D-Fuzzy Metrics

1.INTRODUCTION L. Zadeh introduced in [1] fuzzy sets, which were extended to L-fuzzy sets by Goguen in[2]. L-fuzzy sets unify fuzzy and usual sets but concern no points. [4]restricted L to D-lattice and so L-fuzzy sets become D-fuzzy ones,which treat of points. D-lattice is not so extended as L but enough to include     

Strongly Extremal Kähler Metrics

On a compact complex manifold (M, J) of the Kähler type, we consider the functional defined by the L²-norm of the scalar curvature with its domain the space of Kähler metrics of fixed total volume. We calculate its critical points, and derive a formula that relates the Kähler and Ricci forms of such metrics on surfaces. If these metrics have a nonzero constant scalar curvature, then they must be Einstein. For surfaces, if the scalar curvature is nonconstant, these critical metrics are conformally equivalent to non-Kähler Einstein metrics on an open dense subset of the manifold. We also calculate the Hessian of the lower bound of the functional at a critical extremal class, and show that, in low dimensions, these classes are weakly stable minima for the said bound. We use this result to discuss some applications concerning the two-points blow-up of CP².     

Fuzzy Metrics Characterized by Molecules and Sets

Erceg in [1] extended the Hausdorff distance function betwen subsets of a set X to fuzzy settheory, and introduced a fuzzy pseudo-quasi-metric(p. q. metric) p: L~x xL~x -[0,∞] where L is acompletely distributive lattice, and the associated family of neighborhood mappings {Dr |r>O}. Thefuzzy metric space in Erceg's sense is denoted by (L~x, p, Dr) since there exists a one to one correspondence between fuzzy p. q. metrics and associatcd families of neighborhood mapping, a family{Dr|r>0}satisfying certain conditions is also callcd a (standard) fuzzy p. q metric. In [2] Liang defimed apointwise fuzzy p. q. metric d: P(L~x)×P(L~x)-[0, ∞) and applicd it to the construction of theproduct fuzzy metric. In this paper, we give axiomatic definitions of molcculewise and fuzzy     

Fréchet Embeddings of Negative Type Metrics

We show that every n-point metric of negative type (in particular, every n-point subset of L ₁) admits a Fréchet embedding into Euclidean space with distortion , a result which is tight up to the O(log log n) factor, even for Euclidean metrics. This strengthens our recent work on the Euclidean distortion of metrics of negative into Euclidean space. S. Arora supported by David and Lucile Packard Fellowship and NSF grant CCR-0205594. J.R. Lee supported by NSF grant CCR-0121555, NSF 0514993, NSF 0528414 and an NSF Graduate Research Fellowship.     

Connected Sum Construction for σ k -Yamabe Metrics

In this paper we produce families of Riemannian metrics with positive constant σ _k -curvature equal to $2^{-k} {n \choose k}$ by performing the connected sum of two given compact nondegenerate n-dimensional solutions (M ₁,g ₁) and (M ₂,g ₂) of the (positive) σ _k -Yamabe problem, provided 2≤2k<n. The problem is equivalent to solving a second-order fully nonlinear elliptic equation.     

Cohomogeneity-Three HyperKähler Metrics on Nilpotent Orbits

Let \({\cal O}\) be a nilpotent orbit in ?^? where G is a compact, simple group and ? = Lie(G). It is known that \({\cal O}\) carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov–Kostant–Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when \({\cal O}\) is of cohomogeneity three under the action of G. It is found that such a structure lies on a one-parameter family of hyperKähler metrics with G-invariant Kähler potentials if and only if ? is Sp3, su6, So7, So12 or e7 and otherwise is the unique G-invariant hyperKähler metric with G-invariant Kähler potential.     

Scalar Curvature Functions of Almost-Kähler Metrics

For a closed smooth manifold M admitting a symplectic structure, we define a smooth topological invariant Z(M) using almost-Kähler metrics, i.e., Riemannian metrics compatible with symplectic structures. We also introduce \(Z(M, [[\omega ]])\) depending on symplectic deformation equivalence class \([[\omega ]]\). We first prove that there exists a 6-dimensional smooth manifold M with more than one deformation equivalence class with different signs of \(Z(M, [[\omega ]] )\). Using Z invariants, we set up a Kazdan–Warner type problem of classifying symplectic manifolds into three categories. We finally prove that on every closed symplectic manifold \((M, \omega )\) of dimension \(\ge \!\!4\), any smooth function which is somewhere negative and somewhere zero can be the scalar curvature of an almost-Kähler metric compatible with a symplectic form which is deformation equivalent to \(\omega \).     

Locally Conformal K¨ahler and Hermitian Yang-Mills Metrics

The author shows that if a locally conformal K¨ahler metric is Hermitian YangMills with respect to itself with Einstein constant c ≤ 0, then it is a K¨ahler-Einstein metric.In the case of c > 0, some identities on torsions and an inequality on the second Chern number are derived.     

Minimal Complex Surfaces with Levi–Civita Ricci-flat Metrics

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli–Chern class on compact complex manifolds, and proved that the(1, 1) curvature form of the Levi–Civita connection represents the first Aeppli–Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi–Civita Ricci-flat metrics and classify minimal complex surfaces with Levi–Civita Ricci-flat metrics.More precisely, we show that minimal complex surfaces admitting Levi–Civita Ricci-flat metrics are K¨ahler Calabi–Yau surfaces and Hopf surfaces.     

Nonnegatively Curved Metrics on S 2 × ℝ2

We classify the complete metrics of nonnegative sectional curvature on M ² × ², where M ² is any compact 2-manifold.     

Curve Flows and Solitonic Hierarchies Generated by Einstein Metrics

We investigate bi-Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non-stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric-induced nonlinear connection (N-connection) structure. On spacetime manifolds, we consider a nonholonomic splitting of dimensions and define a new class of liner connections which are ‘N-adapted’, metric compatible and uniquely defined by the metric structure. We prove that for such a linear connection, one yields couples of generalized sine-Gordon equations when the corresponding geometric curve flows result in solitonic hierarchies described in explicit form by nonholonomic wave map equations and mKdV analogs of the Schrödinger map equation. All geometric constructions can be re-defined for the Levi-Civita connection but with “noholonomic mixing” of solitonic interactions. Finally, we speculate why certain methods and results from the geometry of nonholonmic manifolds and solitonic equations have general importance in various directions of modern mathematics, geometric mechanics, fundamental theories in physics and applications, and briefly analyze possible nonlinear wave configurations for modeling gravitational interactions by effective continuous media effects.     

Extremal Almost-Kähler Metrics and Seiberg–Witten Theory

We show that there exist compact non-Kähler almost-Kähler4-manifolds whose metrics minimize L ²-norm of(2/3) s + 2w among all metrics compatible with a fixeddecomposition H ²(M, = H ⁺ H ^–, where s is the scalar curvature and w is the lowest eigenvalue of self-dual Weyl curvature at each point. In particular, the moduli space of such metrics modulo diffeomorphisms is infinite dimensional. This example also shows that LeBrun's estimate of L ²-norm of (1 – )s + · 6won a compact oriented Riemannian4-manifold with a nontrivial Seiberg–Witten invariant cannot beextended over = 1/3.     

K¨ahler Finsler Metrics Are Actually Strongly K¨ahler

In this paper, the Kahler conditions of the Chern-Finsler connection in complex Finsler geometry are studied, and it is proved that Kahler Finsler metrics are actually strongly Kahler.     

An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics

This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher–Rao metric and is averaged with the transportation term. This gives rise to a convex variational problem defining the new metric. Our first contribution is a proof of the existence of geodesics (i.e., solutions to this variational problem). We then show that (generalized) optimal transport and Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. Lastly, we propose a numerical scheme making use of first-order proximal splitting methods and we show an application of this new distance to image interpolation.     

On the Curvature of Conic Kähler–Einstein Metrics

We prove a regularity result for Monge–Ampère equations degenerate along smooth divisor on Kähler manifolds in Donaldson’s spaces of \(\beta \)-weighted functions. We apply this result to study the curvature of Kähler metrics with conical singularities and give a geometric sufficient condition on the divisor for its boundedness.     

Rigidity Theorems for Einstein–Thorpe Metrics Dedicated to my parents

We will prove that every Einstein–Thorpe metric on T ⁸ must be flat and that on compact oriented hyperbolic manifolds of dimension 8, every Einstein–Thorpe metric is a hyperbolic metric up to rescalings and diffeomorphisms.     

Invariant Kahler Metrics and Curvatures on a Class of Pseudoconvex Domains

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Approximation Characteristics of the Spaces S p ϕ in Different Metrics

We continue the investigation of the approximation characteristics of the spaces introduced earlier. In particular, we establish direct and inverse theorems on the approximation of elements of these spaces. We also determine the exact values of upper bounds of m-term approximations of q-ellipsoids in the spaces in the metrics of the spaces .