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.
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.
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
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
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
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
In this paper,the authors give a comparison version of Pythagorean theo-rem to judge the lower or upper bound of the curvature of Alexandrov spaces(including Riemannian manifolds).
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 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.
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.
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This work was supported by the National Natural Science Foundation of China(No.11971057)BNSF Z190003.
文摘In this paper,the authors give a comparison version of Pythagorean theo-rem to judge the lower or upper bound of the curvature of Alexandrov spaces(including Riemannian manifolds).
