lifespan theorem for constrained surface diffusion flows we consider closed immersed hypersurfaces in mathbb r 3 and mathbb r 4 evolving by a class of constrained surface diffusion flows our result similar to earlier results for the willmore flow gives both a positive lower bound on the time for which a smooth solution exists and a small upper bound on a power of the total curvature during this time by phrasing the theorem in terms of the concentration of curvature in the initial surface our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows

noncommutative ricci flow in a matrix geometry we study noncommutative ricci flow in a finite dimensional representation of a noncommutative torus it is shown that the flow exists and converges to the flat metric we also consider the evolution of entropy and a definition of scalar curvature in terms of the ricci flow

quantifying the connectivity of a network the network correlation function method networks are useful for describing systems of interacting objects where the nodes represent the objects and the edges represent the interactions between them the applications include chemical and metabolic systems food webs as well as social networks lately it was found that many of these networks display some common topological features such as high clustering small average path length small world networks and a power law degree distribution scale free networks the topological features of a network are commonly related to the network s functionality however the topology alone does not account for the nature of the interactions in the network and their strength here we present a method for evaluating the correlations between pairs of nodes in the network these correlations depend both on the topology and on the functionality of the network a network with high connectivity displays strong correlations between its interacting nodes and thus features small world functionality we quantify the correlations between all pairs of nodes in the network and express them as matrix elements in the correlation matrix from this information one can plot the correlation function for the network and to extract the correlation length the connectivity of a network is then defined as the ratio between this correlation length and the average path length of the network using this method we distinguish between a topological small world and a functional small world where the latter is characterized by long range correlations and high connectivity clearly networks that share the same topology may have different connectivities based on the nature and strength of their interactions the method is demonstrated on metabolic networks but can be readily generalized to other types of networks

parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

deforming three manifolds with positive scalar curvature in this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three manifold is path connected the proof uses the ricci flow with surgery the conformal method and the connected sum construction of gromov and lawson the work of perelman on hamilton s ricci flow is fundamental as one of the applications we prove the path connectedness of the space of trace free asymptotically flat solutions to the vacuum einstein constraint equations on r 3

the long time behavior of 3 dimensional ricci flow on certain topologies de gruyter 2017 in this paper we analyze the long time behavior of 3 dimensional ricci flow with surgery we prove that under the topological condition that the initial manifold only has non aspherical or hyperbolic components in its geometric decomposition there are only finitely many surgeries and the curvature is bounded by ct 1 for large t this proves a conjecture of perelman for this class of initial topologies the proof of this fact illustrates the fundamental ideas that are used in the subsequent papers of the author

perelman s entropy functional at type i singularities of the ricci flow we study blow ups around fixed points at type i singularities of the ricci flow on closed manifolds using perelman s w functional first we give an alternative proof of the result obtained by naber 18 and enders muller topping 8 that blow up limits are non flat gradient shrinking ricci solitons our second and main result relates a limit w density at a type i singular point to the entropy of the limit gradient shrinking soliton obtained by blowing up at this point in particular we show that no entropy is lost at infinity during the blow up process

eigenmodes of isospectral drums recently it was proved that there exist nonisometric planar regions that have identical laplace spectra that is one cannot hear the shape of a drum the simplest isospectral regions known are bounded by polygons with reentrant corners while the isospectrality can be proven mathematically analytical techniques are unable to produce the eigenvalues themselves furthermore standard numerical methods for computing the eigenvalues such as adaptive finite elements are highly inefficient physical experiments have been performed to measure the spectra but the accuracy and flexibility of this method are limited we describe an algorithm due to descloux and tolley comput methods appl mech engrg 39 1983 pp 37 53 that blends singular finite elements with domain decomposition and show that with a modification that doubles its accuracy this algorithm can be used to compute efficiently the eigenvalues for polygonal regions we present results accurate to 12 digits for the most famous pair of isospectral drums as well as results for another pair

collapsing irreducible 3 manifolds with nontrivial fundamental group let m be a closed orientable irreducible non simply connected 3 manifold we prove that if m admits a sequence of riemannian metrics which volume collapses and whose sectional curvature is locally controlled then m is a graph manifold this is the last step in perelman s proof of thurston s geometrisation conjecture

a modified kahler ricci flow in this note we study a kahler ricci flow modified from the classic version in the non degenerate case strong convergence at infinite time is achieved the main focus should be on degenerate case where some partial results are presented

coupling of brownian motions and perelman s l functional abstract we show that on a manifold whose riemannian metric evolves under backwards ricci flow two brownian motions can be coupled in such a way that their normalized l distance is a supermartingale as a corollary we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non decreasing function and the normalized l distance in particular it provides a new proof of a recent result of topping p topping l optimal transportation for ricci flow j reine angew math 636 2009 93 122

geometrization of 3 dimensional orbifolds this paper is devoted to the proof of the orbifold theorem if o is a compact connected orientable irreducible and topologically atoroidal 3 orbifold with nonempty ramification locus then o is geometric i e has a metric of constant curvature or is seifert fibred as a corollary any smooth orientationpreserving nonfree finite group action on s3 is conjugate to an orthogonal action

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

complete gradient shrinking ricci solitons with pinched curvature we prove that any n dimensional complete gradient shrinking ricci soliton with pinched weyl curvature is a finite quotient of mathbb r n mathbb r times mathbb s n 1 or mathbb s n in particular we do not need to assume the metric to be locally conformally flat

ricci soliton homogeneous nilmanifolds we study a notion weakening the einstein condition on a left invariant riemannian metric g on a nilpotent lie groupn we consider those metrics satisfying ric g ci d for some c in mathbb r and some derivationd of the lie algebra mathfrak n ofn where ric g denotes the ricci operator of n g this condition is equivalent to the metric g to be a ricci soliton we prove that a ricci soliton left invariant metric on n is unique up to isometry and scaling the following characterization is also given n g is a ricci soliton if and only if n g admits a metric standard solvable extension whose corresponding standard solvmanifold s tilde g is einstein this gives several families of new examples of ricci solitons by a variational approach we furthermore show that the ricci soliton homogeneous nilmanifolds n g are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set

the second twisted betti number and the convergence of collapsing riemannian manifolds let mi overset d gh longrightarrow x denote a sequence of n manifolds converging to a compact metric space x in the gromov hausdorff topology such that the sectional curvature is bounded in absolute value and dim x n we prove the following stability result if the fundamental groups of mi are torsion groups of uniformly bounded exponents and the second twisted betti numbers of mi vanish then there is a manifold m and a sequence of diffeomorphisms from m to a subsequence of mi such that the distance functions of the pullback metrics converge to a pseudo metric in c0 norm furthermore m admits a foliation with leaves diffeomorphic to flat manifolds not necessarily compact such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics these results lead to a few interesting applications

convergence of kahler ricci flow in this paper we prove a theorem on convergence of kahler ricci flow on a compact kahler manifold which admits a kahler ricci soliton

structural and functional networks in complex systems with delay functional networks of complex systems are obtained from the analysis of the temporal activity of their components and are often used to infer their unknown underlying connectivity we obtain the equations relating topology and function in a system of diffusively delay coupled elements in complex networks we solve exactly the resulting equations in motifs directed structures of three nodes and in directed networks the mean field solution for directed uncorrelated networks shows that the clusterization of the activity is dominated by the in degree of the nodes and that the locking frequency decreases with increasing average degree we find that the exponent of a power law degree distribution of the structural topology ensuremath gamma is related to the exponent of the associated functional network as ensuremath alpha 2 ensuremath ensuremath gamma ensuremath 1 for ensuremath gamma l2

wave character of metrics and hyperbolic geometric flow in this paper we illustrate the wave character of the metrics and curvatures of manifolds and introduce a new understanding tool the hyperbolic geometric flow this kind of flow is new and very natural to understand certain wave phenomena in the nature as well as the geometry of manifolds it possesses many interesting properties from both mathematics and physics several applications of this method have been found

hyperbolic structures on 3 manifolds i deformation of acylindrical manifolds this is the first in a series of papers showing that haken manifolds have hyperbolic structures this first was published the second two have existed only in preprint form and later preprints were never completed this eprint is only an approximation to the published version which is the definitive form for part i and is provided for convenience only all references and quotations should be taken from the published version since the theorem numbering is different and not all corrections have been incorporated into the present version parts ii and iii will be made available as eprints shortly

conformal deformations of integral pinched 3 manifolds abstract in this paper we prove that under an explicit integral pinching assumption between the l 2 norm of the ricci curvature and the l 2 norm of the scalar curvature a closed 3 manifold with positive scalar curvature admits a conformal metric of positive ricci curvature in particular using a result of hamilton this implies that the manifold is diffeomorphic to a quotient of s 3 the proof of the main result of the paper is based on ideas developed in an article by m gursky and j viaclovsky

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

ricci flow on quasi projective manifolds we consider the kahler ricci flow on complete finite volume metrics that live on the complement of a divisor in a compact kahler manifold x assuming certain spatial asymptotics on the initial metric we compute the singularity time in terms of cohomological data on x we also give a sufficient condition for the singularity if there is one to be type ii

high order eigenvalues for the helmholtz equation in complicated non tensor domains through richardson extrapolation of second order finite differences we apply second order finite differences to calculate the lowest eigenvalues of the helmholtz equation for complicated non tensor domains in the plane using different grids which sample exactly the border of the domain we show that the results obtained applying richardson and pade richardson extrapolations to a set of finite difference eigenvalues corresponding to different grids allow us to obtain extremely precise values when possible we have assessed the precision of our extrapolations comparing them with the highly precise results obtained using the method of particular solutions our empirical findings suggest an asymptotic nature of the fd series in all the cases studied we are able to report numerical results which are more precise than those available in the literature

the kahler ricci flow through singularities we prove the existence and uniqueness of the weak kahler ricci flow on projective varieties with log terminal singularities we also show that the weak kahler ricci flow can be uniquely continued through divisorial contractions and flips if they exist finally we propose an analytic version of the minimal model program with ricci flow

solving the helmholtz equation for membranes of arbitrary shape numerical results i calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on little sinc functions the matrix representation of the pde obtained using this method is explicit and does not require the calculation of integrals to illustrate the virtues of this approach i have considered a large number of examples part of them are taken from the literature and part of them new when possible i have tested the accuracy of these results by comparing them with the exact results when available or with results from the literature in particular in the case of the l shaped membrane the first example discussed in the paper i show that it is possible to extrapolate the results obtained with different grid sizes to obtain highly precise results finally i also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results

inverse mean curvature flow with forced term abstract in this paper we study the evolution of convex hypersurfaces by inverse mean curvature minus an external force field c we prove that the flow will preserve the convexity for any c if c 1 h on the initial surface we prove that the flow will expand the hypersurface for all time and after rescaling the hypersurface will converge to a sphere if c 1 h on the initial surface we show that the maximum existence time of the flow is finite and the hypersurface will contract to a point when approaching the final time

volume collapsed three manifolds with a lower curvature bound in this paper we determine the topology of three dimensional complete orientable riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small

real analyticity of solutions of hamilton s equation il existe des constantes positives cn et sn dependant seulement de la dimension n de la variete m telles que pour une metrique initiale g 0 avec r g 0 1 l equation de hamilton a une solution jusqu a t sn et l estimation suivante est vraie pour la metrique solution en t o t sn k 0 t k k 1 12 k r 2 c n

numerical investigation of isospectral cavities built from triangles we present computational approaches as alternatives to a recent microwave cavity experiment by s sridhar and a kudrolli phys rev lett 72 2175 1994 on isospectral cavities built from triangles a straightforward proof of isospectrality is given based on the mode matching method our results show that the experiment is accurate to 0 3 for the first 25 states the level statistics resemble those of a gaussian orthogonal ensemble when the integrable part of the spectrum is removed

mapping from structure to dynamics a unified view of dynamical processes on networks although it is unambiguously agreed that structure plays a fundamental role in shaping the collective dynamics of complex systems how structure determines dynamics exactly still remains unclear we investigate a general computational transformation by which we can map the network topology directly to the dynamical patterns emergent on it independent of the nature of the dynamical processes remarkably we find that many seemingly different dynamical processes on networks such as coupled oscillators ensemble neuron firing epidemic spreading and diffusion can all be understood and unified through this same procedure utilizing the inherent multiscale nature of this structure dynamics transformation we further define a multiscale complexity measure which can quantify the functional diversity a general network can support at different organization levels using only its structure we find that a wide variety of topological features observed in real networks such as modularity hierarchy degree heterogeneity and mixing all result in higher complexity this result suggests that the demand for functional diversity is driving the structural evolution of physical networks

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

topological structural classes of complex networks we use theoretical principles to study how complex networks are topologically organized at large scale using spectral graph theory we predict the existence of four different topological structural classes of networks these classes correspond respectively to highly homogenous networks lacking structural bottlenecks networks organized into highly interconnected modules with low inter community connectivity networks with a highly connected central core surrounded by a sparser periphery and networks displaying a combination of highly connected groups quasicliques and groups of nodes partitioned into disjoint subsets quasibipartites here we show by means of the spectral scaling method that these classes really exist in real world ecological biological informational technological and social networks we show that neither of three network growth mechanisms random with uniform distribution preferential attachment and random with the same degree sequence as real network is able to reproduce the four structural classes of complex networks these models reproduce two of the network classes as a function of the average degree but completely fail in reproducing the other two classes of networks

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

on the positive mass theorem for manifolds with corners we study the positive mass theorem for certain non smooth metrics following p miao s work our approach is to smooth the metric using the ricci flow as well as improving some previous results on the behaviour of the adm mass under the ricci flow we extend the analysis of the zero mass case to higher dimensions

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface