On the mechanical equilibrium in anisotropic heterogeneous systems

The conditions for equilibrium of liquid surfaces dividing two anisotropic bulk phases have been deduced by means of variational analysis.It has been shown that the presence of an anisotropic tensor in the bulk requires a gradient of surface tension in a state of equilibrium.     

Stability of hyperbolic manifolds with cusps under Ricci flow

We show that every finite volume hyperbolic manifold of dimension greater than or equal to 3 is stable under rescaled Ricci flow, i.e. that every small perturbation of the hyperbolic metric flows back to the hyperbolic metric again. Note that we do not need to make any decay assumptions on this perturbation.     

The first eigenvalue of Finsler p-Laplacian

The eigenvalues and eigenfunctions of p-Laplacian on Finsler manifolds are defined to be critical values and critical points of its canonical energy functional. Based on it, we generalize some eigenvalue comparison theorems of p-Laplacian on Riemannian manifolds, such as Lichnerowicz type estimate, Obata type theorem and Mckean type theorem, to the Finsler setting. Not only that, the Lichnerowicz type estimate we obtained is even better than the corresponding one in Riemannian geometry.     

On Kähler-Norden manifolds

This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of Riemannian curvature tensors and curvature scalars of Kähler-Norden manifolds using the theory of Tachibana operators is presented.     

Kähler–Ricci Flow With Small Initial Energy

In this paper, we prove that the K?hler–Ricci flow converges to a K?hler–Einstein metric when E ₁ energy is small. We also prove that E ₁ is bounded from below if and only if the K-energy is bounded from below in the canonical class. The first named author is partially supported by a NSF grant, while the third author was partially supported by a NSF supplement grant.     

Three- and Four-Dimensional Einstein-like Manifolds and Homogeneity

The aim of this paper is to study three- and four-dimensional Einstein-like Riemannian manifolds which are Ricci-curvature homogeneous, that is, have constant Ricci eigenvalues. In the three-dimensional case, we present the complete classification of these spaces while, in the four-dimensional case, this classification is obtained in the special case where the manifold is locally homogeneous. We also present explicit examples of four-dimensional locally homogeneous Riemannian manifolds whose Ricci tensor is cyclic-parallel (that is, are of type A) and has distinct eigenvalues. These examples are invalidating an expectation stated by F. Podestá and A. Spiro, and illustrating a striking contrast with the three-dimensional case (where this situation cannot occur). Finally, we also investigate the relation between three- and four-dimensional Einstein-like manifolds of type A and D'Atri spaces, that is, Riemannian manifolds whose geodesic symmetries are volume-preserving (up to sign).     

Eigenvalue estimate on a compact Riemann manifold

LetM be a compact Riemann manifold with the Ricci curvature ≽ - R(R = const. > 0) . Denote by d the diameter ofM. Then the first eigenvalue λ₁ ofM satisfies . Moreover if , then      

Strain gradient solution for a finite-domain Eshelby-type anti-plane strain inclusion problem

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing an anti-plane strain inclusion of arbitrary cross-sectional shape prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for anti-plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite anti-plane strain elastic body. The solution reduces to that of the infinite-domain anti-plane strain inclusion problem when the boundary effect is not considered. The problem of a circular cylindrical inclusion embedded concentrically in a finite cylindrical elastic matrix undergoing anti-plane strain deformations is analytically solved by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical linear elasticity-based Eshelby tensor for the circular cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is small and that the boundary effect can dominate when the inclusion volume fraction is high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.     

Polarization tensors of planar domains as functions of the admittivity contrast

(Electric) polarization tensors describe part of the leading order term of asymptotic voltage perturbations caused by low volume fraction inhomogeneities of the electrical properties of a medium. They depend on the geometry of the support of the inhomogeneities and on their admittivity contrast. Corresponding asymptotic formulas are of particular interest in the design of reconstruction algorithms for determining the locations and the material properties of inhomogeneities inside a body from measurements of current flows and associated voltage potentials on the body’s surface. In this work, we consider the two-dimensional case only and provide an analytic representation of the polarization tensor in terms of spectral properties of the double layer integral operator associated with the support of simply connected conductivity inhomogeneities. Furthermore, we establish that an (infinitesimal) simply connected inhomogeneity has the shape of an ellipse, if and only if the polarization tensor is a rational function of the admittivity contrast with at most two poles whose residues satisfy a certain algebraic constraint. We also use the analytic representation to provide a proof of the so-called Hashin–Shtrikman bounds for polarization tensors; a similar approach has been taken previously by Golden and Papanicolaou and Kohn and Milton in the context of anisotropic composite materials.