Mikhail G. Katz
We prove a new inequality relating volume to length of closed geodesics on area minimizers for generic metrics on the complex projective plane. We exploit recent regularity results for area minimizers by Moore and White, and the Kronheimer--Mrowka proof of the Thom conjecture.
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We determine non-Hopf hypersurfaces with constant mean curvature in the complex projective plane which attain equality in a basic inequality between the maximum Ricci curvature and the squared mean curvature.
In this short note, we prove the Miyaoka-Yau inequality for minimal projective $n$-manifolds of general type by using Kähler-Ricci flow.
The first paper in systolic geometry was published by Loewner's student P. M. Pu over half a century ago. Pu proved an inequality relating the systole and the area of an arbitrary metric on the real projective plane. We prove a stronger version of Pu's systolic inequality with a remainder term.
We prove entropy rigidity for finite volume strictly convex projective manifolds in dimensions $\geq 3$, generalizing the work of arXiv:1708.03983 to the finite volume setting. The rigidity theorem uses the techniques of Besson, Courtois, and Gallot's entropy rigidity theorem. It implies uniform lower bounds on the volume of any finite volume strictly convex projective manifold in dimensions $\geq 3$.
We prove an isoperimetric inequality for the second non-zero eigenvalue of the Laplace-Beltrami operator on the real projective plane. For a metric of the unit area this eigenvalue is not greater than 20π. This value is attained in the limit by a sequence of metrics of area one on the projective plane. The limiting metric is singular and could be realized as a union of the projective plane and the sphere touching at a point, with standard metrics and the ratio of the areas 3:2. It is also proven that the multiplicity of the second non-zero eigenvalue on the projective plane is at most 6.
A sharp quantitative Alexandrov inequality and applications to volume preserving geometric flows in 3D
本文研究三维保体积平均曲率流和 Mullins-Sekerka 流的渐近行为，并建立了带周长界的 $C^2$ 集合的尖锐定量 Alexandrov 不等式，为其应用提供理论基础。
In this paper, we prove that principal circle bundles over the complex projective space equipped with the standard Sasakian structures are volume rigid among all $K$-contact manifolds satisfying positivity conditions of tensors involing the Tanaka-Webster curvature.
We prove a conjecture saying that complex projective space has maximal volume (degree) among all toric Kaehler-Einstein manifolds of dimension n. The proof is inspired by our recent work on sharp Moser-Trudinger and Brezis-Merle type inequalities for the complex Monge-Ampere operator, but is essentially self-contained.
Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in \cite{BG}, then the following properties hold: 1 The volume doubling property; 2 The Poincaré inequality; 3 The parabolic Harnack inequality. The key ingredient is the study of dimensional reverse log-Sobolev inequalities for the heat semi…
To a complex projective structure $Σ$ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\|ϕ_Σ\|_\infty$ and $\|ϕ_Σ\|_2$ of the quadratic differential $ϕ_Σ$ of $Σ$ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well known results for convex cocompact hyperbolic structures on 3-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination.…
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.
We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k\subset \mathbb{RP}^n$ (starting with points). This confirms a conjecture of Burago and Zalgaller from 1988 and extends to higher dimensions previous result of M. Ritoré and A. Ros on $\mathbb{RP}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in $\mathbb{S}^n$.
We generalize an inequality of Besson-Courtois-Gallot about volume and simplicial volume of closed manifolds to the $\ell_1$-norm of all the homology classes of complete manifolds. The inequality involves the critical exponent of the fundamental group of the manifold and the mass of the homology classes.
We give an elementary proof to the asymptotic expansion formula of Rochon-Zhang for the unique complete Kähler-Einstein metric of Cheng-Yau, Kobayashi, Tian-Yau and Bando on quasi-projective manifolds. The main tools are the solution formula for second order ODE's with constant coefficients and spectral theory for Laplacian operator on a closed manifold.
Let $(M,g)$ be a closed Riemannian manifold and $\{ω_p\}_{p=1}^{\infty}$ be the volume spectrum of $(M,g)$. We will show that $ω_{k+m+1}\leq ω_k+ω_m+W$ for all $k,m\geq 0$, where $ω_0=0$ and $W$ is the one-parameter Almgren-Pitts width of $(M,g)$. We will also prove the similar inequality for the $\varepsilon$-phase-transition spectrum $\{c_{\varepsilon}(p)\}_{p=1}^{\infty}$ using the Allen-Cahn approach.
An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.
On the continuity of Weil-Petersson volumes of the moduli space weighted points on the projective line
本文证明了射影直线上加权点的模空间中，从Fano几何趋近Calabi-Yau几何时，Weil-Petersson体积连续，且局部化计算的CM体积收敛到McMullen的几何体积。
Projective transformations of convex bodies and volume product
本文研究了体积积泛函在固定凸体小射影变形空间中的变分性质，简短证明了 Klartag 定理，并在二维情形下展示了一族包含该泛函鞍点的临界凸体。
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.
We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to $\R^{n}\setminus Ω, n\ge 3$, and so that their boundary is a minimal hypersurface. (Here, $Ω\subset \R^{n}$ is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by $(V/β_{n})^{(n-2)/n}$, where $V$ is the Euclidean volume of $Ω$ and $β_{n}$ is the volume of the Euclidean unit $n$-ball. This gives a partial proof to a conjecture of Bray and Iga \cite{brayiga}. Surprisingly, we do not require the boundary to be outermost.
Given a warped product space $\mathbb{R} \times_{f} N$ with logarithmically convex warping function $f$, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given a $k$--dimensional region $F \subset \{b\} \times N$, and the horizontal graph $C \subset \mathbb{R} \times_{f} N$ of an almost everywhere differentiable map over $F$, we prove that the $k$--volume of $C$ is always at least the $k$--volume of the smooth constant height graph ove…
Behaviour of the Schwarzian derivative on long complex projective tubes
本文研究Schwarzian导数在长复射影管上的行为，得到其实部与无穷小地震及移植配对的估计，并由此推导出重整化体积在复地震路径下的变化界及压缩曲线时的渐近行为。
We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.
In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic $\mathbb{R}P^{2m}$ in $\mathbb{C}P^n$. Then it minimizes volume among the isotropic submanifolds in the same $\mathbb{Z}/2$ homology class in $\mathbb{C}P^n$ (but not among all submanifolds in this $\mathbb{Z}/2$ homology class). Also the totally geodesic $\mathbb{R}P^{2m-1}$ minimizes volume in its Hamiltonian deformation class in $\mathbb{C}P^n$. As a corollary we'll give estimates for volumes of Lagrangian submanifolds in complete intersections in $\mathbb{C}P^n$.
We explore connections furnished by the Funk metric, a relative of the Hilbert metric, between projective geometry, billiards, convex geometry and affine inequalities. We first show that many metric invariants of the Funk metric are invariant under projective transformations as well as projective duality. These include the Holmes-Thompson volume and surface area of convex subsets, and the length spectrum of their boundary, extending results of Holmes-Thompson and Álvarez Paiva on Schäffer's dual girth conjecture. We explore in particular Funk billiards, which generalize hyperbolic billiards in…
Two geodesically (projectively) equivalent Finsler metrics determine a set of invariant volume forms on the projective sphere bundle. Their proportionality factors are geodesically invariant functions and hence they are first integrals. Being 0-homogeneous functions, the first integrals are common for the entire projective class. In Theorem 1.1 we provide a practical and easy way of computing these first integrals as the coefficients of a characteristic polynomial.
In this thesis we describe how minimal surface techniques can be used to prove the Penrose inequality in general relativity for two classes of 3-manifolds. We also describe how a new volume comparison theorem involving scalar curvature for 3-manifolds follows from these same techniques.
On the $L^2$ volume of Bergman spaces
本文证明了射影流形与射影空间乘积上Bergman空间的Calabi体积和Mabuchi体积为无限，受Shiffman-Zelditch猜想启发。
We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent $n \ge 3$, then it has exactly the $n$-dimensional volume growth. As an application, if an $n$-dimensional Finsler manifold of non-negative $n$-Ricci curvature satisfies the Caffarelli-Kohn-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space.
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。
This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz $φ$-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz $φ$-radial addition of two star bodies, we derive a formula for the Orlicz $L_ϕ$-dual mixed volume. Moreover, a dual Orlicz-Minkowski inequality for the Orlicz $L_ϕ$-dual mixed volume, a dual Orlicz isoperimetric inequality for the Orlicz $L_ϕ$-dual surface area and a dual Orlicz-Urysohn inequality for the Orlicz $L_ϕ$-harmonic mean radius are…
We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are obtained. We deduce these results by proving a general central limit theorem for the weighted volume of the convex hull of random points chosen from the boundary of a smooth convex body according to a positive and continuous density in Euclidean space. In the background are geometric estimates for weighted surface bodies and Berry-Esseen bounds for functionals of i…
We prove that a bounded open set U in Euclidean n-space has k-width less than C(n) Volume(U)^{k/n}. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in Euclidean space. In particular, we estimate the smallest (n-1)-dilation of any degree 1 map between two n-dimensional rectangles. For any pair of rectangles, our estimate is accurate up to a dimensional constant C(n). We give examples in which the (n-1)-dilation of the linear map is bigger than the optimal value by an arbitrarily large factor.
We prove an inequality of Bonnesen type for the real projective plane, generalizing Pu's systolic inequality for positively-curved metrics. The remainder term in the inequality, analogous to that in Bonnesen's inequality, is a function of R-r (suitably normalized), where R and r are respectively the circumradius and the inradius of the Weyl-Lewy Euclidean embedding of the orientable double cover. We exploit John ellipsoids of a convex body and Pogorelov's ridigity theorem.
The paper investigates higher dimensional analogues of Burago's inequality bounding the area of a closed surface by its total curvature. We obtain sufficient conditions for hypersurfaces in 4-space that involve the Ricci curvature. We get semi-local variants of the inequality holding in any dimension that involve domains with non-vanishing Gauss-Kronecker curvature. The paper also contains inequalities of isoperimetric type involving the total curvature, as well as a "reverse" isoperimetric inequality for spaces with constant curvature.
We generalize an inequality of E. Heintze and H. Karcher [8] for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for $k$-Ricci curvature. Even in the case of a pointwise bound, this generalizes the classical inequality by replacing a sectional curvature bound with a $k$-Ricci bound. This work is motivated by the estimates of Petersen-Shteingold-Wei for the volume of tubes around a geodesic [12] and generalizes their result. Using similar ideas we also prove a Hessian comparison theorem for $k$-Ricci curvature which generalizes the usual Hessian and Lap…
In an earlier paper \cite{mazeng} the authors introduced the notion of curvature entropy, and proved the plane log-Minkowski inequality of curvature entropy under the symmetry assumption. In this paper we demonstrate the plane log-Minkowski inequality of curvature entropy for general convex bodies. The equivalence of the uniqueness of cone-volume measure, the log-Minkowski inequality of volume, and the log-Minkowski inequality of curvature entropy for general convex bodies in $\mathbb R^{2}$ are shown.
The article treats some questions around Gromov's filling area conjecture. It intended to show that any filling with volume $< 2 π$ would not be attained and to show a local systolic inequality, implying an a priori lower estimate on the total volume. The proof of both assertions, the second of which is wrong, suffered from a computational mistake.
By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincaré inequality and Harnack inequality. As a consequence, we obtain that the dimension of space of harmonic functions on graphs with polynomial growth is fi…
Lipschitz-Volume rigidity and Sobolev coarea inequality for metric surfaces
本文证明闭度量曲面到同面积闭黎曼曲面的1-Lipschitz满射必为等距映射，建立度量曲面上连续Sobolev函数的余面积不等式（推广Esmayli-Ikonen-Rajala结果），并研究非光滑目标空间下映射的正则性。
It was proved in Chen's paper \cite{chen} that every real hypersurface in the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$ δ(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature and $δ(2)$ is a $δ$-invariant introduced by him. In this paper, we study non-Hopf real hypersurfaces satisfying the equality case of the inequality under the condition that the mean curvature is constant along each integral curve of the Reeb vector field. We describe how to obtain all such hypersurfaces.
Illumination Bodies in Projective Geometries
本文将照明体的概念推广到常曲率黎曼空间和投影芬斯勒几何，证明其体积导数定义了这些背景下凸体的表面积，推广了欧氏空间中的仿射表面积。
Based on the isoperimetric inequality, G. Huisken proposed a definition of total mass in general relativity that is equivalent to the ADM mass for (smooth) asymptotically flat 3-manifolds of nonnegative scalar curvature, but that is well-defined in greater generality. In a similar vein, we use the isocapacitary inequality (bounding capacity from below in terms of volume) to suggest a new definition of total mass. We prove an inequality between it and the ADM mass, and prove the reverse inequality with harmonically flat asymptotics, or, with general asymptotics, for exhaustions by balls (as opp…
Mass, Conformal Capacity, and the Volumetric Penrose Inequality
本文证明了渐近平坦流形的 ADM 质量与共形容量之间的尖锐不等式，导出了体积 Penrose 不等式及其稳定性结果，核心方法是基于 3-调和函数等值集的单调性公式。
Rigidity of projective area widths and a weighted Crofton formula on $RP^3$
本文证明 $RP^3$ 上黎曼度量是曲面 Zoll 度量当且仅当四个射影面积宽度相等，并建立加权 Crofton 公式，导出 systole、体积与宽度公共值的不等式。
We determine the expected curvature polynomial of random real projective varieties given as the zero set of independent random polynomials with Gaussian distribution, whose distribution is invariant under the action of the orthogonal group. In particular, the expected Euler characteristic of such random real projective varieties is found. This considerably extends previously known results on the number of roots, the volume, and the Euler characteristic of the solution set of random polynomial equations
The Extremals of Minkowski's Quadratic Inequality
本文通过将任意凸体的混合体积表示为与高度退化椭圆算子相关联的Dirichlet形式，利用算子的定量刚性性质，完全解决了Minkowski二次不等式的极值问题并证实了R. Schneider的猜想。
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