Yi Wang
This paper continues the study of Alexandrov-Fenchel inequalities for quermassintegrals for $k$-convex domains. It focuses on the application to the Michael-Simon type inequalities for $k$-curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a domain.
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Michael-Simon Sobolev inequalities in Euclidean space
本文利用 Alexandrov-Bakelman-Pucci (ABP) 方法，证明了欧几里得空间中紧致子流形上光滑对称一致正定(0, 2)-张量场的 Michael-Simon 型不等式。
The Michael-Simon-Sobolev inequality on manifolds for positive symmetric tensor fields
本研究利用ABP方法，在非负截面曲率黎曼流形的紧子流形上，建立了光滑对称一致正定(0,2)张量场的Michael-Simon-Sobolev不等式，推广了S. Brendle的已有结果。
The anisotropic Michael-Simon inequality for surfaces in every codimension
该研究针对任意余维数的R^N中2-varifold，在各向异性接近面积泛函时证明Michael-Simon不等式，引入各向异性应力测度投影方法得到适用于任意维数和余维数的几何不等式。
The anisotropic Michael-Simon inequality
本文证明了任意维数和余维数varifolds的各向异性Michael-Simon不等式，结合相关定理给出超曲面正则性，还得出varifolds的密度界与紧性性质，证明依赖投影方法及消失质量猜想的解决。
Proof of the Michael-Simon-Sobolev inequality using optimal transport
本文用最优输运方法重新证明了 Michael-Simon-Sobolev 不等式，并指出该不等式在余维数为 2 的子流形情形下达到最优。
Optimal Transport Approach to Michael-Simon-Sobolev Inequalities in Manifolds with Intermediate Ricci Curvature Lower Bounds
该研究将最优输运的McCann定理推广至子流形场景，证明了满足中间Ricci曲率下界条件的流形内子流形的Michael-Simon-Sobolev不等式，涵盖了已有相关研究结论的变体情形。
Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space
本文利用局部约束的平均曲率流和逆曲率流，在双曲空间中针对具有较弱凸性条件的超曲面，成功证明了关于平均曲率及高阶平均曲率的精确Michael-Simon型不等式。
Scale-invariant bounds on thin sets for measures satisfying a first-order PDE and the anisotropic Michael-Simon inequality
本文证明了受偏微分方程约束的测度在σ-有限H^m测度集合上的质量估计，并据此推导出满足原子条件(AC1)的积分项下具有有界各向异性一阶变分的varifolds的Michael-Simon不等式。
Michael-Simon type inequalities in hyperbolic space $\mathbb{H}^{n+1}$ via Brendle-Guan-Li's flows
本文利用 Brendle-Guan-Li 流，在 $h$-凸条件下建立了双曲空间中平均曲率及第 $k$ 个平均曲率的锐化 Michael-Simon 不等式，并推广了 Minkowski 型及加权 Alexandrov-Fenchel 不等式。
Sobolev inequalities in manifolds with nonnegative intermediate Ricci curvature
本文利用Alexandrov-Bakelman-Pucci方法，将Michael-Simon型Sobolev不等式推广到具有非负中间Ricci曲率的Riemannian流形中的子流形上，并由此导出极小子流形的等周不等式。
Michael-Simon inequality for anisotropic energies close to the area via multilinear Kakeya-type bounds
本文研究了近似面积的各向异性能量的Michael-Simon不等式，通过多线性Kakeya型界和凸性条件，证明了在特定条件下该不等式成立，并探讨了其在不同积分算子中的应用。
Sobolev and Michael-Simon inequalities via the ABP method beyond Euclidean volume growth
本文通过改进Brendle的接触集论证，在体积非坍塌假设下发展了ABP方法，证明了包含低阶项的Michael-Simon型不等式和Varopoulos的L1-Sobolev不等式，并给出了等周轮廓的下界。
Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvature
该研究在渐近非负曲率流形上证明等周不等式成立，在紧集外非负曲率流形上推导Michael-Simon不等式，采用ABP方法并结合几何与谱性质完成证明，拓展了相关不等式的适用条件。
In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface is of smoothly compact starshaped, then the solution of the flow (1.5) exists for all time and converges to a sphere in smooth topology. Following this flow argument, not only do we achieve a new proof of the celebrated sharp Michael-Simon inequality for mean curvature in (n+1) dimensional Euclidean space, but we also get the necessary and sufficient condition for the establishment of the equality. In the second part of this paper, we study a mean curvature type fl…
The sharp $L^2$-Michael--Simon inequality
该研究证明了针对 $n \geq 3$ 的等距浸入的尖锐 $L^2$-Michael--Simon 不等式，刻画取等条件，利用共形不变性推广到其他流形子流形，并推导 Chen--Fenchel--Willmore 型不等式。
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 volume ratio of the ambient manifold.
Sobolev inequalities involving 2-tensor fields in manifolds with nonnegative sectional curvature
本文应用ABP方法，在非负截面曲率流形中为光滑对称一致正定(0,2)张量场建立了Log Sobolev型及Michael Simon Sobolev不等式。
A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the case of Lagrangian submanifolds or affine hypersurfaces.
Sobolev Inequalities in Spacelike Submanifolds of Minkowski Space
本文将ABP估计方法应用于Minkowski空间$\mathbb R^{n,1}$中的类空子流形，导出了Michael-Simon型不等式，并在超曲面平均凸的条件下得到了不含平均曲率项的Sobolev不等式。
Regularity of varifolds with bounded anisotropic first variation
本文证明各向异性被积函数下具有$L^p$平均曲率的varifolds的$\varepsilon$-正则性定理，建立任意余维数的各向异性Allard正则性，并给出满足条件但不接近椭球范数的各向异性实例。
We apply our abstract gradient inequalities developed by the authors in arXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz--Simon gradient inequalities for the harmonic map energy function generalize those of Kwon (2002), Liu and Yang (2010), Simon (1983, 1985), and Topping (1997).
Geometric and Analytic Aspects of Simon-Lojasiewicz Inequalities on Vector Bundles
本文研究向量丛上Simon-Lojasiewicz不等式的几何与解析性质，概述向量丛上的泛函与变分问题理论，并将其应用于任意实解析泛函及$S^{n-1}$上的能量泛函。
Geometric inequalities and the Alexandrov-Bakelman-Pucci technique
本文基于Alexandrov-Bakelman-Pucci技术，统一讨论了几何不等式的证明框架，涵盖欧几里得空间的等周不等式、子流形的均曲率不等式及Sobolev不等式等核心方法与贡献。
In this sequel to arXiv:1510.03817, we apply our abstract Lojasiewicz-Simon gradient inequality to prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions generalize that of the pure Yang-Mills energy function due to the first author (Theorems 23.1 and 23.17 in arXiv:1409.1525) for base manifolds of dimensions two, three and four and due to Rade (1992) for dimensions two and thre…
In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle, and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by two results which, once casted in the light of this relation, are almost self evident. The first is a genuine, 5-lines proof, for the isoperimetric inequality for $k$-cycles in $\mathbb{R}^n$, with a constant differing from the optimal constant by a factor of only $\sqrt{k}$, as o…
We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasiewicz proved by Simon (1983). We also prove that the optimal exponent of the Lojasiewicz-Simon gradient inequality is obtained when the functional is Morse-Bott, improving on similar results due to Chill (2003, 2006), Haraux and Jendoubi (2007), and Simon (1996). In our article ar…
Łojasiewicz-Simon inequalities for minimal networks: stability and convergence
本文证明平面三重点网络的Łojasiewicz-Simon梯度不等式，并由此建立极小网络在曲率运动下的稳定性与平滑收敛性，同时构造了收敛到退化网络的实例。
Alexandrov's Theorem for Integral Varifolds and Applications to Geometric Inequalities
本文通过最优质量输运，为积分varifolds建立了等周和Michael-Simon-Sobolev不等式，推广了Brendle-Eichmair结果，并推广了Alexandrov定理，常数在余维2时最优。
Scale-Separated Curvature Estimates for Strongly Stable Hypersurfaces
本文通过将Young不等式替换为Hölder不等式，简化了曲率估计的论证过程，得到了更小的常数，并自然扩展到常均曲率（CMC）设定中。研究还证明了在强稳定的CMC超曲面中，Hölder机制产生了一个具有两个竞争项的积分曲率估计，并在特定条件下将CMC估计转化为最小曲面形式。
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。
For any compact Lie group $G$ and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\geq 2$, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal $G$-bundle over $X$ supporting a connection with $L^p$-small curvature, when $p>d/2$, to the case of a connection with $L^{d/2}$-small curvature. We prove an optimal Lojasiewicz-Simon gradient inequality for abstract Morse-Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis in arXiv:1510.03817. We apply this result to prove the optimal Lojasiewicz-Si…
Jacobian estimates and geometric inequalities under intermediate Ricci curvature
本文证明了一个适用于任意余维子流形的 Jacobian 估计，无需曲率符号假设，并由此在中间 Ricci 曲率条件下推导出定量 Fenchel--Willmore、Sobolev 和等周不等式，所需条件仅与子流形维数相关。
In this note we generalize an extension theorem in [5] and [9] of the mean curvature flow to the H^{k} mean curvature flow under some extra conditions. The main difficult problem in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the H^{k} mean curvature flow, and to do a suitable Moser iteration process. These two problems are overcame by imposing some extra conditions which may be weakened or removed in our forthcoming paper [7]. On the other hand, we derive some estimates for the generalized mean curvature flow, which have their own interesting.
Normal Scalar Curvature Inequality on a Class of Austere Submanifolds
本文研究了一类绝热子流形的正常标量曲率不等式，通过改进的DDVV型不等式得出新结果，并给出了达到等号的例子及Simons型间隙定理。
In this note, we use Warren-Yuan's super isoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two dimensional Lagrangian mean curvature equation. We assume the Lagrangian phase to be supercritical with bounded second derivatives. Unlike the previous approach, the simplified approach in this proof does not require the Michael-Simon mean value and Sobolev inequalities on generalized submanifolds of $\mathbb{R}^n$.
In this note we prove uniqueness of nondegenerate compact blowups for the motion by curvature of planar networks. The proof follows ideas introduced in "Lojasiewicz-Simon inequalities for minimal networks: stability and convergence" for the study of stability properties of critical points of the length functional and it is based on the application of a Lojasiewicz-Simon gradient inequality.
On the Geometry of Complete Spacelike LW-Submanifolds in Locally Symmetric Semi-Riemannian Spaces
研究完全类空LW子流形在局部对称半黎曼空间中的几何特性，通过刚性结果和分析技术推导出关于曲率的尖锐不等式，并统一了已知的分类定理。
The PDE-ODI principle and cylindrical mean curvature flows
本文提出PDE-ODI原理，用于分析柱状模型的均值曲率流古解和奇点，通过将抛物型微分方程转化为常微分不等式系统，获得高阶渐近展开，统一了经典结果并证明了碗溶iton的唯一性。
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