geometric inequalities

Title: Some geometric inequalities by the ABP method Authors: Doanh Pham Categories: math.DG math.AP Comments: to appear in International Mathematics Research Notices \\ In this paper, we apply the so-called Alexandrov-Bakelman-Pucci (ABP) method to establish some geometric inequalities. We first prove a logarithmic Sobolev inequality for closed $n$-dimensional minimal submanifolds $\Sigma$ of $\mathbb S^{n+m}$. As a consequence, it recovers the classical result that $|\mathbb S^n| \leq |\Sigma|$ for $m = 1,2$. Next, we prove a Sobolev-type inequality for positive symmetric … Continue reading “Title Some geometric inequalities by the ABP method…”