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Metric measure limits of spheres and complex projective spaces  [PDF]
Takashi Shioya
Mathematics , 2014,
Abstract: We study the limits of sequences of spheres and complex projective spaces with unbounded dimensions. A sequence of spheres (resp. complex projective spaces) either is a Levy family, infinitely dissipates, or converges to (resp. the Hopf quotient of) a virtual infinite-dimensional Gaussian space, depending on the size of the spaces. These are the first discovered examples with the property that the limits are drastically different from the spaces in the sequence. For the proof, we introduce a metric on Gromov's compactification of the space of metric measure spaces.
Fixed point theorems for a class of mappings depending of another function and defined on cone metric spaces  [PDF]
José R. Morales,Edixon Rojas
Mathematics , 2009,
Abstract: In this paper we study the existence and uniqueness of fixed points of a class of mappings defined on complete, (sequentially compact) cone metric spaces, without continuity conditions and depending on another function.
On the Existence of Fixed Points of Contraction Mappings Depending of Two Functions on Cone Metric Spaces  [PDF]
José R. Morales,Edixon Rojas
Mathematics , 2009,
Abstract: In this paper, we study the existence of fixed points for mappings defined on complete, (sequentially compact) cone metric spaces, satisfying a general contractive inequality depending of two additional mappings.
Hyperbolic groupoids: metric and measure  [PDF]
Volodymyr Nekrashevych
Mathematics , 2011,
Abstract: We construct Patterson-Sullivan measure and a natural metric on the unit space of a hyperbolic groupoid. In particular, this gives a new approach to defining SRB measures on Smale spaces using Gromov hyperbolic graphs.
Regularity of the solutions to SPDEs in metric measure spaces  [PDF]
Elena Issoglio,Martina Z?hle
Mathematics , 2014, DOI: 10.1007/s40072-015-0048-8
Abstract: In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernels. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique solution is H\"older continuous in time with values in a suitable fractional Sobolev space. As this analysis is done via a-priori estimates, we can apply this result to stochastic PDEs on metric measure spaces and solve the equation in a pathwise sense for almost all paths. The main example of noise term is of fractional Brownian type and the metric measure spaces can be classical as well as given by various fractal structures. The whole approach is low dimensional and works for spectral dimensions less than 4.
Almost isometric embeddings of metric spaces  [PDF]
Menachem Kojman,Saharon Shelah
Mathematics , 2004,
Abstract: We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum separable metric spaces on aleph_1 so that every separable metric space is almost isometrically embedded into one of them when the continuum hypothesis fails. (3) There is no collection of fewer than continuum metric spaces of cardinality aleph_2 so that every ultra-metric space of cardinality aleph_2 is almost isometrically embedded into one of them if aleph_2<2^{aleph_0}. We also prove that various spaces X satisfy that if a space X is almost isometric to X than Y is isometric to X.
Marked metric measure spaces  [PDF]
Andrej Depperschmidt,Andreas Greven,Peter Pfaffelhuber
Mathematics , 2011,
Abstract: A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r) is a complete and separable metric space and mu is a probability measure on XxI for some Polish space I of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed I. It arises as state space in the construction of Markov processes which take values in random graphs, e.g. tree-valued dynamics describing randomly evolving genealogical structures in population models. We derive here the topological properties of the space of mmm-spaces needed to study convergence in distribution of random mmm-spaces. Extending the notion of the Gromov-weak topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish space. We give a characterization of tightness for families of distributions of random mmm- spaces and identify a convergence determining algebra of functions, called polynomials.
Metric measure geometry  [PDF]
Takashi Shioya
Mathematics , 2014,
Abstract: In this book, we study Gromov's metric geometric theory on the space of metric measure spaces, based on the idea of concentration of measure phenomenon due to L\'evy and Milman. Although most of the details are omitted in the original article of Gromov, we present complete and detailed proofs for some main parts, in which we prove several claims that are not mentioned in any literature. We also discuss concentration with a lower bound of curvature, originally studied by Funano and the author.
The Intrinsic Geometry of Almost Contact Metric Manifolds  [PDF]
Sergey V. Galaev
Mathematics , 2011,
Abstract: In this paper the notion of the intrinsic geometry of an almost contact metric manifold is introduced. Description of some classes of spaces with almost contact metric structures in terms of the intrinsic geometry is given. A new type of almost contact metric spaces, more precisely, Hermitian almost contact metric spaces, is introduced.
Harmonic functions on metric measure spaces  [PDF]
Bobo Hua,Martin Kell,Chao Xia
Mathematics , 2013,
Abstract: In this paper, we study harmonic functions on metric measure spaces with Riemannian Ricci curvature bounded from below, which were introduced by Ambrosio-Gigli-Savar\'e. We prove a Cheng-Yau type local gradient estimate for harmonic functions on these spaces. Furthermore, we derive various optimal dimension estimates for spaces of polynomial growth harmonic functions on metric measure spaces with nonnegative Riemannian Ricci curvature.
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