Sobolev Inequality In Manifolds With Lower Quadratic Curvature Decay

TL;DR

The paper establishes Sobolev-type inequalities for compact domains and compact submanifolds in complete Riemannian manifolds under the assumption of lower quadratic curvature decay, extending the ABP method of Cabré and Brendle to curved settings. The authors define the decay framework via a nonincreasing function $\lambda$, construct comparison functions $h_1,h_2$ solving $h_i''=\lambda h_i$, and derive explicit constants involving the asymptotic volume ratio $\theta$, the curvature-decay parameter $B$, and $b_1$, along with the radius parameter $r_0$. For domains, they obtain an inequality that bounds boundary and gradient terms against an $L^{\frac{n}{n-1}}$ norm of $f$; for submanifolds, a parallel inequality includes mean curvature $|H|$. The results yield isoperimetric-type consequences and broaden the scope of Sobolev inequalities in noncompact manifolds with curvature decay, with precise constants dictated by geometric data.

Abstract

By using the ABP method developed by Cabré and Brendle, we establish some Sobolev inequalities for compact domains and submanifolds in a complete Riemannian manifold with lower quadratic curvature decay

Theorems & Definitions (20)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof