Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under th...Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.展开更多
In this paper, Yau’s conjecture on harmonic functions in Riemannian manifolds is generalized to Alexandrov spaces. It is proved that the space of harmonic functions with polynomial growth of a fixed rate is finite di...In this paper, Yau’s conjecture on harmonic functions in Riemannian manifolds is generalized to Alexandrov spaces. It is proved that the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional and strong Liouville theorem holds in Alexandrov spaces with nonnegative curvature.展开更多
Let X be a complete Alexandrov space with curvature ≥1 and radius 〉 π/2. We prove that any connected, complete, and locally convex subset without boundary in X also has the radius 〉 π/2.
We introduce quasi-convex subsets in Alexandrov spaces with lower curvature bound,which include not only all closed convex subsets without boundary but also all extremal subsets.Moreover,we explore several essential p...We introduce quasi-convex subsets in Alexandrov spaces with lower curvature bound,which include not only all closed convex subsets without boundary but also all extremal subsets.Moreover,we explore several essential properties of such kind of subsets including a generalized Liberman theorem.It turns out that the quasi-convex subset is a nice and fundamental concept to illustrate the similarities and differences between Riemannian manifolds and Alexandrov spaces with lower curvature bound.展开更多
We show that a closed piecewise fiat 2-dimensional Alexandrov space Σ can be bi-Lipschitz embedded into a Euclidean space such that the embedded image of Σ has a tubular neighborhood in a generalized sense. As an ap...We show that a closed piecewise fiat 2-dimensional Alexandrov space Σ can be bi-Lipschitz embedded into a Euclidean space such that the embedded image of Σ has a tubular neighborhood in a generalized sense. As an application, we show that for any metric space sufficiently close to Σ in the Gromov-Hausdorff topology, there is a Lipschitz Gromov-Hausdorff approximation.展开更多
This paper is mainly devoted to proving the four equivalent defining properties of a CBA(κ)space.The proof is based on an interesting tool we established which describes the cyclical five-step deformation procedure f...This paper is mainly devoted to proving the four equivalent defining properties of a CBA(κ)space.The proof is based on an interesting tool we established which describes the cyclical five-step deformation procedure for quadrangles in the model 2-plane S^(2)_(κ),including the limit shape of each step.As a byproduct we give a complete list of cyclical deformation procedures for all kinds of quadrangles on S21.At last we make a contrast of geometric properties of CBA with CBB spaces,including a comparison between their defining properties and a discussion about Alexandrov’s Lemma.展开更多
In the present paper system and the solutions to the the solvability condition of the linearized Gauss-Codazzi homogenous system are given. In the meantime, the ＇solvability of a relevant linearized Darboux equation...In the present paper system and the solutions to the the solvability condition of the linearized Gauss-Codazzi homogenous system are given. In the meantime, the ＇solvability of a relevant linearized Darboux equation is given. The equations are arising in a geometric problem which is concerned with the realization of the Alexandrov＇s positive annulus in R^3.展开更多
Let X∈Alex^(n)(−1)be an n-dimensional Alexandrov space with curvature≥−1.Let the r-scale(k,ε)-singular set S_(ε,r)^(k)(X)be the collection of x∈X so that B_(r)(x)is notr-close to a ball in any splitting spaceℝ^(k...Let X∈Alex^(n)(−1)be an n-dimensional Alexandrov space with curvature≥−1.Let the r-scale(k,ε)-singular set S_(ε,r)^(k)(X)be the collection of x∈X so that B_(r)(x)is notr-close to a ball in any splitting spaceℝ^(k+1)×Z.We show that there exists C(n,ε)>0 and𝛽(n,ε)>0,independent of the volume,so that for any disjoint collection{B_(ri)(xi)∶x_(i)∈S_(ε,βri)^(k)(X)∩B_(1),r_(i)≤1,the packing estimateΣr_(i)^(k)≤C holds.Consequently,we obtain the Hausdorff measure estimates H^(k)(S_(ε)^(k)(X))∩B_(1))≤C and H^(n)(B_(r)(S_(ε,βri)^(k))∩B_(1))≤C rn−k.This answers an open question in Kapovitch et al.(Metric-measure boundary and geodesic flow on Alexandrov spaces.arXiv:1705.04767(2017)).We also show that the k-singular set S^(k)(X)=⋃ε>0⋂r>0 S_(ε,r)^(k)𝜖,ris k-rectifiable and construct examples to show that such a structure is sharp.For instance,in the k=1 case we can build for any closed set T⊆S^(1)andε>0 a space Y∈Alex^(3)(0)with S_(ε)^(1)(Y)=Ф(T),whereФ∶S^(1)→Y is a bi-Lipschitz embedding.Taking T to be a Cantor set it gives rise to an example where the singular set is a 1-rectifiable,1-Cantor set with positive 1-Hausdorff measure.展开更多
In this paper,we shall prove that any minimizer of Ginzburg-Landau functional from an Alexandrov space with curvature bounded below into a nonpositively curved metric cone must be locally Lipschitz continuous.
This is a survey about our recent works on the Gauss-Bonnet-Chern （GBC） mass for asymptotically flat and asymptotically hyperbolic manifolds. We first introduce the GBC mass, a higher order mass, for asymptotically ...This is a survey about our recent works on the Gauss-Bonnet-Chern （GBC） mass for asymptotically flat and asymptotically hyperbolic manifolds. We first introduce the GBC mass, a higher order mass, for asymptotically flat and for asymptotically hyperbolic manifolds, respectively, by using a higher order scalar curvature. Then we prove its positivity and the Penrose inequality for graphical manifolds. One of the crucial steps in the proof of the Penrose inequality is the use of an Alexandrov-Fenchel inequality, which is a classical^inequality in the Euclidean space. In the hyperbolic space, we have established this new Alexandrov-Fenchel inequality. We also have a similar work for asymptotically locally hyperbolic manifolds. At the end, we discuss the relation between the GBC mass and Chern＇s magic form.展开更多
Let M be a C^(2)-smooth Riemannian manifold with boundary and X be a metric space with non-positive curvature in the sense of Alexandrov.Let u:M→X be a Sobolev mapping in the sense of Korevaar and Schoen.In this shor...Let M be a C^(2)-smooth Riemannian manifold with boundary and X be a metric space with non-positive curvature in the sense of Alexandrov.Let u:M→X be a Sobolev mapping in the sense of Korevaar and Schoen.In this short note,we introduce a notion of p-energy for u which is slightly different from the original definition of Korevaar and Schoen.We show that each minimizing p-harmonic mapping(p≥2)associated to our notion of p-energy is locally Holder continuous whenever its image lies in a compact subset of X.展开更多
基金supported by NSFC (10831008)NKBRPC(2006CB805905)
文摘Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.
文摘In this paper, Yau’s conjecture on harmonic functions in Riemannian manifolds is generalized to Alexandrov spaces. It is proved that the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional and strong Liouville theorem holds in Alexandrov spaces with nonnegative curvature.
基金Acknowledgements The authors would like to show their respect to the referees for their suggestions, especially on the form of the conclusion ＇rad（N） ≥rad（X） 〉 π/2＇ in Main Theorem （in the original version of the paper, the conclusion is ＇rad（N） 〉 π/2＇）. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11001015, 11171025）.
文摘Let X be a complete Alexandrov space with curvature ≥1 and radius 〉 π/2. We prove that any connected, complete, and locally convex subset without boundary in X also has the radius 〉 π/2.
基金supported in part by the National Natural Science Foundation of China(Grant No.11971057)Beijing Natural Science Foundation(No.Z190003).
文摘We introduce quasi-convex subsets in Alexandrov spaces with lower curvature bound,which include not only all closed convex subsets without boundary but also all extremal subsets.Moreover,we explore several essential properties of such kind of subsets including a generalized Liberman theorem.It turns out that the quasi-convex subset is a nice and fundamental concept to illustrate the similarities and differences between Riemannian manifolds and Alexandrov spaces with lower curvature bound.
基金the National Natural Science Foundation of China (Grant No. 11501258).
文摘We show that a closed piecewise fiat 2-dimensional Alexandrov space Σ can be bi-Lipschitz embedded into a Euclidean space such that the embedded image of Σ has a tubular neighborhood in a generalized sense. As an application, we show that for any metric space sufficiently close to Σ in the Gromov-Hausdorff topology, there is a Lipschitz Gromov-Hausdorff approximation.
基金Supported by China Scholarship Council (Grant No. 201708120014)。
文摘This paper is mainly devoted to proving the four equivalent defining properties of a CBA(κ)space.The proof is based on an interesting tool we established which describes the cyclical five-step deformation procedure for quadrangles in the model 2-plane S^(2)_(κ),including the limit shape of each step.As a byproduct we give a complete list of cyclical deformation procedures for all kinds of quadrangles on S21.At last we make a contrast of geometric properties of CBA with CBB spaces,including a comparison between their defining properties and a discussion about Alexandrov’s Lemma.
基金Project supported by the National Natural Science Foundation of China (No. 11101068)the Fundamental Research Funds for the Central Universities (No. ZYGX2010J109)the Sichuan Youth Science and Technology Foundation (No. 2011JQ0003)
文摘In the present paper system and the solutions to the the solvability condition of the linearized Gauss-Codazzi homogenous system are given. In the meantime, the ＇solvability of a relevant linearized Darboux equation is given. The equations are arising in a geometric problem which is concerned with the realization of the Alexandrov＇s positive annulus in R^3.
文摘Let X∈Alex^(n)(−1)be an n-dimensional Alexandrov space with curvature≥−1.Let the r-scale(k,ε)-singular set S_(ε,r)^(k)(X)be the collection of x∈X so that B_(r)(x)is notr-close to a ball in any splitting spaceℝ^(k+1)×Z.We show that there exists C(n,ε)>0 and𝛽(n,ε)>0,independent of the volume,so that for any disjoint collection{B_(ri)(xi)∶x_(i)∈S_(ε,βri)^(k)(X)∩B_(1),r_(i)≤1,the packing estimateΣr_(i)^(k)≤C holds.Consequently,we obtain the Hausdorff measure estimates H^(k)(S_(ε)^(k)(X))∩B_(1))≤C and H^(n)(B_(r)(S_(ε,βri)^(k))∩B_(1))≤C rn−k.This answers an open question in Kapovitch et al.(Metric-measure boundary and geodesic flow on Alexandrov spaces.arXiv:1705.04767(2017)).We also show that the k-singular set S^(k)(X)=⋃ε>0⋂r>0 S_(ε,r)^(k)𝜖,ris k-rectifiable and construct examples to show that such a structure is sharp.For instance,in the k=1 case we can build for any closed set T⊆S^(1)andε>0 a space Y∈Alex^(3)(0)with S_(ε)^(1)(Y)=Ф(T),whereФ∶S^(1)→Y is a bi-Lipschitz embedding.Taking T to be a Cantor set it gives rise to an example where the singular set is a 1-rectifiable,1-Cantor set with positive 1-Hausdorff measure.
文摘In this paper,we shall prove that any minimizer of Ginzburg-Landau functional from an Alexandrov space with curvature bounded below into a nonpositively curved metric cone must be locally Lipschitz continuous.
基金Acknowledgements The most part of this survey was talked in the conference ＂Metric Riemannian Geometry Workshop＂ held in Shanghai Jiao Tong University, Shanghai, China. The authors would like to take this opportunity to thank the organizers both from China and from Germany. This work was partly supported by SFB/TR71 ＂Geometric partial differential equations＂ of DFG. JW was supported by the National Natural Science Foundation of China （Grant No. 11401553） and CX in part by the Fundamental Research Funds for the Central Universities （Grant No. 20720150012） and the National Natural Science Foundation of China （Grant No. 11501480）.
文摘This is a survey about our recent works on the Gauss-Bonnet-Chern （GBC） mass for asymptotically flat and asymptotically hyperbolic manifolds. We first introduce the GBC mass, a higher order mass, for asymptotically flat and for asymptotically hyperbolic manifolds, respectively, by using a higher order scalar curvature. Then we prove its positivity and the Penrose inequality for graphical manifolds. One of the crucial steps in the proof of the Penrose inequality is the use of an Alexandrov-Fenchel inequality, which is a classical^inequality in the Euclidean space. In the hyperbolic space, we have established this new Alexandrov-Fenchel inequality. We also have a similar work for asymptotically locally hyperbolic manifolds. At the end, we discuss the relation between the GBC mass and Chern＇s magic form.
基金supported by the Qilu funding of Shandong University (62550089963197)financially supported by the National Natural Science Foundation of China (11701045)the Yangtze Youth Fund (2016cqn56)
文摘Let M be a C^(2)-smooth Riemannian manifold with boundary and X be a metric space with non-positive curvature in the sense of Alexandrov.Let u:M→X be a Sobolev mapping in the sense of Korevaar and Schoen.In this short note,we introduce a notion of p-energy for u which is slightly different from the original definition of Korevaar and Schoen.We show that each minimizing p-harmonic mapping(p≥2)associated to our notion of p-energy is locally Holder continuous whenever its image lies in a compact subset of X.
