TITLE

Weak convergence of currents and cancellation

AUTHOR(S)
Sormani, Christina; Wenger, Stefan
PUB. DATE
May 2010
SOURCE
Calculus of Variations & Partial Differential Equations;May2010, Vol. 38 Issue 1/2, p183
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in terms of topology of the supports which ensure that no cancellation occurs and that the support of the weak limit agrees with the Hausdorff limit of the supports. We use our results to prove countable Hm-rectifiability of the Gromov-Hausdorff limit of sequences of Lipschitz manifolds Mn all of which are ?-linearly locally contractible up to some scale r0. In the Appendix, we show that the Gromov-Hausdorff limit need not be countably Hm-rectifiable if the Mn have a common local geometric contractibility function which is only concave (and not linear).We also relate our results towork of Cheeger-Colding on the limits of noncollapsing sequences of manifolds with nonnegative Ricci curvature.
ACCESSION #
49024227

 

Related Articles

  • Semicontinuity of eigenvalues under intrinsic flat convergence. Portegies, Jacobus // Calculus of Variations & Partial Differential Equations;Oct2015, Vol. 54 Issue 2, p1725 

    We use the theory of rectifiable metric spaces to define a Dirichlet energy of Lipschitz functions defined on the support of integral currents. This energy is obtained by integration of the square of the norm of the tangential derivative, or equivalently of the approximate local dilatation, of...

  • Compactness for manifolds and integral currents with bounded diameter and volume. Wenger, Stefan // Calculus of Variations & Partial Differential Equations;Mar2011, Vol. 40 Issue 3/4, p423 

    By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class of oriented k-dimensional Riemannian...

  • Lipschitz Functions and Ekeland's Theorem. Beer, Gerald; Ceniceros, Jose // Journal of Optimization Theory & Applications;Mar2012, Vol. 152 Issue 3, p652 

    As shown by F. Sullivan (Proc. Am. Math. Soc. 83:345-346, ), validity of the weak Ekeland variational principle implies completeness of the underlying metric space. In this note, we show that what really forces completeness in Sullivan's argument is an even simpler geometric property of lower...

  • Uniqueness for Ricci flow with unbounded curvature in dimension 2. Qing Chen; Yajun Yan // Annals of Global Analysis & Geometry;Oct2010, Vol. 38 Issue 3, p293 

    We consider the uniqueness of Ricci flow with the initial curvature bounded from above, but not necessarily bounded from below, on a 2-dimensional complete noncompact manifold.

  • Unavoidable sigma-porous sets.  // Journal of the London Mathematical Society;Oct2007, Vol. 76 Issue 2, p467 

    We prove that every separable metric space which admits an ℓ1-tree as a Lipschitz quotient has a σ-porous subset which contains every Lipschitz curve up to a set of one-dimensional Hausdorff measure zero. This applies to any Banach space containing ℓ1. We also obtain an...

  • On Hölder maps of cubes. Shchepin, E. V. // Mathematical Notes;Jun2010, Vol. 87 Issue 5/6, p757 

    A map of metric spaces f: X → Y satisfying the inequality for some C and α and all x, y ∈ X is called a Hölder map with exponent α. V. I. Arnold posed the following problem: Does there exist a Höldermap from the square onto the cube with exponent 2/3? The firstmain...

  • Tikhonov regularization of metrically regular inclusions. Gaydu, Micha�l; Geoffroy, Michel // Positivity;Jun2009, Vol. 13 Issue 2, p385 

    We present a Tikhonov regularization method for inclusions of the form $$T(x) \ni 0$$ where T is a set-valued mapping defined on a Banach space that enjoys metric regularity properties. We investigate, subsequently, the case when the mapping T is metrically regular, strongly metrically regular,...

  • ON FIXED POINTS OF GENERALIZED SET-VALUED CONTRACTIONS. BENAHMED, S.; AZÉ, D. // Bulletin of the Australian Mathematical Society;Feb2010, Vol. 81 Issue 1, p16 

    Using a variational method introduced in [D. Azé and J.-N. Corvellec, 'A variational method in fixed point results with inwardness conditions', Proc. Amer. Math. Soc. 134(12) (2006), 3577-3583], deriving directly from the Ekeland principle, we give a general result on the existence of a fixed...

  • On Some Lacunary Generalized Difference Sequence Spaces Defined by a Modulus Function in a Locally Convex Space. Bhardwaj, Vinod K.; Bala, Indu // Southeast Asian Bulletin of Mathematics;2009, Vol. 33 Issue 4, p665 

    In this paper, we introduce some new lacunary generalized difference sequence spaces defined by a modulus function in a locally convex Hausdorff topological linear space whose topology is determined by a finite set Q of seminorms. Various algebraic and topological properties of these spaces, and...

Share

Read the Article

Courtesy of VIRGINIA BEACH PUBLIC LIBRARY AND SYSTEM

Sign out of this library

Other Topics