Minkowski inequalities and constrained inverse curvature flows in warped spaces

Abstract

This paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows, we prove long-time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases, new Minkowski type inequalities are the consequence. In higher dimensions, we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.

On an inverse curvature flow in two-dimensional space forms

“…The isoperimetric inequality is another typical application (also one that we perform here, for both convex and non-convex cases -see Theorem 5.2). There has been recently a burst of activity in the area, including inequalities involving quermassintegrals, Minkowski-type inequalities, Alexandrov-Fenchel inequalities and more general weighted inequalities [7,8,9,10,11,17,18,29,30,21,22,27,28,35].…”