Tootfinder

This https://arxiv.org/abs/2211.15575 has been replaced.
link: https://scholar.google.com/scholar?q=a

Hyperbolicity and bounded-valued cohomology
We generalise a theorem of Gersten on surjectivity of the restriction map in $\ell^{\infty}$-cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and $\ell^{\infty}$-cohomology calculations for some well-known classes of groups. Along the way, we obtain hyperbolicity criteria for groups of type $FP_2(\mathbb Q)$ and for those satisfying a rational homological linear isoperimetric inequality, answering a ques…

Sharp embedding results and geometric inequalities for H\"{o}rmander vector fields
Hua Chen, Hong-Ge Chen, Jin-Ning Li
https://arxiv.org/abs/2404.19393 https://arxiv.org/pdf/2404.19393
arXiv:2404.19393v1 Announce Type: new
Abstract: Let $U$ be a connected open subset of $\mathbb{R}^n$, and let $X=(X_1,X_{2},\ldots,X_m)$ be a system of H\"{o}rmander vector fields defined on $U$. This paper addresses sharp embedding results and geometric inequalities in the generalized Sobolev space $\mathcal{W}_{X,0}^{k,p}(\Omega)$, where $\Omega\subset\subset U$ is a general open bounded subset of $U$. By employing Rothschild-Stein's lifting technique and saturation method, we prove the representation formula for smooth functions with compact support in $\Omega$. Combining this representation formula with weighted weak-$L^p$ estimates, we derive sharp Sobolev inequalities on $\mathcal{W}_{X,0}^{k,p}(\Omega)$, where the critical Sobolev exponent depends on the generalized M\'{e}tivier index. As applications of these sharp Sobolev inequalities, we establish the isoperimetric inequality, logarithmic Sobolev inequalities, Rellich-Kondrachov compact embedding theorem, Gagliardo-Nirenberg inequality, Nash inequality, and Moser-Trudinger inequality in the context of general H\"{o}rmander vector fields.

This https://arxiv.org/abs/2002.00914 has been replaced.
link: https://scholar.google.com/scholar?q=a

The relative isoperimetric inequality for minimal submanifolds with free boundary in the Euclidean space
In this paper, we mainly consider the relative isoperimetric inequalities for minimal submanifolds with free boundary. We first generalize ideas of restricted normal cones introduced by Choe-Ghomi-Ritoré in \cite{CGR06} and obtain an optimal area estimate for generalized restricted normal cones. This area estimate, together with the ABP method of Cabré in \cite{Cabre2008}, provides a new proof of the relative isoperimetric inequality obtained by Choe-Ghomi-Ritoré in \cite{CGR07}. Furthermore…

Phase transitions in isoperimetric problems on the integers
Joseph Briggs, Chris Wells
https://arxiv.org/abs/2402.14087 https://arxiv…

Phase transitions in isoperimetric problems on the integers
Barber and Erde asked the following question: if $B$ generates $\mathbb Z^n$ as an additive group, then must the extremal sets for the vertex/edge-isoperimetric inequality on the Cayley graph $\operatorname{Cay}(\mathbb Z^n,B)$ form a nested family? We answer this question negatively for both the vertex- and edge-isoperimetric inequalities, specifically in the case of $n=1$. The key is to show that the structure of the cylinder $\mathbb Z\times(\mathbb Z/k\mathbb Z)$ can be mimicked in certain …

This https://arxiv.org/abs/2403.08075 has been replaced.
link: https://scholar.google.com/scholar?q=a

Several isoperimetric inequalities of Dirichlet and Neumann eigenvalues of the Witten-Laplacian
In this paper, by mainly using the rearrangement technique and suitably constructing trial functions, under the constraint of fixed weighted volume, we can successfully obtain several isoperimetric inequalities for the first and the second Dirichlet eigenvalues, the first nonzero Neumann eigenvalue of the Witten-Laplacian on bounded domains in space forms. These spectral isoperimetric inequalities extend those classical ones (i.e. the Faber-Krahn inequality, the Hong-Krahn-Szegő inequality and…

Willmore-type inequalities for closed hypersurfaces in weighted manifolds
Guoqiang Wu, Jia-Yong Wu
https://arxiv.org/abs/2404.16286 https://

Willmore-type inequalities for closed hypersurfaces in weighted manifolds
In this paper, we prove some Willmore-type inequalities for closed hypersurfaces in weighted manifolds with nonnegative Bakry-Émery Ricci curvature. In particular, we give a sharp Willmore-like inequality in shrinking gradient Ricci solitons. These results can be regarded as generalizations of Agostiniani-Fogagnolo-Mazzieri's Willmore-type inequality in weighted manifolds. As applications, we derive some isoperimetric type inequalities under certain existence assumptions of isoperimetric regio…

This https://arxiv.org/abs/2305.11799 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Estimates for the lowest Neumann eigenvalues of parallelograms and domains of constant width
We prove sharp upper bounds for the first and second non-trivial eigenvalues of the Neumann Laplacian in two classes of domains: parallelograms and domains of constant width. This gives in particular a new proof of an isoperimetric inequality for parallelograms recently obtained by A. Henrot, A. Lemenant and I. Lucardesi.

This https://arxiv.org/abs/2403.05712 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions
The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric inequalities relating the volumes of a convex body and its difference body and polar projection body, respectively. Following a classical work by Schneider, both inequalities have been extended to the so-called higher-order setting. In this work, we establish the higher-order analogues for these inequalities in the setting of log-concave functions. In particular, this extends the Zhang's inequality for…

This https://arxiv.org/abs/2209.13842 has been replaced.
link: https://scholar.google.com/scholar?q=a

Isoperimetric inequalities for Neumann eigenvalues on bounded domains in rank-1 symmetric spaces
In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space [13], and Szegö-Weinberger inequality in rank-1 symmetric spaces obtained by Aithal and Santhanam [1].

Brock-type isoperimetric inequality for Steklov eigenvalues of the Witten-Laplacian
Jing Mao, Shijie Zhang
https://arxiv.org/abs/2404.07412 https://…

Brock-type isoperimetric inequality for Steklov eigenvalues of the Witten-Laplacian
In this paper, by imposing suitable assumptions on the weighted function, (under the constraint of fixed weighted volume) a Brock-type isoperimetric inequality for Steklov-type eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven. This conclusion is actually an interesting extension of F. Brock's classical result about the isoperimetric inequality for Steklov eigenvalues of the Laplacian given in the influential paper [Z. Angew. Math.…

Finding Fibonacci in the Hyperbolic Plane
MurphyKate Montee
https://arxiv.org/abs/2404.04389 https://arxiv.org/pdf/2404.04389<…

Finding Fibonacci in the Hyperbolic Plane
We use a combinatorial approximation of the hyperbolic plane to investigate properties of hyperbolic geometry such as exponential growth of perimeter and area of disks, and the linear isoperimetric inequality. This calculations give a surprising link to Fibonacci numbers.

Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions
Dylan Langharst, Francisco Mar\'in Sola, Jacopo Ulivelli
https://arxiv.org/abs/2403.05712

Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions
The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric inequalities relating the volumes of a convex body and its difference body and polar projection body, respectively. Following a classical work by Schneider, both inequalities have been extended to the so-called higher-order setting. In this work, we establish the higher-order analogues for these inequalities in the setting of log-concave functions. In particular, this extends the Zhang's inequality for…

This https://arxiv.org/abs/2403.08070 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-Laplacian
An isoperimetric inequality for lower order nonzero Neumann eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven in this paper. About this conclusion, we would like to point out two things: It strengthens the well-known Szegő-Weinberger inequality for nonzero Neumann eigenvalues of the classical free membrane problem given in [J. Rational Mech. Anal. 3 (1954) 343-356] and [J. Rational Mech. Anal. 5 (1956) 633-636]; Recently, Xia-…

An isoperimetric inequality for the first Robin-Dirichlet eigenvalue of the Laplacian
Nunzia Gavitone, Gianpaolo Piscitelli
https://arxiv.org/abs/2404.06607

An isoperimetric inequality for the first Robin-Dirichlet eigenvalue of the Laplacian
In this paper, we study the first eigenvalue of the Laplacian on doubly connected domains when Robin and Dirichlet conditions are imposed on the outer and the inner part of the boundary, respectively. We provide that the spherical shell reaches the maximum of the first eigenvalue of this problem among the domains with fixed measure, outer perimeter and inner $(n-1)^{th}$ quermassintegral.

This https://arxiv.org/abs/2404.06607 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An isoperimetric inequality for the first Robin-Dirichlet eigenvalue of the Laplacian
In this paper, we study the first eigenvalue of the Laplacian on doubly connected domains when Robin and Dirichlet conditions are imposed on the outer and the inner part of the boundary, respectively. We provide that the spherical shell reaches the maximum of the first eigenvalue of this problem among the domains with fixed measure, outer perimeter and inner $(n-1)^{th}$ quermassintegral.

This https://arxiv.org/abs/2404.06607 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An isoperimetric inequality for the first Robin-Dirichlet eigenvalue of the Laplacian
In this paper, we study the first eigenvalue of the Laplacian on doubly connected domains when Robin and Dirichlet conditions are imposed on the outer and the inner part of the boundary, respectively. We provide that the spherical shell reaches the maximum of the first eigenvalue of this problem among the domains with fixed measure, outer perimeter and inner $(n-1)^{th}$ quermassintegral.