Rigidity and mean curvature flow via harmonic Gauss maps

We investigate properties of harmonic Gauss maps and their applications to Lawson-Osserman’s problem, to the rigidity of space-like submanifolds in a pseudo-Euclidean space and to the mean curvature flow.     

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.     

Optimal transport maps for Monge-Kantorovich problem on loop groups

Let G be a compact Lie group, and consider the loop group LeG:={?∈C([0,1],G); ?(0)=?(1)=e}. Let ν be the heat kernel measure at the time 1. For any density function F on LeG such that Entν(F)<∞, we shall prove that there exists a unique optimal transportation map which pushes ν forward to Fν.     

Pseudo-linear superposition principle for the Monge-Ampère equation based on generated pseudo-operations

Through this paper, we consider generated pseudo-operations of the following form: x⊕y=g⁻¹(g(x)+g(y)), x⊙y=g⁻¹(g(x)g(y)), where g is a continuous generating function. Pseudo-linear superposition principle, i.e., the superposition principle with this type of pseudo-operations in the core, for the Monge-Ampère equation is investigated.     

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

Monge's transport problem on a Riemannian manifold

Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures , find the measure-preserving map between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold -- and assuming absolute continuity of -- an optimal map will be shown to exist. Aspects of its uniqueness are also established.

     

Solution of the Monge-Ampère equation on Wiener space for general log-concave measures

In this work we prove that the unique 1-convex solution of the Monge-Kantorovitch measure transportation problem between the Wiener measure and a target measure which has an H-log-concave density, in the sense of Feyel and Üstünel [J. Funct. Anal. 176 (2000) 400-428], w.r.t the Wiener measure is also the strong solution of the Monge-Ampère equation in the frame of infinite-dimensional Fréchet spaces. We further enhance the polar factorization results of the mappings which transform a spread measure to another one in terms of the measure transportation of Monge-Kantorovitch and clarify the relation between this concept and the Itô-solutions of the Monge-Ampère equation.     

The first initial-boundary value problem for fully nonlinear parabolic equations generated by functions of the eigenvalues of the Hessian

For a class of elliptic Hessian operators raised by Caffarelli-Nirenberg-Spruck, the corresponding parabolic Monge-Ampère equation was studied, the existence and uniqueness of the admissible solution to the first initial-boundary value problem for the equation were established, which extended a result of Ivochkina-Ladyzhenskaya.     

Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs

Given two densities on with the same total mass, the Monge transport problem is to find a Borel map rearranging the first distribution of mass onto the second, while minimizing the average distance transported. Here distance is measured by a norm with a uniformly smooth and convex unit ball. This paper gives a complete proof of the existence of optimal maps under the technical hypothesis that the distributions of mass be compactly supported. The maps are not generally unique. The approach developed here is new, and based on a geometrical change-of-variables technique offering considerably more flexibility than existing approaches.

     

Classification of Isometric Immersions of the Hyperbolic Space H 2 into H 3

We transform the problem of determining isometric immersions from H ²(-1) into H ³(c) (c of solving an elliptic Monge--Ampère equation on the unit disc. Then we classify isometric immersions which possess bounded principal curvatures.     

Sobolev estimates for optimal transport maps on Gaussian spaces

In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and target measures, with the Sobolev norm of the optimal transport map will be given. As applications, we will construct strong solutions to Monge–Ampère equations in finite dimension, as well as on the Wiener space, when the target measure satisfies the strong log-concavity condition. A result on the regularity on the optimal transport map on the Wiener space will be obtained.     

An efficient approach for the numerical solution of the Monge-Ampère equation

In this paper, we present a new method to compute the numerical solution of the elliptic Monge-Ampère equation. This method is based on solving a parabolic Monge-Ampère equation for the steady state solution. We study the problem of global existence, uniqueness, and convergence of the solution of the fully nonlinear parabolic PDE to the unique solution of the elliptic Monge-Ampère equation. Some numerical experiments are presented to show the convergence and the regularity of the numerical solution.     

Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space

Let (W,,H) be an abstract Wiener space assume two _i ,i=1,2 probabilities on (W,(W)). We give some conditions for the Wasserstein distance between ₁ and ₂ with respect to the Cameron-Martin space to be finite, where the infimum is taken on the set of probability measures on W×W whose first and second marginals are ₁ and ₂. In this case we prove the existence of a unique (cyclically monotone) map T=I _W +, with :WH, such that T maps ₁ to ₂. Moreover, if ₂, then T is stochastically invertible, i.e., there exists S:WW such that ST=I _{W 1} a.s. and TS=I _{W 2} a.s. If, in addition, ₁=, then there exists a 1-convex function in the Gaussian Sobolev space such that =. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from can be written as the composition of a transport map T and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions and Ito-type solutions of the Monge-Ampère equation on W. Mathematics Subject Classification (2000)60H07, 60H05,60H25, 60G15, 60G30, 60G35, 46G12, 47H05, 47H1, 35J60, 35B65,35A30, 46N10, 49Q20, 58E12, 26A16, 28C20cf. Theorem 6.1 for the precise hypothesis about ₁ and ₂.In fact this hypothesis is too strong, cf. Theorem 6.1. AcknowledgementThe authors are grateful to Françoise Combelles for all the bibliographical help that she has supplied for the realization of this research. We thank also the anonymous referee for his particular attention and valuable remarks.     

Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space

It is proved that, in Minkowski 3-space, a CSM-helicoidal surface, i.e., a helicoidal surface under cubic screw motion is isometric to a rotation surface so that helices on the helicoidal surface correspond to parallel circles on the rotation surface. By distinguishing a CSM-helicoidal surface as three cases, that is, the case of type I, the case of type II with negative and positive pitch, the relations are discussed between the mean curvatures or Gauss maps of a pair of isometric helicoidal and rotation surface. A CSM-helicoidal surface of Case 1 or 2 and its isometric rotation surface with null axis have same mean curvatures (resp. Gauss maps) if and only if they are minimal. But each pair of isometric CSM-helicoidal surface of Case 3 and rotation surface with spacelike axis have different Gauss maps.     

Boundedness of maximal function related to the Monge-Ampère equation

In this paper, we establish the boundedness of maximal function on Morrey spaces related to the Monge-Ampère equation.     

A research into the numerical method of Dirichlet's problem of complex Monge-Ampère equation on Cartan-Hartogs domain of the third type

Monge-Ampère equation is a nonlinear equation with high degree, therefore its numerical solution is very important and very difficult. In present paper the numerical method of Dirichlet's problem of Monge-Ampère equation on Cartan-Hartogs domain of the third type is discussed by using the analytic method. Firstly, the Monge-Ampère equation is reduced to the nonlinear ordinary differential equation, then the numerical method of the Dirichlet problem of Monge-Ampère equation becomes the numerical method of two point boundary value problem of the nonlinear ordinary differential equation. Secondly, the solution of the Dirichlet problem is given in explicit formula under the special case, which can be used to check the numerical solution of the Dirichlet problem.     

The value distribution of the hyperbolic Gauss map

In this paper, we investigate the hyperbolic Gauss map of a complete CMC-1 surface in , and prove that it cannot omit more than four points unless the surface is a horosphere.

     

Regularity estimates of solutions to complex Monge-Ampère equations on Hermitian manifolds

In this paper, we obtain the Bedford-Taylor interior C² estimate and local Calabi C³ estimate for the solutions to complex Monge-Ampère equations on Hermitian manifolds.