Differentiability properties of the isoperimetric profile in complete noncompact Riemannian manifolds with $$C^0$$ C 0
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fferentiability properties of the isoperimetric profile in complete noncompact Riemannian manifolds with C 0‑locally asymptotic bounded geometry Abraham Enrique Muñoz Flores1 Accepted: 23 September 2020 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract For a complete noncompact connected Riemannian manifold with bounded geometry M n , we prove that the isoperimetric profile function IMn is twice differentiable almost everywhere. Moreover, we show that a differential inequality is satisfied by IM ; extending in this way well-known results for compact manifolds due to Bavard and Pansu (Ann Sci École Norm Sup :479–490, 1986), to this class of noncompact complete Riemannian manifolds with bounded geometry. Here for C0-locally asymptotic bounded geometry we mean that for all pointed sequences pj ∈ M diverging at infinity the sequence of pointed Riemannian manifolds (M, pj , g) sub-converge in C0 topology to a limit manifold (M∞ , g∞ , p∞ ) that we assume to be at least of class C2. Keywords Differentiability of isoperimetric profile · C0-Locally bounded geometry · Finite perimeter sets Mathematics Subject Classification 49Q20 · 58E99 · 53A10 · 49Q05
1 Introduction Along this paper, we always assume that all the Riemannian manifolds (M n , g) considered are smooth with smooth Riemannian metric g. We denote by V the canonical Riemannian measure induced on M by g, and by A the (n − 1)-Hausdorff measure associated to the canonical Riemannian length space distance d of M. When it is Communicated by Claudio Gorodski. Partially supported by Capes. * Abraham Enrique Muñoz Flores [email protected] 1
Departamento de Geometria e Representação Gráfica, Instituto de Matemática e Estatística, UERJ-Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil
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Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
already clear from the context, explicit mention of the metric g will be suppressed in what follows. We give here the basic definitions of BV-functions and finite perimeter sets on a manifold. Definition 1.1 Let (M n , g) be a Riemannian manifold of dimension n, U ⊆ M an open subset, 𝔛c (U) the set of smooth vector fields with compact support on U. Given a function u ∈ L1 (M) , define the variation of u by } { udivg (X)dVg ∶ X ∈ 𝔛c (M), ||X||g,∞ ≤ 1 , |Du|g (M) ∶= sup (1) �M { } where ||X||g,∞ ∶= sup |Xp |gp ∶ p ∈ M and |Xp |gp is the norm of the vector Xp in
the metric gp on Tp M . We say that a function u ∈ L1 (M) , has bounded variation, if |Du|g (M) < ∞ and we define the set of all functions of bounded variations on M by BVg (M) ∶= {u ∈ L1 (M, g) ∶ |Du|g (M) < +∞}.
Definition 1.2 Let (M n , g) be a Riemannian manifold of dimension n. Given E ⊂ M measurable with respect to the Riemannian measure, U ⊆ M an open subset, the perimeter of E in U , Pg (E, U) ∈ [0, +∞] , is defined as } { 𝜒 div (X)dVg ∶ X ∈ 𝔛c (U), ||X||g,∞ ≤ 1 , Pg (E, U) ∶= sup (2) �U E g { } where ||X||g,∞ ∶= sup |Xp |gp ∶ p ∈ M and |Xp |gp is the norm of the vector Xp in
the metric gp on
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