曲面上的曲率在理论物理中的一些应用

In this survey article,we present two applications of surface curvatures in theoretical physics.The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy.In this formalism,the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics.We first show that there is an obstruction,arising from the spontaneous curvature,to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori.We then propose a scale-invariant anisotropic bending energy,which extends the Canham energy,and show that it possesses a unique toroidal energy minimizer,up to rescaling,in all parameter regime.Furthermore,we establish some genus-dependent topological lower and upper bounds,which are known to be lacking with the Helfrich energy,for the proposed energy.We also present the shape equation in our context,which extends the Helfrich shape equation.The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings.In this formalism,gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector.This setting provides a lucid exhibition of the interplay of the underlying geometry,matter energy,and topological characterization of the system.In both areas of applications,we encounter highly challenging nonlinear partial differential equation problems.We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.

Some(Anti-)Self Duality Solutions on Six Manifolds

The bundle of the 2-forms on 6-manifold decomposes into three subbundles such that Λ~2(R^6) = Λ~2_1⊕Λ~2_6⊕Λ~2_8 with dimensions 1,6 and 8,respectively.The self and anti self duality solutions of the 2-forms,called Φ-duality,are handled and these solutions show that the anti self dual gauge fields live on the subbundles Λ~2_1 and Λ~2_6 while the self ones equations on Λ28.Also the solution on Λ~2_1 presents a flat connection.In addition,the curvatures of the connections on Λ~2_6 and Λ~2_8 have Tr[F^3] = 0,and so the topological invariants determined by the Chern classes,i.e.topological charge,consist only on the second Chern class.In the result of this case,the anti self and self Φ-dual gauge invariant Lagrangians of defined on both subbundles are bounded by the same topological charge.Also,one gives a quantization case to be relating to the instanton number.

Topological bound states in elastic phononic plates induced by disclinations

Topological defects(including disclinations and dislocations)which commonly exist in various materials have shown an amazing ability to produce excellent mechanical and physical properties of matters.In this paper,disclinations are introduced into topological nontrivial elastic phononic plates.The deformation of the lattice yielded by disclinations produces a pentagonal core with the local five-fold symmetry.The topological bound states are well localized around the boundaries of the pentagonal cores with and without hollow regions.The topological bound states immunize against the finite sizes and the moderate imperfects of plates,essentially differing from the trivial defect states.The discovery of topological bound states unveils a new horizon in topological mechanics and physics,and it provides a novel platfonn to implement large-scale elastic devices with topologically protected resonances.