Communications in Analysis and Geometry

Volume 25 (2017)

Number 3

ABP estimate and geometric inequalities

Pages: 685 – 708



Chao Xia (School of Mathematical Sciences, Xiamen University, Xiamen, China)

Xiangwen Zhang (Department of Mathematics, University of California at Irvine)


In this paper, we present the proofs for several geometric inequalities using the Alexandrov–Bakelman–Pucci (ABP) estimate on Riemannian manifolds. First, we give new proofs of the Heintze–Karcher inequality for mean convex domain on manifolds with nonnegative Ricci curvature and the classical Minkowski inequality for mixed volumes. Then, we prove the anisotropic Heintze–Karcher inequality. Along the new approach, we also establish an anisotropic version of ABP estimate which may be of independent interest.

Research of the first author is supported in part by the Fundamental Research Funds for the Central Universities (Grant No.20720150012), NSFC (Grant No.11501480) and CRC Postdoc Fellowship. Research of the second author is supported in part by NSF Grant: DMS-1308136 and DMS-1605968.

Received 18 September 2015

Published 13 September 2017