on the long time behavior of type iii ricci flow solutions we show that three dimensional homogeneous ricci flow solutions that admit finite volume quotients have long time limits given by expanding solitons we show that the same is true for a large class of four dimensional homogeneous solutions we give an extension of hamilton s compactness theorem that does not assume a lower injectivity radius bound in terms of riemannian groupoids using this we show that the long time behavior of type iii ricci flow solutions is governed by the dynamics of an mathbb r action on a compact space

a higher order tv type variational problem related to the denoising and inpainting of images abstract we give a comprehensive survey on a class of higher order variational problems which are motivated by applications in mathematical imaging the overall aim of this note is to investigate if and in which manner results from the first author s previous work on variants of the tv regularization model see e g bildhauer and fuchs 2012 2014 2015 and fuchs and tietz 2015 can be extended to functionals which involve higher derivatives this seems to be not only of theoretical interest but also relevant to applications since higher order tv denoising appears to maintain the advantages of the classical model as introduced by rudin osher and fatemi in rudin et al 1992 while avoiding the unpleasant staircasing effect see e g bredies et al 2010 or lysaker et al 2003 our paper features results concerning generalized solutions in spaces of functions of higher order bounded variation dual solutions as well as partial regularity of minimizers

elasticities of krull domains with finite divisor class group abstract fix any positive integer b and consider the set z b of all values r where r is a krull domain with divisor class group z b and r denotes the elasticity of r we focus on the case b 2p showing that 2p 1 p z 2p and z p z 2p we also present a precise description of the set i 1 k z 2

laplacian flow of homogeneous g2 structures and its solitons we use the bracket flow algebraic soliton approach to study the laplacian flow of g2 structures and its solitons in the homogeneous case we prove that any homogeneous laplacian soliton is equivalent to a semi algebraic soliton that is a g invariant g2 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

on the moment map for the variety of lie algebras we consider the moment map m pvn iu n for the action of gl n on vn 2 cn cn the critical points of the functional fn m 2 pvn r are studied in order to understand the stratification of ln pvn defined by the negative gradient flow of fn where ln is the algebraic subset of all lie algebras we obtain a description of the critical points which lie in ln in terms of those which are nilpotent as well as the minima and maxima of fn ln r a characterization of the critical points modulo isomorphism as the union of categorical quotients of suitable actions is considered and some applications to the study of ln are given

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

finding einstein solvmanifolds by a variational method we use a variational approach to prove that any nilpotent lie algebra having a codimension one abelian ideal and anyone of dimension leq 5 admits a rank one solvable extension which can be endowed with an einstein left invariant riemannian metric a curve of 8 dimensional einstein solvmanifolds is also given

homogeneous einstein metrics and simplicial complexes on compact homogeneous spaces we investigate the hilbert action restricted to the space of homogeneous metrics of volume one based on a detailed understanding of the high energy levels of this action we assign to a compact homogeneous space a simplicial complex whose non contractibility is a sufficient condition for the existence of homogeneous einstein metrics

an analysis using the zaks skula constant of element factorizations in dedekind domains abstract let d be a dedekind domain with finite class group g let be a nonzero nonunit of d and suppose 1 s 1 t are irreducibles of d such that 1 s 1 t two recent papers j steffan 1986 j algebra 102 229 236 r j valenza 1990 j number theory 38 212 218 have studied the quotient s t and both conclude that given d there exists a smallest real number d such that s t d while it is easy to show that d g 2 we use the zaks skula constant to find a sharper upper bound on d we then use this upper bound to explore the behavior of the function which counts the number of different lengths of certain products on two specific classes of dedekind domains

einstein solvmanifolds existence and non existence questions the aim of this paper is to study the problem of which solvable lie groups admit an einstein left invariant metric the space mathcal n of all nilpotent lie brackets on mathbb r n parametrizes a set of n 1 dimensional rank one solvmanifolds s mu mu in mathcal n containing the set of all those which are einstein in that dimension the moment map for the natural gl n action on mathcal n evaluated at mu in mathcal n encodes geometric information on s and suggests the use of strong results from geometric invariant theory for instance the functional on mathcal n whose critical points are precisely the einstein s s is the square norm of this moment map we use a gl n invariant stratification for the space mathcal n and show that there is a strong interplay between the strata and the einstein condition on the solvmanifolds s as an application we obtain criteria to decide whether a given nilpotent lie algebra can be the nilradical of a rank one einstein solvmanifold or not we find several examples of mathbb n graded even 2 step nilpotent lie algebras which are not a classification in the 7 dimensional 6 step case and an existence result for certain 2 step algebras associated to graphs are also given

immortal homogeneous ricci flows we show that for an immortal homogeneous ricci flow solution any sequence of parabolic blow downs subconverges to a homogeneous expanding ricci soliton this is established by constructing a new lyapunov function based on curvature estimates which come from real geometric invariant theory

regularity results for equilibria in a variational model for fracture in recent years models describing interactions between fracture and damage have been proposed in which the relaxed energy of the material is given by a functional involving bulk and interfacial terms of the form f vu dx i n x jn k where q is an open bounded subset of r q 1 g l r a 0 0 the bulk energy density f is quasiconvex k c r is closed and the admissible deformation u q f r is c in i k one of the main issues has to do with regularity properties of the crack site k for a minimizing pair k u in the scalar case i e when u o f r similar models were adopted to image segmentation problems and the regularity of the edge set k has been successfully resolved for a quite broad class of convex functions f with growth p 1 at infinity in turn this regularity entails the existence of classical solutions the methods thus used cannot be carried out to the vectorial case except for a very restrictive class of integrands in this paper we deal with a vector valued case on the plane obtaining regularity for minimizers of q corresponding to polyconvex bulk energy densities of the form where the convex function h grows linearly at infinity 1991 mathematics subject classification amer math soc 35j20 49q20 49j45 49n60

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

einstein solvmanifolds attached to two step nilradicals a riemannian einstein solvmanifold possibly any noncompact homogeneous einstein space is almost completely determined by the nilradical of its lie algebra a nilpotent lie algebra which can serve as the nilradical of an einstein metric solvable lie algebra is called an einstein nilradical we give a classification of two step nilpotent einstein nilradicals with two dimensional center informally the defining matrix pencil must have no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity we also show that the dual to a two step einstein nilradical is not in general an einstein nilradical

noncompact homogeneous einstein manifolds attached to graded lie algebras in this paper we study the nilradicals of parabolic subalgebras of semisimple lie algebras and the natural one dimensional solvable extensions of them we investigate the structures curvatures and einstein conditions of the associated nilmanifolds and solvmanifolds we show that our solvmanifold is einstein if the nilradical is two step new examples of einstein solvmanifolds with three step and four step nilradicals are also given

inequalities defining orbit spaces the orbit space of a representation of a compact lie group has a natural semialgebraic structure we describe explicit ways of finding the inequalities defining this structure and we give some applications

conformally parallel g2 structures on a class of solvmanifolds starting from a 6 dimensional nilpotent lie group n endowed with an invariant su 3 structure we construct a homogeneous conformally parallel g 2 metric on an associated solvmanifold we classify all half flat su 3 structures that endow the rank one solvable extension of n with a conformally parallel g 2 structure by suitably deforming the su 3 structures obtained we are able to describe the corresponding non homogeneous ricci flat metrics with holonomy contained in g 2 in the process we also find a new metric with exceptional holonomy

a compactness result for kahler ricci solitons abstract in this paper we prove a compactness result for compact kahler ricci gradient shrinking solitons if m i g i is a sequence of kahler ricci solitons of real dimension n 4 whose curvatures have uniformly bounded l n 2 norms whose ricci curvatures are uniformly bounded from below and g i 1 2 a where is perelman s functional there is a subsequence m i g i converging to a compact orbifold m g with finitely many isolated singularities where g is a kahler ricci soliton metric in an orbifold sense satisfies a soliton equation away from singular points and smoothly extends in some gauge to a metric satisfying kahler ricci soliton equation in a lifting around singular points

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

willmore orbits for isometric lie actions in this work we study the willmore submanifolds in a closed connected riemannian manifold which are orbits for the isometric action of a compact connected lie group we call them homogeneous willmore submanifolds or willmore orbits the criteria for these special willmore submanifolds is much easier than the general theory which requires a complicated euler lagrange equation our main theorem claims when the orbit type stratification for the group action satisfies certain conditions then we can find a willmore orbit in each stratified subset some classical examples of special importance like willmore torus veronese surface etc can be interpreted as willmore orbits and easily verified with our method our theorems provide a large number of new examples for willmore submanifolds as well as estimates for their numbers which are sharp in some classical cases

computing the elasticity of a krull monoid abstract if s is a krull monoid with finitely generated divisor class group such that only finitely many divisor classes of s contain prime divisors then we construct an algorithm to compute the elasticity of s

a step towards the alekseevskii conjecture we refine existing structure results for non compact homogeneous einstein manifolds and provide a reduction in the classification problem of such spaces using this work we verify the generalized alekseevskii conjecture for a large class of homogeneous spaces

the ricci flow on solvmanifolds of real type abstract we show that for any solvable lie group of real type any homogeneous ricci flow solution converges in cheeger gromov topology to a unique non flat solvsoliton which is independent of the initial left invariant metric as an application we obtain results on the isometry groups of non flat solvsoliton metrics and einstein solvmanifolds

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

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

einstein solvmanifolds and the pre einstein derivation an einstein nilradical is a nilpotent lie algebra which can be the nilradical of a metric einstein solvable lie algebra the classification of riemannian einstein solvmanifolds possibly of all noncompact homogeneous einstein spaces can be reduced to determining which nilpotent lie algebras are einstein nilradicals and to finding for every einstein nilradical its einstein metric solvable extension for every nilpotent lie algebra we construct an essentially unique derivation the pre einstein derivation the solvable extension by which may carry an einstein inner product using the pre einstein derivation we then give a variational characterization of einstein nilradicals as an application we prove an easy to check convex geometry condition for a nilpotent lie algebra with a nice basis to be an einstein nilradical and also show that a typical two step nilpotent lie algebra is an einstein nilradical the theory of riemannian homogeneous spaces with an einstein metric splits into three very different cases depending on the sign of the einstein constant the scalar curvature among them the picture is complete only in the ricci flat case by the result of ak every ricci flat homogeneous space is flat the major open conjecture in the case of negative scalar curvature is the alekseevski conjecture al1 asserting that a noncompact einstein homogeneous space admits a simply transitive solvable isometry group this is equivalent to saying that any such space is a solvmanifold a solvable lie group with a left invariant riemannian metric satisfying the einstein condition by a deep result of j lauret la5 any einstein solvmanifold is standard this means that the metric solvable lie algebra s of such a solvmanifold has the following property the orthogonal complement to the derived algebra of s is abelian the systematic study of standard einstein solvmanifolds and the term standard originated from the paper of j heber heb on the lie algebra level all the metric einstein solvable lie algebras can be obtained as the result of the following construction heb la1 la5 lw one starts with the three pieces of data a nilpotent lie algebra n a semisimple derivationof n and an inner product h i n on n with respect to which is symmetric an extension of n byis a solvable lie algebra s rh n as a linear space with adh n the inner product on s is defined by h h ni 0 k hk 2 tr and coincides with the existing one on n the resulting metric solvable lie algebra s h i is einstein provided n is nice and the derivationand the inner product h i n are chosen in the correct way note however that these conditions are expressed by a system of algebraic equations which could hardly be analyzed directly see section 2 metric einstein solvable lie algebras of higher rank with the codimension of the nilradical greater than one having the same nilradical n can be obtained from s via a known procedure by further adjoining to n semisimple derivation commuting with it turns out that the structure of an einstein metric solvable lie algebra is completely encoded in its nilradical in the following sense given a nilpotent lie algebra n there is no more than one possibly none choice ofand of h i n up to conjugation by aut n and scaling which may result in an einstein metric solvable lie algebra s h i definition 1 a nilpotent lie algebra is called an einstein nilradical if it is the nilradical of an einstein metric solvable lie algebra a derivationof an einstein nilradical n and an inner product h i n for which the metric solvable lie algebra s h i is einstein are called an einstein derivation and a nilsoliton inner product respectively

collapsing riemannian manifolds to ones of lower dimensions on considere l effondrement d une suite de varietes de riemann mi vers un espace de dimension inferieure x on etudie le cas ou l espace limite x est une variete de riemann et les courbures sectionnelles de mi sont bornees

parabolic subgroups of semisimple lie groups and einstein solvmanifolds in this paper we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple lie algebras these solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type we show that the ricci curvatures of our solvmanifolds coincide with the restrictions of the ricci curvatures of the ambient symmetric spaces consequently all of our solvmanifolds are einstein which provide a large number of new examples of noncompact homogeneous einstein manifolds we also show that our solvmanifolds are minimal but not totally geodesic submanifolds of symmetric spaces

einstein solvmanifolds are standard we study einstein manifolds admitting a transitive solvable lie group of isometries solvmanifolds it is conjectured that these exhaust the class of noncompact homogeneous einstein manifolds j heber has shown that under a simple algebraic condition he calls such a solvmanifold standard einstein solvmanifolds have many remarkable structural and uniqueness properties in this paper we prove that any einstein solvmanifold is standard by applying a stratification procedure adapted from one in geometric invariant theory due to f kirwan

on solvable lie groups of negative ricci curvature we study solvable lie groups which admit a left invariant metric of strictly negative ricci curvature we obtain necessary and sufficient conditions of the existence of such a metric for lie groups the nilradical of whose lie algebra is either abelian or heisenberg or standard filiform and discuss some open questions

solvable lie algebras and graphs abstract we define a solvable extension of the graph 2 step nilpotent lie algebras of 5 by adding elements corresponding to the 3 cliques of the graph we study some of their basic properties and we prove that two such lie algebras are isomorphic if and only if their graphs are isomorphic we also briefly discuss some metric properties providing examples of homogeneous spaces with nonpositive curvature operator and solvsolitons

existence and non existence of homogeneous einstein metrics on demontre un theoreme d existence general pour des metriques d einstein homogenes et on presente des espaces homogenes compacts simplement connexes qui ne portent pas une metrique d einstein homogene

on high order denoising models and fast algorithms for vector valued images variational techniques for gray scale image denoising have been deeply investigated for many years however little research has been done for the vector valued denoising case and the very few existent works are all based on total variation regularization it is known that total variation models for denoising gray scaled images suffer from staircasing effect and there is no reason to suggest this effect is not transported into the vector valued models high order models on the contrary do not present staircasing in this paper we introduce three high order and curvature based denoising models for vector valued images their properties are analyzed and a fast multigrid algorithm for the numerical solution is provided ams subject classifications 68u10 65f10 65k10

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

hermitian curvature flow on unimodular lie groups and static invariant metrics we investigate the hermitian curvature flow hcf of leftinvariant metrics on complex unimodular lie groups we show that in this setting the flow is governed by the ricci flow type equation tgt ric gt the solution gt always exists for all positive times and 1 t gt converges as t in cheeger gromov sense to a nonflat left invariant soliton g g moreover up to homotheties on each of these groups there exists at most one left invariant soliton solution which is a static hermitian metric if and only if the group is semisimple in particular compact quotients of complex semisimple lie groups yield examples of compact non k nahler manifolds with static hermitian metrics we also investigate the existence of static metrics on nilpotent lie groups and we generalize a result of enrietti fino and the third author j symplectic geom 10 2012 no 2 203 223 for the pluriclosed flow in the last part of the paper we study the hcf on lie groups with abelian complex structures

the alekseevskii conjecture in low dimensions the long standing alekseevskii conjecture states that a connected homogeneous einstein space g k of negative scalar curvature must be diffeomorphic to mathbb r n this was known to be true only in dimensions up to 5 and in dimension 6 for non semisimple g in this work we prove that this is also the case in dimensions up to 10 when g is not semisimple for arbitrary g besides 5 possible exceptions we show that the conjecture holds up to dimension 8

ricci flow of homogeneous manifolds we present in this paper a general approach to study the ricci flow on homogeneous manifolds our main tool is a dynamical system defined on a subset mathcal h q n of the variety of q n dimensional lie algebras parameterizing the space of all simply connected homogeneous spaces of dimension n with a q dimensional isotropy which is proved to be equivalent in a precise sense to the ricci flow the approach is useful to better visualize the possible nonflat pointed limits of ricci flow solutions under diverse rescalings as well as to determine the type of the possible singularities ancient solutions arise naturally from the qualitative analysis of the evolution equation we develop two examples in detail a 2 parameter subspace of mathcal h 1 3 reaching most of 3 dimensional geometries and a 2 parameter family in mathcal h 0 n of left invariant metrics on n dimensional compact and non compact semisimple lie groups

boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains boundary behavior and the martin boundary problem for p harmonic functions in lipschitz domains

construction of nice nilpotent lie groups abstract we illustrate an algorithm to classify nice nilpotent lie algebras of dimension n up to a suitable notion of equivalence applying the algorithm we obtain complete listings for n 9 on every nilpotent lie algebra of dimension 7 we determine the number of inequivalent nice bases which can be 0 1 or 2 we show that any nilpotent lie algebra of dimension n has at most countably many inequivalent nice bases

einstein locally conformal calibrated g_2 structures we study locally conformal calibrated g 2 structures whose underlying riemannian metric is einstein showing that in the compact case the scalar curvature cannot be positive as a consequence a compact homogeneous 7 manifold cannot admit an invariant einstein locally conformal calibrated g 2 structure unless the underlying metric is flat in contrast to the compact case we provide a non compact example of homogeneous manifold endowed with a locally conformal calibrated g 2 structure whose associated riemannian metric is einstein and non ricci flat the homogeneous einstein metric is a rank one extension of a ricci soliton on the 3 dimensional complex heisenberg group endowed with a left invariant coupled mathrm su 3 structure omega psi i e such that d omega c mathrm re psi with c in mathbb r 0 nilpotent lie algebras admitting a coupled mathrm su 3 structure are also classified

a characterization of bounded symmetric domains by curvature this paper will prove that a bounded homogeneous domain is symmetric if and only if in the bergman metric all sectional curvatures are nonpositive introduction it is well known that a bounded symmetric domain has nonpositive sectional curvature in the bergman metric this paper is devoted to the converse namely that a bounded homogeneous domain equivalently homogeneous siegel domain which has nonpositive sectional curvature in the bergman metric must be a symmetric domain the proof uses the techniques of normal j algebras 13 as well as results of vinberg 16 17 the first author 1 3 and indirectly dorfmeister 5 6 the body of the paper is in two sections in the first we derive certain relations between the dimensions of the root spaces in an irreducible normal j algebra which in the presence of the additional curvature assumption show the equality of the dimensions of all root spaces corresponding to certain roots for which no multiple is a root and the equality of the dimensions of all root spaces corresponding to roots which are half of another root in the second section we show that the first dimension result implies that the cone of the corresponding siegel domain is self dual while the two dimension results together imply that the domain is quasi symmetric the proof is then finished because 3 a quasi symmetric domain with nonpositive sectional curvature in the bergman metric is known to be symmetric we remark that some of the subsidiary results mentioned above are not essentially new however for consistency of presentation it is necessary to have them in the language of normal jalgebras and for that reason they are included there is also overlap here with work of zelow lundquist 21 1 throughout this paper s j will denote a normal j algebra with admissible form w this means that s is a finite dimensional real split solvable lie algebra with almost complex structure j s s such that x y j jx y j x jy jx jy and w is a linear form on s such that the bilinear form o jx y is symmetric positive definite and j invariant note that the assumption that s is real split was incorrectly omitted in 2 let n s s and let a be the orthogonal complement of n in s by the basic structure theorem of pyatetskii shapiro 13 a is a commutative received by the editors july 13 1981 1980 mathematics subiect classification primary 32mi0 53c30 1983 american mathematical society 0002 9947 82 000 0759 09 00