Currents in metric spaces

By Luigi Ambrosio, B. Kirchheim. 2000

448 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b903bab0dd50281ad45f3ac8cc30a8d3da2d50f5

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Well posedness of Lagrangian flows and continuity equations in metric measure spaces (2014, methodology): “ch is dense in Vand which plays the role of the C∞ c functions in the theory of distributions. Using A, we can deﬁne in Section 3 “vector ﬁelds” as derivations, in the same spirit of [We00] (see also [AK00] for parallel developments in the theory of currents): a derivation b is a linear map from A t”
Regular Hypersurfaces, Intrinsic Perimeter and Implicit Function Theorem in Carnot Groups (2003, methodology): “We only mention some of them: De Giorgi [14], [15], [16], Gromov [28], [29], Preiss and Tisˇ er [44], Kirchheim [33], David & Semmes [11], Cheeger [9] and Ambrosio and Kirchheim [3], [ 4 ].”
Gromov hyperbolic spaces and the sharp isoperimetric constant (2006, methodology): “In the following we adopt the notation of [ 2 ] and refer to it for precise definit ions.”
Area Minimizing Discs in Metric Spaces (2015, background): “The theory of integral currents, developed by Feder r-Fleming [FF60] in the setting of Euclidean spaces, has been generalized by Ambrosio-Kirchheim [AK00a] to the setting of arbitrary complete metric spaces.”
A notion of rectifiability modeled on Carnot groups (2000, unknown): “In recent years, there has been significant interest and progress in the study of rectifiable sets not only in Euclidean space but in more general metric spaces as well (see, for example, [Amb], [AK99b], [AK99a], [Che99], [DS97], [FSSC99], [GN96], [Kir94], [Mag00], [Pan89], [Whi98].”
Functions of bounded higher variation (2002, methodology): “Elements of our argument were later used by Ambrosio and Kirchheim [ 3 ] in developing a general theory of rectiable currents in metric spaces.”
Surface measures in Carnot-Carathéodory spaces (2001, background): “An interesting proposal of a general Geometric Measure Theory in metric spaces was given in [20] and the program has been developed in [4] and [ 5 ].”
Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups (2009, background): “Kirchheim and Serra-Cassano [39] have constructed an H1-regular hypersurface in H1 whose Euclidean Hausdorff dimension is 2.5, even locally at every point.”
Bilipschitz Embeddings of Metric Spaces into Space Forms (2001, background): “In this generality, a suitable notion of k-chain is provided by the new theory of metric currents due to L. Ambrosio and B. Kirchheim [ AmK ].”
Local Currents in Metric Spaces (2011, background): “This will also establish the connection to the metric currents of [3].”
Smooth Convergence Away from Singular Sets (2012, background): “where ∂T is the boundary of the integral current defined as in [1] so that it satisfies Stoke’s Theorem.”
The Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current Spaces (2010, methodology): “Stronger versions of locality and continuity, as well as product and chain rules are proven in [2][Theorem 3.”
The Jacobian and the Ginzburg-Landau energy (2002, result): “A much more general version of this result was later established by Ambrosio andKirchheim [3].”
Unrectifiability and rigidity in stratified groups (2004, background): “The purpose of developing a metric theory of currents has been successfully achieved in [2], where k-rectifiable sets (see 3.”