Diffusion processes on complete riemannian manifolds

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On the Existence and Regularity of Mass-Minimizing Currents with an Elastic Boundary

We study the following variational problem. For a compact manifold S₀ embedded in the Euclidean space we consider deformations of S₀. They are represented by Lipschitz continuous homeomorphisms of S₀ whose images are embedded manifolds. We introduce an energy of a deformation which depends on the first derivative of the curvature of (S₀) and the mass of a mass minimizing current which is bounded by (S₀). In this paper it is shown that an energy minimizing deformation of (S₀) exists. Moreover, in the case that S₀ has codimension 1, (S₀) is an embedded C^3a -submanifold, if is of the class C^2,1.     

On twistor Gauss maps of surfaces in 4-spheres

In this paper we study surfaces in S⁴ and their twistor Gauss maps. Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given. We find many examples of nonisotropic harmonic maps from a surface to P ³.Supported by the National Natural Science Foundation of China and the Science Foundation of Zhejiang Province.     

The boundary of iterates in Euclidean growth models

This paper defines a general Euclidean growth model via a translation invariant, monotone and local transformation on Borel subsets of . The main result gives a geometric condition for the boundary curvature of the iterates to go to 0. Consequences include estimates for the speed of convergence to the asymptotic shape, and a result about survival of Euclidean deterministic forest fires.

     

On the validity of an einstein relation in models of interface dynamics

We consider models of interface dynamics derived from Ising systems with Kac interactions and we prove the validity of the Einstein relation=, where is the proportionality coefficient in the motion by curvature, is the interface mobility, and is the surface tension.     

Curvature from spin

We show that, considering the dislocation defect induced by torsion in spacetime, which behaves like a string with tension, we are lead also to defect angle and then to curvature of spacetime. The space with torsion and curvature is then equivalent to an elastic continuum which has undergone plastic deformations and, following Sakharov idea of the spacetime as a elastic continuum, we are lead to a gravitational constant, which occurs in the Einstein action, as the metrical elasticity of spacetime with the exact value without introducing any arbitrary cutoff, when also torsion is considered.     

Mathematical Models of Triple Junctions

Including triple junctions in rigorous mathematical models had very limited success until about 25 years ago. This paper surveys the mathematical results on triple junctions since then. Anisotropic surface energies, zero triple line energy, positive triple line energy, and negative triple line energy are all considered, in both sharp and diffuse interfaces. In particular, the problems with negative line junction energies in sharp interface theory are presented. Both equilibrium and growth models are reviewed, in 2 and 3 dimensions.     

de Sitter空间中具有常均曲率的类空超曲面

设M是deSitter空间中具有常均曲率的类空超曲面,本文给出M是全脐的或是等参的一些条件．     

Evolution of hypersurfaces in $ {\Bbb R}^N$ by Gaussian curvature

In this paper we first prove short time existence of a classical solution for the problem which describes the evolution by Gaussian curvature of a strictly convex hypersurface in . Then we give a proof of the existence of a viscosity solution for this problem in such a way as to define a generalized motion existing for each time. Received November 24, 1997     

Surfaces with harmonic inverse mean curvature in space forms

We define surfaces with harmonic inverse mean curvature in space forms and generalize a theorem due to Lawson by which surfaces of constant mean curvature in one space form isometrically correspond to those in another. We also obtain an immersion formula, which gives a deformation family for these surfaces.