Generalized existence of isoperimetric regions in non-compact Riemannian manifolds and applications to the isoperimetric profile. (English) Zbl 1297.49081

Let \((M,g)\) be a complete Riemannan manifold. 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 metric \(d\) of \(M\). The isoperimetric profile function \(I_M:[0,V(M)[\to [0,\infty[\) is defined by \(I_M(v)=\inf\{A(\partial \Omega):\Omega\in \tau_M, V(\Omega)=v\}\) for \(v\neq 0\) and \(I_M(0)=0\), where \(\tau_M\) denotes the set of relatively compact open subsets of \(M\) with smooth boundary. The author considers instead of \(\tau_M\) the subsets of finite perimeter. The main result of the present paper is Theorem 2 which provides a generalized existence result for isoperimetric regions in a noncompact Riemannian manifold satisfying the condition of bounded geometry. In the general case solutions do not exist in the original ambient manifold, but rather in the disjoint union of a finite number of pointed limit manifolds \(M_{1,\infty},\dots M_{N,\infty}\), obtained as limit of \(N\) sequences of pointed manifolds \((M,p_{ij},g)_j, i\in\{1,\dots,N\}\). After the proof of Theorem 2, one obtains a decomposition lemma for the thick part of a subsequence of an arbitrary minimizing sequence, which extracts exactly the structure of such sets, important for the study of the isometric profile.


49Q20 Variational problems in a geometric measure-theoretic setting
49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58E99 Variational problems in infinite-dimensional spaces
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