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Collapsing Riemannian Metrics to Sub-Riemannian and the Geometry of Hypersurfaces in Carnot Groups

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Around the Research of Vladimir Maz'ya I

Part of the book series: International Mathematical Series ((IMAT,volume 11))

Abstract

Given a Carnot group with step two G, we study the limit as ε → 0 of a family of left-invariant Riemannian metrics g ε on G. In such metrics, neither the Ricci tensor nor the sectional curvatures are bounded from below. Nonetheless, our main result shows that the Riemannian first and second variation formulas in the g ε-metrics converge to the corresponding sub-Riemannian ones. This testifies of an intrinsic stability of some deli- cate cancellation processes and makes the method of collapsing Riemannian metrics a possible tool for attacking various fundamental open questions in sub-Riemannian geometry. Finally, we mention that the restriction to groups of step two is purely for ease of exposition and that the same method can be applied to Carnot groups of arbitrary step.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Nicola Garofalo

  2. Università di Padova, 35131, Padova, Italy

    Nicola Garofalo

  3. Physics Laboratory, The Johns Hopkins University, Baltimore, MD, USA

    Christina Selby

Authors
  1. Nicola Garofalo
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  2. Christina Selby
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Correspondence to Nicola Garofalo .

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Editors and Affiliations

  1. Department of Mathematics, Imperial College, Queen's Gate 180, London, SW7 2AZ, United Kingdom

    Ari Laptev

Additional information

Dedicated to Professor Maz’ya with affection and admiration

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Garofalo, N., Selby, C. (2010). Collapsing Riemannian Metrics to Sub-Riemannian and the Geometry of Hypersurfaces in Carnot Groups. In: Laptev, A. (eds) Around the Research of Vladimir Maz'ya I. International Mathematical Series, vol 11. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1341-8_7

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