The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate onthe boundary is contained As some app...The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate onthe boundary is contained As some applications the smooth isometric embeddings of positivelyand nonnegatively curved disks into R^3 are constructed.展开更多
A new method to solve the Gauss-Codazzi system is given in which we transform the linearized system to a partial differential equation of second order, and by the method we solve the problem of semi-global isometric e...A new method to solve the Gauss-Codazzi system is given in which we transform the linearized system to a partial differential equation of second order, and by the method we solve the problem of semi-global isometric embedding of surfaces with Gaussian curvature changing sign cleanly.展开更多
In the paper, the authors show that any abstract smooth surface can be locally isometricallyembedded into a class of 3-dimensional spaces Nρ0 (ρ0>0) with the non-positively sectionalcurvature being fixed sufficie...In the paper, the authors show that any abstract smooth surface can be locally isometricallyembedded into a class of 3-dimensional spaces Nρ0 (ρ0>0) with the non-positively sectionalcurvature being fixed sufficiently small.展开更多
This paper studies two isometric problems between unit spheres of Banach spaces.In the first part,we introduce and study the Figiel type problem of isometric embeddings between unit spheres.However,the classical Figie...This paper studies two isometric problems between unit spheres of Banach spaces.In the first part,we introduce and study the Figiel type problem of isometric embeddings between unit spheres.However,the classical Figiel theorem on the whole space cannot be trivially generalized to this case,and this is pointed out by a counterexample.After establishing this,we find a natural necessary condition required by the existence of the Figiel operator.Furthermore,we prove that when X is a space with the T-property,this condition is also sufficient for an isometric embedding T:S_(X)→S_(Y) to admit the Figiel operator.This answers the Figiel type problem on unit spheres for a large class of spaces.In the second part,we consider the extension of bijectiveε-isometries between unit spheres of two Banach spaces.It is shown that every bijectiveε-isometry between unit spheres of a local GL-space and another Banach space can be extended to be a bijective 5ε-isometry between the corresponding unit balls.In particular,whenε=0,this recovers the MUP for local GL-spaces obtained in[40].展开更多
Haberl and Ludwig introduced the L_p-intersection body I_pK for an originsymmetric star body K in R^n,where p < 1 and p ≠ 0.In this paper,we consider the Busemann-Petty's problem for L_p-intersection bodies I_...Haberl and Ludwig introduced the L_p-intersection body I_pK for an originsymmetric star body K in R^n,where p < 1 and p ≠ 0.In this paper,we consider the Busemann-Petty's problem for L_p-intersection bodies I_pK and I_pL.That is,whether I_pK ■ IpL implies Vol_n(K) ≤ Vol_n(L).We obtain that for two origin-symmetric star bodies K and L in R^n,such that(R^n,||·||K) embeds in L_p and I_pK ■ IpL,then vol_n(K) ≤ vol_n(L) for 0 < p < 1 and vol_n(K) ≥ vol_n(L) for p < 0.展开更多
In this paper,we give a number of characterizations for a Banach space X which is isometric to a subspace of c_(0),or,c_(0)(Г),successively,in terms of extreme points of its dual unit ball Bx*,Frechet and Gateaux der...In this paper,we give a number of characterizations for a Banach space X which is isometric to a subspace of c_(0),or,c_(0)(Г),successively,in terms of extreme points of its dual unit ball Bx*,Frechet and Gateaux derivatives of its norm,or,in terms ofω^(*)-strongly exposed points andω^(*)-exposed points of Bx*.展开更多
文摘The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate onthe boundary is contained As some applications the smooth isometric embeddings of positivelyand nonnegatively curved disks into R^3 are constructed.
基金supported by the Fundamental Research Funds for the Central Universities(Grant No. ZYGX2010J109)National Natural Science Foundation of China (Grant No. 11101068)the Sichuan Youth Science and Technology Foundation (Grant No. 2011JQ0003)
文摘A new method to solve the Gauss-Codazzi system is given in which we transform the linearized system to a partial differential equation of second order, and by the method we solve the problem of semi-global isometric embedding of surfaces with Gaussian curvature changing sign cleanly.
文摘In the paper, the authors show that any abstract smooth surface can be locally isometricallyembedded into a class of 3-dimensional spaces Nρ0 (ρ0>0) with the non-positively sectionalcurvature being fixed sufficiently small.
基金the National Nature Science Foundation of China(11671214,11971348,12071230)the Hundred Young Academia Leaders Program of Nankai University(63223027,ZB22000105)+1 种基金the Undergraduate Education and Teaching Project of Nankai University(NKJG2022053)the National College Students’Innovation and Entrepreneurship Training Program of Nankai University(202210055048)。
文摘This paper studies two isometric problems between unit spheres of Banach spaces.In the first part,we introduce and study the Figiel type problem of isometric embeddings between unit spheres.However,the classical Figiel theorem on the whole space cannot be trivially generalized to this case,and this is pointed out by a counterexample.After establishing this,we find a natural necessary condition required by the existence of the Figiel operator.Furthermore,we prove that when X is a space with the T-property,this condition is also sufficient for an isometric embedding T:S_(X)→S_(Y) to admit the Figiel operator.This answers the Figiel type problem on unit spheres for a large class of spaces.In the second part,we consider the extension of bijectiveε-isometries between unit spheres of two Banach spaces.It is shown that every bijectiveε-isometry between unit spheres of a local GL-space and another Banach space can be extended to be a bijective 5ε-isometry between the corresponding unit balls.In particular,whenε=0,this recovers the MUP for local GL-spaces obtained in[40].
基金Supported by the NNSF of China(11161019)Supported by the Foundation of the Education Department of Gansu Province(1009B-09)
文摘Haberl and Ludwig introduced the L_p-intersection body I_pK for an originsymmetric star body K in R^n,where p < 1 and p ≠ 0.In this paper,we consider the Busemann-Petty's problem for L_p-intersection bodies I_pK and I_pL.That is,whether I_pK ■ IpL implies Vol_n(K) ≤ Vol_n(L).We obtain that for two origin-symmetric star bodies K and L in R^n,such that(R^n,||·||K) embeds in L_p and I_pK ■ IpL,then vol_n(K) ≤ vol_n(L) for 0 < p < 1 and vol_n(K) ≥ vol_n(L) for p < 0.
基金Supported by National Natural Science Foundation of China(Grant No.11731010)。
文摘In this paper,we give a number of characterizations for a Banach space X which is isometric to a subspace of c_(0),or,c_(0)(Г),successively,in terms of extreme points of its dual unit ball Bx*,Frechet and Gateaux derivatives of its norm,or,in terms ofω^(*)-strongly exposed points andω^(*)-exposed points of Bx*.
