Related papers: Geometric inequalities and the Alexandrov-Bakelman…

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…

We prove Michael-Simon type Sobolev inequalities for $n$-dimensional submanifolds in $(n+m)$-dimensional Riemannian manifolds with nonnegative $k$-th intermediate Ricci curvature by using the Alexandrov-Bakelman-Pucci method. Here…

We establish the validity of the isoperimetric inequality (or equivalently, an $L^1$ Euclidean-type Sobolev inequality) on manifolds with asymptotically non-negative sectional curvature. Unlike previous results in the literature, our…

We present the proof of several inequalities using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, we give a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan…

In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in…

In this paper, we uncover a novel connection between the Fenchel-Willmore inequality and a new logarithmic Sobolev inequality for mean-convex submanifolds immersed in non-negatively curved manifolds with Euclidean volume growth. Building on…

Inspired by [1, 13], we prove Michael-Simon type inequalities for smooth symmetric uniformly positive define (0, 2)-tensor fields on compact submanifolds in Euclidean space by the Alexandrov-Bakelman-Pucci (ABP) method.

We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.

We prove the Michael-Simon-Sobolev inequality for smooth symmetric uniformly positive definite (0, 2)-tensor fields on compact submanifolds with or without boundary in Riemannian manifolds with nonnegative sectional curvature by the…

We establish a sharp Fenchel-Willmore inequality for closed submanifolds of arbitrary dimension and codimension immersed in a complete Riemannian manifold with non-negative intermediate Ricci curvature and Euclidean volume growth. In the…

We review recent results regarding the problem of the stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds. We shall describe techniques and methods from smooth and non-smooth geometry, the fruitful…

In this paper we establish higher-order Sobolev and Rellich-type inequalities on non-compact Riemannian manifolds supporting an isoperimetric inequality. We highlight two notable settings: manifolds with non-negative Ricci curvature and…

We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic…

By using optimal mass transport theory we prove a sharp isoperimetric inequality in ${\sf CD} (0,N)$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for…

For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of…

We prove a sharp Log-Sobolev inequality for submanifolds of a complete non-compact Riemannian manifold with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth. Our work extends a result of Dong-Lin-Lu which…

We prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifold with nonnegative sectional curvature of arbitrary dimension and codimension, while the ambient manifold needs to…

In this paper, we investigate the reverse improvement property of Sobolev inequalities on manifolds with quadratically decaying Ricci curvature. Specifically, we establish conditions under which the uniform decay of the heat kernel implies…

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian…

Using the ABP-method as in a recent work by Brendle, we establish some sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds in a complete Riemannian manifold with asymptotically nonnegative curvature. These…