothing is achieved by combining together the Laplacian flow and a mesh evolution by a function of the mean curvature. To enhance salient ridge and ravine structures we use a coupled nonlinear diffusion of the mesh normals and vertices.

Defects in wetting fronts: Experimental and theoretical results

We study flow in a porous medium in the presence of a discontinuous perturbation (defect) that perturbs its parameters. The discontinuous perturbation is at the interface between regions of different porosities, produced in the experiment by uniformly depositing black toner with a laser printer. When water propagates in such a medium the wetting front develops a nontrivial, curved steady state solution. We present experimental measurements of the profiles and their propagation rate, as well as the characteristics of the steady state solution. A naive attempt to use perturbation theory fails due to discontinuities at the ``defect.'' We proceed to develop a theoretical treatment based on Laplacian flow, that is good to all orders and predicts most of the features that can be observed. We find good quantitative agreement between theory and experiment, without any fit parameters.

A new analytical solution for Laplacian porous-media flow with arbitrary boundary shapes and conditions

Solutions to the Laplace equation are required in many branches of groundwater hydrology. The most difficult class of problems involve generating solutions to the Laplace equation when the domain is an irregular shape and/or has a free surface. In this case, solutions can be generated by numerical Methods, e.g., finite difference, finite element, or boundary integer element methods. However, due to the computational requirements of these methods, they are not often incorporated into “practical” models of subcatchment hydrology. This paper, illustrates an alternative method to finite difference and finite elements for generating solutions to saturated and partiallysaturated porous media flow problems with mixed and irregular boundaries. The solution strategy is based on approximating the boundary conditions with orthonormal sequence expansions and developing appropriate recurrence relations for the coefficients of the expansion. In practice, the solution methodology involves truncation of the sequence to meet a desired accuracy. In past research, analytical series solutions were limited to problems with simple domains. However, we show that this method with modifications, can also be applied to complex domains. We also show that the method is computationally very efficient, and rapid solutions can be generated on basic PC computers. Several examples will be discussed.