Inverse curvature flows in Riemannian warped products

Abstract

The long-time existence and umbilicity estimates for compact, graphical solutions to expanding curvature flows are deduced in Riemannian warped products of a real interval with a compact fibre. Notably we do not assume the ambient manifold to be rotationally symmetric, nor the radial curvature to converge, nor a lower bound on the ambient sectional curvature. The inverse speeds are given by powers p ≤ 1 of a curvature function satisfying few common properties.

Stability of the PMT and RPI for asymptotically hyperbolic manifolds foliated by IMCF

“…For our purposes here the results of Scheuer [29,30] and the author [3] are particularly significant. In [29,30] Scheuer gives long time existence and asymptotic analysis results for rotationally symmetric metrics with nonpositive radial curvature and later generalizing to warped products. The asymptotic results therein imply that after rescaling you will find thatΣ t converges to a C 2,α hypersurface but you cannot conclude it is a sphere, as expected for aymptotically hyperbolic spaces.…”

“…The outward pointing normal vector will be well defined in our case since we have in mind considering M 3 to by asymptotically hyperbolic manifolds with one end. For a glimpse of long time existence and asymptotic analysis results for smooth IMCF in various ambient manifolds see [1,3,12,15,16,29,30,31,33]. For our purposes here the results of Scheuer [29,30] and the author [3] are particularly significant.…”

“…For a glimpse of long time existence and asymptotic analysis results for smooth IMCF in various ambient manifolds see [1,3,12,15,16,29,30,31,33]. For our purposes here the results of Scheuer [29,30] and the author [3] are particularly significant. In [29,30] Scheuer gives long time existence and asymptotic analysis results for rotationally symmetric metrics with nonpositive radial curvature and later generalizing to warped products.…”

A class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold

“…With some assumptions on the initial hypersurface, he proved that the flow exists for all time and the hypersurfaces become more and more umbilic. Furthermore, Brendle et al [1] and Scheuer [12] showed the above facts still hold in the antide Sitter-Schwarzschild (AdS-Schwarzschild) manifold and in a class of warped product manifolds. The inverse curvature flows are useful tools in proving geometric inequalities.…”

Minkowski inequalities and constrained inverse curvature flows in warped spaces

“…The idea on how to exploit its monotonicity properties comes from [5], also see [19], in which the hyperbolic case is treated. The convergence of the inverse curvature flows in general warped products was proven in [31], also see [40]. However note that the latter work does not cover the required asymptotics in the ambient spaces we are considering here.…”