Abstract
AbstractFor the quermassintegral inequalities of horospherically convex hypersurfaces in the $$(n+1)$$
(
n
+
1
)
-dimensional hyperbolic space, where $$n\ge 2$$
n
≥
2
, we prove a stability estimate relating the Hausdorff distance to a geodesic sphere by the deficit in the quermassintegral inequality. The exponent of the deficit is explicitly given and does not depend on the dimension. The estimate is valid in the class of domains with upper and lower bound on the inradius and an upper bound on a curvature quotient. This is achieved by some new initial value-independent curvature estimates for locally constrained flows of inverse type.
Funder
Engineering and Physical Sciences Research Council
Johann Wolfgang Goethe-Universität, Frankfurt am Main
Publisher
Springer Science and Business Media LLC
Reference16 articles.
1. Borisenko, A., Miquel, V.: Total curvatures of convex hypersurfaces in hyperbolic space. Ill. J. Math. 43(1), 61–78 (1999)
2. De Rosa, A., Gioffré, S.: Quantitative stability for anisotropic nearly umbilical hypersurfaces. J. Geom. Anal. 29(3), 2318–2346 (2019)
3. Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182(1), 167–211 (2010)
4. Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168(3), 941–980 (2008)
5. Gerhardt, C.: Curvature problems, Series in Geometry and Topology, vol. 39. International Press of Boston Inc., Sommerville (2006)