energy functionals and canonical kahler metrics yau y2 has conjectured that a fano manifold admits a kahler einstein metric if and only if it is stable in the sense of geometric invariant theory there has been much progress on this conjecture by tian t1 t2 donaldson do1 do2 and others the mabuchi energy functional plays a central role in these ideas we study the ek functionals introduced by x x chen and g tian ct1 which generalize the mabuchi energy we show that if a fano manifold admits a kahler einstein metric then the functional e1 is bounded from below and modulo holomorphic vector fields is proper this answers affirmatively a question raised by chen c2 in fact we show that e1 is proper if and only if there exists a kahler einstein metric giving a new analytic criterion for the existence of this canonical metric with possible implications for the study of stability we also show that on a fano kahler einstein manifold all of the functionals ek are bounded below on the space of metrics with nonnegative ricci curvature

on energy functionals kahler einstein metrics and the moser trudinger onofri neighborhood abstract we prove that the existence of a kahler einstein metric on a fano manifold is equivalent to the properness of the energy functionals defined by bando chen ding mabuchi and tian on the set of kahler metrics with positive ricci curvature we also prove that these energy functionals are bounded from below on this set if and only if one of them is this answers two questions raised by x x chen as an application we obtain a new proof of the classical moser trudinger onofri inequality on the two sphere as well as describe a canonical enlargement of the space of kahler potentials on which this inequality holds on higher dimensional fano kahler einstein manifolds

the moser trudinger inequality on kahler einstein manifolds we prove the conjecture of tian on the strong form of the moser trudinger inequality for k ahler einstein manifolds with positive first chern class when there are no holomorphic vector fields and more generally when the setting is invariant under a maximal compact subgroup of the automorphism group

long time existence of minimizing movement solutions of calabi flow abstract we recast the calabi flow in degiorgi s language of minimizing movements we establish the long time existence of minimizing movements for k energy with arbitrary initial condition furthermore we establish some a priori regularity results for these solutions and that sufficiently regular minimizing movements are smooth solutions to calabi flow

the space of kahler metrics this paper the second of a series deals with the function space mathcal h of all smooth kahler metrics in any given n dimensional closed complex manifold v these metrics being restricted to a given fixed real cohomology class called a polarization of v this function space is equipped with a pre hilbert metric structure introduced by t mabuchi 10 who also showed that formally this metric has nonpositive curvature in the first paper of this series 4 the second author showed that the same space is a path length space he also proved that mathcal h is geodesically convex in the sense that for any two points of mathcal h there is a unique geodesic path joining them which is always length minimizing and of class c1 1 this partially verifies two conjectures of donaldson 8 on the subject in the present paper we show first of all that the space is as expected a path length space of nonpositive curvature in the sense of a d aleksandrov a second result is related to the theory of extremal kahler metrics namely that the gradient flow in mathcal h of the k energy of v has the property that it strictly decreases the length of all paths in mathcal h except those induced by one parameter families of holomorphic automorphisms of m

ideal trefoil knot a set of self contact points of the most tight parametrically tied trefoil knot is determined the knot is subjected to further tightening procedure based on the shrink on no overlaps algorithm changes in the structure of the set of the self contact points are monitored and the final form of the set is determined

optimal lower bounds for donaldson s j functional abstract in this paper we provide an explicit formula for the optimal lower bound of donaldson s j functional in the sense of finding explicitly the optimal constant in the definition of coercivity which always exists and takes negative values in general this constant is positive precisely if the j equation admits a solution and the explicit formula has a number of applications first this leads to new existence criteria for constant scalar curvature kahler csck metrics in terms of tian s alpha invariant moreover we use the above formula to discuss calabi dream manifolds and an analogous notion for the j equation and show that for surfaces the optimal bound is an explicitly computable rational function which typically tends to minus infinity as the underlying class approaches the boundary of the kahler cone even when the underlying kahler classes admit csck metrics as a final application we show that if the lejmi szekelyhidi conjecture holds then the optimal bound coincides with its algebraic counterpart the set of j semistable classes equals the closure of the set of uniformly j stable classes in the kahler cone and there exists an optimal degeneration for uniform j stability

the gursky streets equations gursky streets introduced a formal riemannian metric on the space of conformal metrics in a fixed conformal class of a compact riemannian four manifold in the context of the sigma 2 yamabe problem the geodesic equation of gursky streets metric is a fully nonlinear degenerate elliptic equation using this geometric structure and the geodesic equation gursky streets proved an important result in conformal geometry that the solution of the sigma 2 yamabe problem is unique the existence of such a solution was known more than a decade ago a key ingredient is the convexity of chang yang s mathcal f functional along the smooth geodesic however gursky streets have only proved uniform c 0 1 regularity for a perturbed equation and it turns out that the uniform c 1 1 regularity is very delicate without such a uniform c 1 1 regularity gursky streets arguments of the uniqueness theorem are very complicated in this paper we establish the uniform c 1 1 regularity of the gursky streets equation in the course of deriving regularity we also obtain very interesting and new convexity regarding matrices in gamma 2 as an application we can establish strictly the convexity of mathcal f functional along c 1 1 geodesic this gives a straightforward proof of the uniqueness of solutions of sigma 2 yamabe problem

a brunn minkowski type inequality for fano manifolds and some uniqueness theorems in kahler geometry for phi a metric on the anticanonical bundle k x of a fano manifold x we consider the volume of x begin aligned int x e phi end aligned in earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on k x our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on x even with very low regularity assumptions on the geodesic as a consequence we get a simplified proof of the bando mabuchi uniqueness theorem for kahler einstein metrics a generalization of this theorem to twisted kahler einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being fano is also given we moreover discuss a generalization of the main result to other bundles than k x and finally use the same method to give a new proof of the theorem of tian and zhu on uniqueness of kahler ricci solitons

remarks on the existence problem of positive kahler einstein metrics on introduit un invariant m qui donne un critere pour l existence de metriques de kahler einstein

the best constant of the moser trudinger inequality on s2 we consider the best constant of the moser trudinger inequality on s 2 under a certain orthogonality condition applying moser s calculation we construct a counterexample to the sharper inequality with the condition

role of flexibility in entanglement entanglement is essential to the function of many physical systems flexibility and length determine the extent to which the system can become entangled given a perfectly flexible unit radius tube several researchers have studied the minimum length needed to tie different types of knots can one obtain the same configurations with less flexible tubing does more flexibility always yield tighter knots we demonstrate a phase change in flexibility beyond which more flexibility adds very little entanglement this level of flexibility is surprisingly low and appears to have a global bound since tensile strength and flexibility act inversely this level of flexibility provides the maximal tensile strength for materials that need to pack tightly this is a basic design principle that should be observable in nature

a family of exact travelling wave solutions to nonlinear evolution and wave equations a family q of the multicomponent special functions is defined for obtaining the exact travelling wave solutions to nonlinear evolution and wave equations it is shown that the functions qn from q for some n 2 4 are closely related to the special unitary groups su n the necessary and sufficient conditions for existence of a family of the exact multicomponent travelling wave solutions to a quasilinear evolution equation are given an efficiency of the method based on q functions is demonstrated on several classes of the nonlinear partial differential equations

high resolution portrait of the ideal trefoil knot the shape of the tightest polygonal trefoil knot with n 200 640 vertices constructed by the authors using an appropriately modified finite element method is carefully analyzed the large number of vertices provide a spatial structure with very precise plots of its curvature and torsion based on the increased accuracy of the data the authors are able to formulate new conjectures concerning the key features of the shape of the ideal trefoil knot

monge ampere equations in big cohomology classes we define non pluripolar products of an arbitrary number of closed positive 1 1 currents on a compact kahler manifold x given a big 1 1 cohomology class on x i e a class that can be represented by a strictly positive current and a positive measure on x of total mass equal to the volume of and putting no mass on pluripolar sets we show that can be written in a unique way as the top degree self intersection in the non pluripolar sense of a closed positive current in we then extend kolodziedj s approach to sup norm estimates to show that the solution has minimal singularities in the sense of demailly if has l 1 e density with respect to lebesgue measure if is smooth and positive everywhere we prove that t is smooth on the ample locus of provided is nef using a fixed point theorem we finally explain how to construct singular kahler einstein volume forms with minimal singularities on varieties of general type

the extended tanh method for abundant solitary wave solutions of nonlinear wave equations the extended tanh method is used to establish abundant solitary wave solutions of nonlinear wave equations the obtained solutions include solitons kinks and plane periodic solutions the study is an extension to the remarkable development by malfliet 1 the extended tanh method presents a wider applicability for handling nonlinear wave equations

on the convergence and singularities of the j flow with applications to the mabuchi energy the j flow of s k donaldson and x x chen is a parabolic flow on kahler manifolds with two kahler metrics it is the gradient flow of the j functional that appears in chen s formula for the mabuchi energy we find a positivity condition in terms of the two metrics that is both necessary and sufficient for the convergence of the j flow to a critical metric we use this result to show that on manifolds with ample canonical bundle the mabuchi energy is proper on all kahler classes in an open neighborhood of the canonical class defined by a positivity condition this improves previous results of chen and of the second author we discuss the implications of this for the problem of the existence of constant scalar curvature kahler metrics we also study the singularities of the j flow and under certain conditions which always hold for dimension 2 derive some estimates away from a subvariety we discuss the conjectural remark of donaldson that if the j flow does not converge on a kahler surface then it should blow up over some curves of negative self intersection 2007 wiley periodicals inc

helical tubes in crowded environments when placed in a crowded environment a semiflexible tube is forced to fold so as to make a more compact shape one compact shape that often arises in nature is the tight helix especially when the tube thickness is of comparable size to the tube length in this paper we use an excluded volume effect to model the effects of crowding this gives us a measure of compactness for configurations of the tube which we use to look at structures of the semiflexible tube that minimize the excluded volume we focus most of our attention on the helix and which helical geometries are most compact we found helices of specific pitch to radius ratio 2 512 to be optimally compact this is the same geometry that minimizes the global curvature of the curve defining the tube we further investigate the effects of adding a bending energy or multiple tubes to begin to explore the more complete space of possible geometries a tube could form

growth of balls of holomorphic sections and energy at equilibrium let l be a big line bundle on a compact complex manifold x given a non pluripolar compact subset k of x and a continuous hermitian metric e on l we define the energy at equilibrium of k as the monge ampere energy of the extremal psh weight associated to k we prove the differentiability of the energy at equilibrium with respect to and we show that this energy describes the asymptotic behaviour as k of the volume of the sup norm unit ball induced by k k on the space of global holomorphic sections h 0 x kl as a consequence of these results we recover and extend rumely s robin type formula for the transfinite diameter we also obtain an asymptotic description of the analytic torsion and extend yuan s equidistribution theorem for algebraic points of small height to the case of a big line bundle

k energy on polarized compactifications of lie groups abstract in this paper we study mabuchi s k energy on a compactification m of a reductive lie group g which is a complexification of its maximal compact subgroup k we give a criterion for the properness of k energy on the space of k k invariant kahler potentials in particular it turns to give an alternative proof of delcroix s theorem for the existence of kahler einstein metrics in case of fano manifolds m we also study the existence of minimizers of k energy for general kahler classes of m

a thermodynamical formalism for monge ampere equations moser trudinger inequalities and kahler einstein metrics we develop a variational calculus for a certain free energy functional on the space of all probability measures on a kahler manifold x this functional can be seen as a generalization of mabuchi s k energy functional and its twisted versions to more singular situations applications to monge ampere equations of mean field type twisted kahler einstein metrics and moser trudinger type inequalities on miller manifolds are given tian s alpha invariant is generalized to singular measures allowing in particular a proof of the existence of kahler einstein metrics with positive ricci curvature that are singular along a given anti canonical divisor which combined with very recent developments concerning miller metrics with conical singularities confirms a recent conjecture of donaldson as another application we show that if the calabi flow in the anti canonical class exists for all times then it converges to a kahler einstein metric when a unique one exists which is in line with a well known conjecture c 2013 elsevier inc all rights reserved

ideal geometry of periodic entanglements three dimensional entanglements including knots knotted graphs periodic arrays of woven filaments and interpenetrating nets form an integral part of structure analysis because they influence va

nonlinear variable separation solutions of some mkdv type equations and the unified construction method from the viewpoint of inverse problem research by using the hirota s bilinear transformation method and direct variable separation ansatz some mkdv type equations with nonlinear variable separation solutions are unified constructed these exact special solutions include several lower dimensional arbitrary functions which can be used to express various soliton structures

link lengths and their growth powers for a certain infinite family of knots or links we study the growth power ratios of their stick number lattice stick number minimum lattice length and minimum ropelength compared with their minimum crossing number c k for every it is known that the stick number and lattice stick number grow between the and linear power of the crossing number and minimum lattice length and minimum ropelength grow with at least the power of crossing number which is called the four thirds power law furthermore the minimal lattice length and minimum ropelength grow at most as o but it is unknown whether any family exhibits superlinear growth for any real number r between and 1 we give an infinite family of non splittable prime links in which the stick number and lattice stick number grow exactly as the rth power of crossing number furthermore for any real number r between and 1 we give another infinite family of non splittable prime links in which the minimum lattice length and minimum ropelength grow exactly as the rth power of crossing number

kahler einstein metrics with edge singularities this article considers the existence and regularity of kahler einstein metrics on a compact kahler manifold m with edge singularities with cone angle 2 along a smooth divisor d we prove existence of such metrics with negative zero and some positive cases for all cone angles 2 2 the results in the positive case parallel those in the smooth case we also establish that solutions of this problem are polyhomogeneous i e have a complete asymptotic expansion with smooth coecients along d for all 2 2

the tanh coth method for solitons and kink solutions for nonlinear parabolic equations abstract the tanh coth method is used to derive solitons and kink solutions for some of the well known nonlinear parabolic partial differential equations the equations include the fisher equation newell whithead equation allen cahn equation fitzhugh nagumo equation fisher s equation and the burgers fisher equation the new tanh coth approach provides abundant solitons and kink solutions in addition to the existing ones the power of this manageable method is confirmed

ricci flow and the metric completion of the space of kahler metrics we consider the space of kahler metrics as a riemannian submanifold of the space of riemannian metrics and study the associated submanifold geometry in particular we show that the intrinsic and extrinsic distance functions are equivalent we also determine the metric completion of the space of kahler metrics making contact with recent generalizations of the calabi yau theorem due to dinew and guedj zeriahi as an application we obtain a new analytic stability criterion for the existence of a kahler einstein metric on a fano manifold in terms of the ricci flow and the distance function we also prove that the kahler ricci flow converges as soon as it converges in the metric sense

some discretizations of geometric evolution equations and the ricci iteration on the space of kahler metrics we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite dimensional spaces we illustrate the fruitfulness of this approach in the context of the ricci flow as well as another flow in kahler geometry we introduce and study dynamical systems related to the ricci operator on the space of kahler metrics that arise as discretizations of these flows we pose some problems regarding their dynamics we point out a number of applications to well studied objects in kahler and conformal geometry such as constant scalar curvature metrics kahler ricci solitons nadel type multiplier ideal sheaves balanced metrics the moser trudinger onofri inequality energy functionals and the geometry and structure of the space of kahler metrics e g we obtain a new sharp inequality strengthening the classical moser trudinger onofri inequality on the two sphere

the weighted monge ampere energy of quasiplurisubharmonic functions abstract we study degenerate complex monge ampere equations on a compact kahler manifold x we show that the complex monge ampere operator d d c n is well defined on the class e x of plurisubharmonic functions with finite weighted monge ampere energy the class e x is the largest class of psh functions on which the monge ampere operator is well defined and the comparison principle is valid it contains several functions whose gradient is not square integrable we give a complete description of the range of the operator d d c n on e x as well as on some of its subclasses we also study uniqueness properties extending calabi s result to this unbounded and degenerate situation and we give applications to complex dynamics and to the existence of singular kahler einstein metrics

unstable polarized del pezzo surfaces we provide new examples of k unstable polarized smooth del pezzo surfaces using a flopped version first used by cheltsov and rubinstein of the test configurations introduced by ross and thomas as an application we provide new obstructions for the existence of constant scalar curvature kahler metrics on polarized smooth del pezzo surfaces

ricci iteration on homogeneous spaces the ricci iteration is a discrete analogue of the ricci flow we give the first study of the ricci iteration on a class of riemannian manifolds that are not kahler the ricci iteration in the non kahler setting exhibits new phenomena among them is the existence of so called ancient ricci iterations as we show these are closely related to ancient ricci flows and provide the first nontrivial examples of riemannian metrics to which the ricci operator can be applied infinitely many times in some of the cases we study these ancient ricci iterations emerge in the gromov hausdorff topology from a collapsed einstein metric and converge smoothly to a second einstein metric in the case of compact homogeneous spaces with maximal isotropy we prove a relative compactness result that excludes collapsing our work can also be viewed as proposing a dynamical criterion for detecting whether an ancient ricci flow exists on a given riemannian manifold as well as a method for predicting its limit

time space integrable decompositions of nonlinear evolution equations separation of the time and space variables of evolution equations is analyzed without using any structure associated with evolution equations the resulting theory provides techniques for constructing time space integrable decompositions of evolution equations which transform an evolution equation into two compatible liouville integrable ordinary differential equations in the time and space variables the techniques are applied to the kdv mkdv and diffusion equations thereby yielding several new time space integrable decompositions of these equations

kahler geometry of horosymmetric varieties and application to mabuchi s k energy functional we introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties we develop kahler geometry on these varieties with applications to canonical metrics in mind as a generalization of the guillemin abreu donaldson geometry of toric varieties namely we associate convex functions with hermitian metrics on line bundles and express the curvature form in terms of this function as well as the corresponding monge ampere volume form and scalar curvature we then provide an expression for the mabuchi functional and derive as an application a combinatorial sufficient condition of properness similar obtained by li zhou and zhu on group compactifications

the mabuchi geometry of finite energy classes abstract we introduce different finsler metrics on the space of smooth kahler potentials that will induce a natural geometry on various finite energy classes e x motivated by questions raised by r berman v guedj and y rubinstein we characterize the underlying topology of these spaces in terms of convergence in energy and give applications of our results to existence of kahler einstein metrics on fano manifolds

relative k stability and modified k energy on toric manifolds in this paper we give a sufficient condition for both the relative k stability and the properness of modified k energy associated to calabi s extremal metrics on toric manifolds in addition several examples of toric manifolds which satisfy the sufficient condition are presented

on the existence of conic kahler einstein metrics abstract in this paper we prove the conic version of ytd conjecture on log fano manifolds

travelling wave solutions of generalized forms of burgers burgers kdv and burgers huxley equations in this work exact travelling wave solutions of generalized forms of burgers burgers kdv and burgers huxley equations are obtained the analysis rests mainly on the standard tanh method the work emphasizes the need for a transformation formula for the case where the parameter m is non integer the approach can be used in a variety of many types of nonlinearity

the existence of kahler einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group on utilise la methode de continuite pour demontrer l existence de metriques de kahler einstein pour des varietes de kahler de fibre linge positif anticanonique sous l hypothese additionnelle de l existence d un bon groupe de symetries fini ou compact

frobenius integrable decompositions for ninth order partial differential equations of specific polynomial type frobenius integrable decompositions are presented for a kind of ninth order partial differential equations of specific polynomial type two classes of such partial differential equations possessing frobenius integrable decompositions are connected with two rational backlund transformations of dependent variables the presented partial differential equations are of constant coefficients and the corresponding frobenius integrable ordinary differential equations possess higher order nonlinearity the proposed method can be also easily extended to the study of partial differential equations with variable coefficients

frobenius integrable decompositions of nonlinear evolution equations with modified term frobenius integrable decompositions are introduced for nonlinear evolution equations with modified term two classes of polynomial equations with modified term are transformed into frobenius integrable ordinary differential equations by two special backlund transformations the resulting solutions are illustrated to describe the solution phenomena shared with the mkdv equation sharma tasso olver equation the modified kawachara equation the fifth order kdv equation and its various forms

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

asymptotically log fano varieties abstract motivated by the study of fano type varieties we define a new class of log pairs that we call asymptotically log fano varieties and strongly asymptotically log fano varieties we study their properties in dimension two under an additional assumption of log smoothness and give a complete classification of two dimensional strongly asymptotically log smooth log fano varieties based on this classification we formulate an asymptotic logarithmic version of calabi s conjecture for del pezzo surfaces for the existence of kahler einstein edge metrics in this regime we make some initial progress towards its proof by demonstrating some existence and non existence results among them a generalization of matsushima s result on the reductivity of the automorphism group of the pair and results on log canonical thresholds of pairs one by product of this study is a new conjectural picture for the small angle regime and limit which reveals a rich structure in the asymptotic regime of which a folklore conjecture concerning the case of a fano manifold with an anticanonical divisor is a special case

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

k stability and kahler einstein metrics we prove that if a fano manifold m is k stable then it admits a kahler einstein metric it affirms a longstanding conjecture for fano manifolds 2015 wiley periodicals inc

conical kahler einstein metrics revisited in this paper we introduce the interpolation degeneration strategy to study kahler einstein metrics on a smooth fano manifold with cone singularities along a smooth divisor that is proportional to the anti canonical divisor by interpolation we show the angles in 0 2 that admit a conical kahler einstein metric form a connected interval and by degeneration we determine the boundary of the interval in some important cases as a first application we show that there exists a kahler einstein metric on mathbb p 2 with cone singularity along a smooth conic degree 2 curve if and only if the angle is in 2 2 when the angle is 2 3 this proves the existence of a sasaki einstein metric on the link of a three dimensional a2 singularity and thus answers a question posed by gauntlett martelli sparks yau as a second application we prove a version of donaldson s conjecture about conical kahler einstein metrics in the toric case using song wang s recent existence result of toric invariant conical kahler einstein metrics

solitary wave solutions of the generalized shallow water wave gsww equation by hirota s method tanh coth method and exp function method abstract the generalized short wave equation gsww is studied by three distinct methods the hirota s bilinear method is used to derive multiple soliton solutions for the completely integrable forms of the gsww equation the tanh coth method is used to obtain single soliton solutions for this equation the exp function method is also applied to derive a variety of travelling wave solutions with distinct physical structures for this non linear evolution equation

a construction of compact and noncompact solutions for nonlinear dispersive equations of even order a construction of compact and noncompact solutions for nonlinear dispersive partial differential equations of even order is considered two model equations of even orders k 4 are adopted to carry out this study the analysis reveals the presence of distinct roots of different indices that play a significant role in the solutions

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations