Heat Kernel Comparison on Alexandrov Spaces with Curvature Bounded Below

In this paper the comparison result for the heat kernel on Riemannian manifolds with lower Ricci curvature bound by Cheeger and Yau (1981) is extended to locally compact path metric spaces (X,d) with lower curvature bound in the sense of Alexandrov and with sufficiently fast asymptotic decay of the volume of small geodesic balls. As corollaries we recover Varadhan's short time asymptotic formula for the heat kernel (1967) and Cheng's eigenvalue comparison theorem (1975). Finally, we derive an integral inequality for the distance process of a Brownian Motion on (X,d) resembling earlier results in the smooth setting by Debiard, Geavau and Mazet (1975).     

Harmonic Maps Between Alexandrov Spaces

In this paper, we shall discuss the existence, uniqueness and regularity of harmonic maps from an Alexandrov space into a geodesic space with curvature \(\leqslant 1\) in the sense of Alexandrov.     

Generalized Liouville Theorem in Nonnegatively Curved Alexandrov Spaces

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.     

Three-Dimensional Alexandrov Spaces with Positive or Nonnegative Ricci Curvature

We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the CD ^?(K,N) sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional CD ^?(2,3)-Alexandrov space must be homeomorphic to a spherical space form or to the suspension of \(\mathbb {R}P^{2}\). We then classify closed three-dimensional CD ^?(0,3)-Alexandrov spaces.     

Jung's Theorem for Alexandrov Spaces of Curvature Bounded Above

The classical Jung theorem gives an optimal upper estimate for the radius of a bounded subset of R ⁿ in terms of its diameter and the dimension. In this note we present an analogue of this result for metric spaces of curvature bounded above in the sense of Alexandrov.     

带位势的p-调和热流

在本文中我们考察从黎曼流形M到具左不变度量的齐次空间N的带位势的 p-调和映射热流,证明存在全局弱解.     

The p-Harmonic Heat Flow with Potential into Homogeneous Spaces

In this paper we prove the existence of global weak solutions of the p-harmonic flow with potential between Riemannian manifolds M and N for arbitrary initial data having finite p-energy in the case when the target N is a homogeneous space with a left invariant metric. Received March 17, 1999, Revised September 22, 1999, Accepted October 15, 1999     

Closed Geodesics in Alexandrov Spaces of Curvature Bounded from Above

In this paper, we show a local energy convexity of W ^1,2 maps into CAT(K) spaces. This energy convexity allows us to extend Colding and Minicozzi’s width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.     

到齐次空间带位势的P-调和热流的非唯一性

对齐次空间的具有位势的P-调和热流的唯一性问题进行了研究,并且证明了如果初始数据是非稳态的具有位势的P-调和映射,则存在无穷多个全局弱解．     

到齐次空间带位势的P-调和热流的非唯一性

对齐次空间的具有位势的P-调和热流的唯一性问题进行了研究,并且证明了如果初始数据是非稳态的具有位势的P-调和映射,则存在无穷多个全局弱解.     

Stationary Critical Points of the Heat Flow in Spaces of Constant Curvature

The paper considers stationary critical points of the heat flowin sphere S^N and in hyperbolic space H^N, and proves severalresults corresponding to those in Euclidean space R^N which havebeen proved by Magnanini and Sakaguchi. To be precise, it isshown that a solution u of the heat equation has a stationarycritical point, if and only if u satisfies some balance lawwith respect to the point for any time. In Cauchy problems forthe heat equation, it is shown that the solution u has a stationarycritical point if and only if the initial data satisfies thebalance law with respect to the point. Furthermore, one point,say x₀, is fixed and initial-boundary value problems are consideredfor the heat equation on bounded domains containing x₀. It isshown that for any initial data satisfying the balance law withrespect to x₀ (or being centrosymmetric with respect to x₀)the corresponding solution always has x₀ as a stationary criticalpoint, if and only if the domain is a geodesic ball centredat x₀ (or is centrosymmetric with respect to x₀, respectively).     

Convex functions on Alexandrov surfaces

We investigate the topological structure of Alexandrov surfaces of curvature bounded below which possess convex functions. We do not assume the continuities of these functions. Nevertheless, if the convex functions satisfy a condition of local nonconstancy, then the topological structures of Alexandrov surfaces and the level sets configurations of these functions in question are determined.

     

Nonlinear Markov Operators,Discrete Heat Flow,and Harmonic Maps Between Singular Spaces

We develop a theory of harmonic maps f:MN between singular spaces M and N. The target will be a complete metric space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov. The domain will be a measurable space (M,) with a given Markov kernel p(x,dy) on it. Given a measurable map f:MN, we define a new map Pf:MN in the following way: for each xM, the point Pf(x)N is the barycenter of the probability measure p(x,f ^–1(dy)) on N. The map f is called harmonic on DM if Pf=f on D. Our theory is a nonlinear generalization of the theory of Markov kernels and Markov chains on M. It allows to construct harmonic maps by an explicit nonlinear Markov chain algorithm (which under suitable conditions converges exponentially fast). Many smoothing and contraction properties of the linear Markov operator P _M,R carry over to the nonlinear Markov operator P=P _M,N . For instance, if the underlying Markov kernel has the strong Lipschitz Feller property then all harmonic maps will be Lipschitz continuous.     

Isospectral Alexandrov spaces

We construct the first non-trivial examples of compact non-isometric Alexandrov spaces which are isospectral with respect to the Laplacian and not isometric to Riemannian orbifolds. This construction generalizes independent earlier results by the authors based on Schüth’s version of the torus method.     

Equidistant sets on Alexandrov surfaces

We examine properties of equidistant sets determined by nonempty disjoint compact subsets of a compact 2-dimensional Alexandrov space (of curvature bounded below). The work here generalizes many of the known results for equidistant sets determined by two distinct points on a compact Riemannian 2-manifold. Notably, we find that the equidistant set is always a finite simplicial 1-complex. These results are applied to answer an open question concerning the Hausdorff dimension of equidistant sets in the Euclidean plane.     

Alexandrov Par Excellence

This is a short tribute to Alexandr Alexandrov (1912–1999).     

The Horizontal Heat Kernel on the Quaternionic Anti-De Sitter Spaces and Related Twistor Spaces

Potential Analysis - The geometry of the quaternionic anti-de Sitter fibration is studied in details. As a consequence, we obtain formulas for the horizontal Laplacian and subelliptic heat kernel...     

Cohomogeneity one Alexandrov spaces

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary we obtain the classification of closed, n-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric T ⁿ⁻¹ action. In contrast to the one- and two-dimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of \mathbbRP² \mathbb{R}{P^2} , which is not a manifold.     

Quantization of the Geodesic Flow on Quaternion Projective Spaces

We study a problem of the geometric quantization for the quaternionprojective space. First we explain a Kähler structure on the punctured cotangent bundleof the quaternion projective space, whose Kähler form coincides withthe natural symplectic form on the cotangent bundle and show thatthe canonical line bundle of this complex structure is holomorphicallytrivial by explicitly constructing a nowhere vanishing holomorphicglobal section. Then we construct a Hilbert space consisting of acertain class of holomorphic functions on the punctured cotangentbundle by the method ofpairing polarization and incidentally we construct an operatorfrom this Hilbert space to the L ₂ space of the quaternionprojective space. Also we construct a similar operator between thesetwo Hilbert spaces through the Hopf fiberation.We prove that these operators quantizethe geodesic flow of the quaternion projective space tothe one parameter group of the unitary Fourier integral operatorsgenerated by the square root of the Laplacian plus suitable constant.Finally we remark that the Hilbert space above has the reproducing kernel.