Title: Some geometric inequalities by the ABP method
\nAuthors: Doanh Pham
\nCategories: math.DG math.AP
\nComments: to appear in International Mathematics Research Notices
\n\\\\
\n In this paper, we apply the so-called Alexandrov-Bakelman-Pucci (ABP) method
\nto establish some geometric inequalities. We first prove a logarithmic Sobolev
\ninequality for closed $n$-dimensional minimal submanifolds $\\Sigma$ of $\\mathbb
\nS^{n+m}$. As a consequence, it recovers the classical result that $|\\mathbb
\nS^n| \\leq |\\Sigma|$ for $m = 1,2$. Next, we prove a Sobolev-type inequality for
\npositive symmetric two-tensors on smooth domains in $\\mathbb R^n$ which was
\nestablished by D. Serre when the domain is convex. In the last application of
\nthe ABP method, we formulate and prove an inequality related to
\nquermassintegrals of closed hypersurfaces of the Euclidean space.
\n\\\\ ( https:\/\/arxiv.org\/abs\/2305.05819 , 17kb)<\/p>\n","protected":false},"excerpt":{"rendered":"