Yi Wang
In this paper, we obtain the isoperimetric inequality on conformally flat manifold with finite total $Q$-curvature. This is a higher dimensional analogue of Li and Tam's result \cite{L-T} on surfaces with finite total Gaussian curvature. The main step in the proof is based on the construction of a quasiconformal map whose Jacobian is suitably bounded.
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The isoperimetric inequality
本文综述了等周不等式和Sobolev不等式的经典与最新证明方法，涵盖几何与分析中的关键结果。
We prove a uniform isoperimetric inequality for all time along the twisted Kähler-Ricci flow on Fano manifolds.
In this note, we show that there is some counterexample for isoperimetric inequality if the condition $(ϕ')^2-ϕ"ϕ\leq 1$ does not hold in warped product space.
An isoperimetric type inequality in De Sitter space
本文证明了de Sitter空间中类空、紧致、星形且$2$-凸超曲面的最优等周不等式。
We prove an isoperimetric-type inequality for maximal, spacelike submanifold in the Minkowski space. The argument is based on the recent work of Brendle.
An isoperimetric inequality on three-dimensional, contact, non-unimodular Lie groups
针对嵌入连通接触非幺模三维李群中的紧致有界曲面体，证明了等周不等式。
In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature, and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature.
We prove that the isoperimetric inequality is satisfied in the cigar steady soliton and in the Bryant steady soliton. Since both of them are Riemannian manifolds with warped product metric, we utilize the result of Guan-Li-Wang to get our conclusion. For the sake of the soliton structure, we believe that the geometric restrictions for manifolds in which the isoperimetric inequality holds are naturally satisfied for steady Ricci solitons.
In this article, we prove an isoperimetric inequality for the harmonic mean of the first $(n-1)$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Hyperbolic space. Our approach to prove this result also gives a similar inequality for the first $n$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Euclidean space.
The Isoperimetric Inequality in Higher-Dimensional Heisenberg Groups
本文证明了高维海森堡群中的锐利水平等周不等式，并刻画了等号成立的条件为Pansu球的左平移和膨胀。
Let $({\M}, g(t))$ be a Kähler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on $({\M}, g(t))$, and a Poincaré inequality without assuming the Ricci curvature is bounded from below.
We prove an Hersch's type isoperimetric inequality for the third positive eigenvalue on $\mathbb S^2$. Our method builds on the theory we developped to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.
The Sharp Isoperimetric Inequality in the First Heisenberg Group
本文证明了第一海森堡群中尖锐水平等周不等式，等号条件为Pansu球的左平移和伸缩。
In this note, an isoperimetric inequality for the harmonic mean of lower order Steklov eigenvalues is proved on bounded domains in noncompact rank-$1$ symmetric spaces. This work extends result of \cite{BR.01} and \cite{V.21} proved for bounded domains in Euclidean space and Hyperbolic space, respectively.
An isoperimetric inequality of minimal hypersurfaces in spheres
本文在单位球面中的闭浸入极小超曲面上建立了一个特殊等周不等式，并在常标量曲率情形下给出与超曲面无关的一致下界，同时建立了 Cheeger 等周常数与高度函数节点集体积之间的关系。
It is well known that simply connected symmetric spaces of non-positive sectional curvature admit a linear isoperimetric filling inequality for cycles of dimension greater than or equal to the rank of the space. In this note we extend that result to homogeneous Hadamard manifolds.
In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderón's and Zygmund's theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.
We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenböck type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a classification result for such metrics.
For the biharmonic Steklov eigenvalue problem considered in this paper, we show that among all bounded Euclidean domains of class $C^{1}$ with fixed measure, the ball maximizes the first positive eigenvalue.
Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow preserves the area enclosed by the evolving curve, and this fact is used to prove a general isoperimetric inequality applicable to radial graphs in warped product surfaces under weak assumptions on the warp potential.
The classical isoperimetric inequality in the Euclidean plane $\mathbb{R}^2$ states that for a simple closed curve $M$ of the length $L_{M}$, enclosing a region of the area $A_{M}$, one gets \begin{align*} L_{M}^2\geqslant 4πA_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which states that if $M$ is a closed regular simple convex curve, then \begin{align*} L_{M}^2\geqslant 4πA_{M}+8π\left|\widetilde{A}_{E_{\frac{1}{2}}(M)}\right|, \end{align*} where $\widetilde{A}_{E_{\frac{1}{2}}(M)}$ is an oriented area of the Wigner caustic of $M$, and the equality holds …
The optimal isoperimetric inequality for varifolds
本文证明了Brendle等周不等式对有限质量与全变差的varifolds成立，允许奇异部分，仅需可积平均曲率，常数在余维数一、二时最优。
In this paper, we study contact forms on the three- dimensional Heisenberg manifold with its standard CR structure. We discover that the $Q'$-curvature, introduced by Branson, Fontana and Morpurgo [BFM13] on the CR three-sphere and then generalized to any pseudo-Einstein CR three manifold by Case and Yang [CY95], controls the isoperimetric inequality on such a CR-manifold. To show this, we first prove that the nonnegative Webster curvature at infinity deduces that the metric is normal, which is analogous to the behavior on a Riemannian four-manifold.
We study, on a weighted Riemannian manifold of Ric$_{N} \geq K > 0$ for $N < -1$, when equality holds in the isoperimetric inequality. Our main theorem asserts that such a manifold is necessarily isometric to the warped product $\mathbb{R} \times_{\cosh(\sqrt{K/(1-N)}t)} Σ^{n-1}$ of hyperbolic nature, where $Σ^{n-1}$ is an $(n-1)$-dimensional manifold with lower weighted Ricci curvature bound and $\mathbb{R}$ is equipped with a hyperbolic cosine measure. This is a similar phenomenon to the equality condition of Poincaré inequality. Moreover, every isoperimetric minimizer set is isometric to a …
In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve with a simple Fourier series proof: length, area of the region included by the curve, area of the domain enclosed by the locus of curvature centers and integral of the radius of curvature. By using our isoperimetric-type inequalities, we also obtain some new geometric Bonnesen-type inequalities. Furthermore we investigate stability properties of such inequa…
Isoperimetric inequality on Finsler metric measure manifolds with non-negative weighted Ricci curvature
本文在具有非负加权 Ricci 曲率的 Finsler 度量测度流形上，通过定义体积熵和第二 Cheeger 常数，证明了精确等周不等式，并以此推导出 Finsler Laplacian 第一特征值的 Cheeger-Buser 型不等式。
Sharp weighted Carleman and Huber isoperimetric inequalities on the unit ball in higher dimensions
研究在高维单位球上建立加权卡尔曼和胡贝尔等周不等式，采用极限方法，建立了新型加权卡尔曼不等式并分类极值函数，在偶数维建立加权胡贝尔等周不等式，推广了相关结果。
We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary.
Comparison geometry for substatic manifolds and a weighted Isoperimetric Inequality
本文研究广义相对论中具有极小边界的次静态 Riemannian 流形，证明了包括尖锐加权等周不等式在内的多种几何结果，该不等式量化了边界面积最小化性质，其证明基于与 CD(0, 1) 度量共形联系的比较理论。
We settle the case of equality for the relative isoperimetric inequality outside any arbitrary convex set with not empty interior.
We prove an sharp anisotropic isoperimetric inequality for a domain outside an Euclidean ball in $\mathbb{R}^n$ for $n\geq 2$. The proof applies the ABP method to a Neumann boundary value problem.
We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in two-dimensional Alexandrov spaces of curvature $\geqslant c$. We also study the equality case.
We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Lojasiewicz-Simon inequality. A main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers.
We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2.
Isoperimetric inequality for nearly spherical domains in the Bergman ball
本文在伯格曼球中研究近球域的等周不等式，证明了相关定理并首次揭示了该领域的等周现象。
We give a new proof of the isoperimetric inequality in the plane, based on Steiner's formula for the area of a convex neighborhood. This proof establishes the isoperimetric inequality directly, without requiring that we separately establish the existence of an optimal domain. In doing so, this proof bypasses the main difficulty in all of the proofs Steiner outlined for the plane isoperimetric inequality.
Isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature
本文证明了具有非负 Ricci 曲率和 Euclidean 体积增长的测度 Finsler 流形的尖锐等周不等式，并在附加假设下证明了其刚性结果，进而推导出 Euclidean 空间中锥的加权各向异性等周不等式在不可逆情形下的刚性。
The sharp isoperimetric inequality for varifolds in codimension two
本文证明了一类varifolds的Michael-Simon不等式，在余维数2时达到尖锐，并由此得到等号仅在圆盘取得的尖锐等周不等式，证明基于Brendle-Eichmair的最优传输方法。
We prove that if $(X,\mathsf d,\mathfrak m)$ is an essentially non-branching metric measure space with $\mathfrak m(X)=1$, having Ricci curvature bounded from below by $K$ and dimension bounded from above by $N \in (1,\infty)$, understood as a synthetic condition called Measure-Contraction property, then a sharp isoperimetric inequality à la Lévy-Gromov holds true. Measure theoretic rigidity is also obtained.
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 varifold theory in the study of diffused surfaces.
We obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths, the variance of its radii, and its deviation from being convex. Our technique involves a functional minimization problem on a suitably constructed compact manifold and is based on the spectral theory for circulant matrices.
We prove a quantitative reverse isoperimetric inequality for embedded surfaces with Willmore energy bounded away from $8π$. We use this result to analyze the negative $L^2$ gradient flow of the Willmore energy plus a positive multiple of the inclosed volume. We show that initial surfaces of Willmore energy less than $8π$ with positive inclosed volume converge to a round point in finite or infinite time.
In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application we obtain a global existence result for the surface diffusion flow, providing that an initial curve is $H^2$-close to a multiply covered circle and sufficiently rotationally symmetric.
A new proof of the Wulff-Gage isoperimetric inequality and its applications
本文通过给出原点对称凸体Wulff-Gage等周不等式的新证明，进而证明了log-Minkowski问题的唯一性，并为二维$C^{2}$边界凸体提供了曲率熵log-Minkowski不等式的新证明。
Inspired by previous work of Kusner and Bauer-Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by Keller-Mondino-Rivière, our strict inequality leads to existence of minimisers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below $8π$. Besides the geometric interest, such a minimisation problem has been studied in the literature as a simplified model in the theory of lipid bilayer cell membranes.
A local isoperimetric inequality for balls with nonpositive curvature
本文研究了球体在非正曲率度量下的局部等周不等式，证明小扰动不影响等周比，并揭示了保持等周比的条件是相似变换。
} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $Δ^2$ on a bounded smooth domain $\Om$ in the Euclidean $n$-space ${\bf R}^n$ ($n\ge2$) and then prove that the corresponding first non-zero eigenvalue $Υ_1(\Om)$ admits the isoperimetric inequality of Szegö-Weinberger type: $Υ_1(\Om)\le Υ_1(B_{\Om})$, where $B_{\Om}$ is a ball in ${\bf R}^n$ with the same volume of $\Om$. The isoperimetric inequality of Szegö-Weinberger type for the first nonzero Neumann eigenvalue of the even-multi-Laplacian operators $Δ^{2m}$ ($m\ge1$) on $\Om$ is also exploited.
We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $\mathrm{Ric}_{\infty} \ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or super-level) set of the associated guiding function (arising from the needle decomposition), in terms of the deficit in Bakry-Ledoux's Gaussian isoperimetric inequality. This is the first quantitative isoperimetric inequality on noncompact spaces besides Euclidean and Gaussian spaces. Our argument makes use of Klartag's needle decomposition (also called local…
A well known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions. In dimension 2, the constant can be controlled by the integral of the positive part of the Gaussian curvature. In this paper, we showed that on simply connected conformal flat manifolds of higher dimensions, the role of the Gaussian curvature can be replaced by the Branson $Q$- curvature. We achieve this by exploring the relationship between $A_p$ weights and integrals of the Q-curvature.
We prove a sharp dimension-free isoperimetric inequality, involving the volume entropy, in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
Our main result gives an improved bound on the filling areas of closed curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu's classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds.
Employing the affine normal flow, we prove a stability version of the $p$-affine isoperimetric inequality for $p\geq1$ in $\mathbb{R}^2$ in the class of origin-symmetric convex bodies. That is, if $K$ is an origin-symmetric convex body in $\mathbb{R}^2$ such that it has area $π$ and its $p$-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, $K$ is close to an ellipse in the Hausdorff distance.
In this paper, we prove a class of weighted isoperimetric inequalities for bounded domains in hyperbolic space by using the isoperimetric inequality with log-convex density in Euclidean space. As a consequence, we remove the horo-convex assumption of domains in a weighted isoperimetric inequality proved by Scheuer-Xia. Furthermore, we prove weighted isoperimetric inequalities for star-shaped domains in warped product manifolds. Particularly, we obtain a weighted isoperimetric inequality for star-shaped hypersurfaces lying outside a certain radial coordinate slice in the anti-de Sitter-Schwarzs…
The purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdré). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetri…
Brock-type isoperimetric inequality for Steklov eigenvalues of the Witten-Laplacian
本文在加权体积固定约束下证明了欧氏与双曲空间中有界域上Witten-Laplacian的Steklov型特征值的Brock型等周不等式，拓展了F. Brock的经典结果，并提出相关开放问题。
In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov b…
We prove a generalized isoperimetric inequality for a domain diffeomorphic to a sphere that replaces filling volume with $k$-dilation. Suppose $U$ is an open set in $\mathbb{R}^n$ diffeomorphic to a Euclidean $n$-ball. We show that in dimensions at least 4 there is a map from a standard Euclidean ball of radius about $vol(\partial U)^{1/(n-1)}$ to $U$, with degree 1 on the boundary, and $(n-1)$-dilation bounded by some constant only depending on $n$. We also give an example in dimension 3 of an open set where no such map with small $(n-1)$-dilation can be found. The generalized isoperimetric i…
The Cartan-Hadamard isoperimetric inequality under controlled variation of curvature
本文在截面曲率受控变化条件下证明了任意维度的Cartan-Hadamard猜想，通过控制商极小化子的凸包并比较度量畸变与双曲等周轮廓盈余，给出关键证明步骤。
A strong quantitative form of the fractional isoperimetric inequality
本文提出分数阶定量等周不等式的强版本，推广了已有局部结果，采用与原方法核心思路不同的正则化流程完成证明，并据此推导得到分数阶Cheeger不等式的稳定性估计。
A proof of the reverse isoperimetric inequality using a geometric-analytic approach
本文通过几何-分析方法证明了爱因斯坦引力中黑洞的反等周不等式，揭示了引力在AdS黑洞视界反等周性质中的核心作用。
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