What question did this study set out to answer?

The research aims to investigate the conditions under which perimeter minimizing sets lead to isometric splitting in Riemannian manifolds.

April 4, 2026Journal für die reine und angewandte Mathematik (Crelles Journal)0 citationsOpen Access

On perimeter minimizing sets in manifolds with quadratic volume growth

ACAlessandro CucinottaMathematical Institute of the Slovak Academy of Sciences MMMattia MagnaboscoMathematical Institute of the Slovak Academy of Sciences

Key Points

The research aims to investigate the conditions under which perimeter minimizing sets lead to isometric splitting in Riemannian manifolds.
Analyzed Riemannian manifolds with non-negative sectional curvature
Examined the implications of quadratic volume growth at infinity
Identified boundary characteristics of perimeter minimizing sets
Establishes a connection between perimeter minimizing sets and isometric splitting in certain manifolds
Demonstrates that non-negative sectional curvature guarantees this relationship
Shows that the boundary of the minimizing set aligns with a slice in the manifold's product structure

Abstract

Abstract This paper studies whether the presence of a perimeter minimizing set in a Riemannian manifold ( M , g ) (M,g) forces an isometric splitting. We show that this is the case when 𝑀 has non-negative sectional curvature and quadratic volume growth at infinity. Moreover, we obtain that the boundary of the perimeter minimizing set is identified with a slice in the product structure of 𝑀.

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