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Chinese Quarterly Journal of Mathematics 2023年第3期38卷 221-253页

作者： YANG Yi-song Courant Institute of Mathematical Sciences New York UniversityNY 10012USA

In this survey article,we present two applications of surface curvatures in theoretical *** 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 *** this formalism,the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological *** 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 *** 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 ***,we establish some genus-dependent topological lower and upper bounds,which are known to be lacking with the Helfrich energy,for the proposed *** also present the shape equation in our context,which extends the Helfrich shape *** second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic *** 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 *** setting provides a lucid exhibition of the interplay of the underlying geometry,matter energy,and topological characterization of the *** both areas of applications,we encounter highly challenging nonlinear partial differential equation *** demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.

关键词： Mean curvature Gauss curvature Bending energy Cell vesicles Topological bounds Shape equations Einstein tensor cosmic strings Harmonic map model Nirenberg’s problem Conical singularities Deficit angle Conformal metric

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