On perimeter minimizing sets in manifolds with quadratic volume growth

Abstract This paper studies whether the presence of a perimeter minimizing set in a Riemannian manifold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (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.