Yevgeny Liokumovich, Davi Maximo
We construct singular foliations of compact three-manifolds $(M^3,h)$ with scalar curvature $R_h\geq Λ_0>0$ by surfaces of controlled area, diameter and genus. This extends Urysohn and waist inequalities of Gromov-Lawson and Marques-Neves.
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On the waist and width inequality in complete 3-manifolds with positive scalar curvature
本文证明了数量曲率有正下界的完备非紧3-流形上存在由面积和直径受控的曲面构成的奇异叶状结构，从而给出腰围与宽度不等式。
Uniform waist inequalities in codimension two for manifolds with Kazhdan fundamental group
研究具有卡扎丹基本群的闭黎曼流形M，已知其有限覆盖在余维数一时满足一致腰不等式，现证明其有限覆盖在余维数二时同样满足该不等式。
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.
Lower curvature bounds on metric surfaces via the Bishop--Gromov inequality
本文利用Bishop--Gromov不等式，为度量曲面上的下曲率界提供了刻画，建立了曲率与度量球体积增长之间的关联。
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。
The waist inequality states that for a continuous map from S^n to R^q, not all fibers can have small (n-q)-dimensional volume. We construct maps for which most fibers have small (n-q)-dimensional volume and all fibers have bounded (n-q)-dimensional volume.
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。
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。
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.
In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study $(κ,N)$-convex functions on metric spaces where $κ$ is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that is provided by $κ$. Then, in the spirit of previous work by Erbar, Kuwada and Sturm \cite{erbarkuwadasturm} we introduce an entropic curvature-dimension condition $CD^e(κ,N)$ for metric measure spaces and lower semi-continuous $κ$. This condition is stable with respect to Gromov …
In this note we prove the Heintze-Karcher inequality in the context of essentially non-branching metric measure spaces satisfying a lower Ricci curvature bound in the sense of Lott-Sturm-Villani. The proof is based on the the needle decomposition technique for metric measure spaces introduced by Cavalletti-Mondino. Moreover, in the class of spaces satisfying a Riemannian curvature-dimension condition with positive curvature the equality case is characterized.
The equivalence between timelike Ricci curvature and the timelike Brunn Minkowski inequality on synthetic Lorentzian spaces
本文研究了合成洛伦兹空间中时序里奇曲率与时序Brunn-Minkowski不等式的关系，证明了在时序q-非分支设定下，q-时序曲率维度条件与时序Brunn-Minkowski条件等价，扩展了在非光滑设定下的研究成果。
We prove a sharp dimension-free isoperimetric inequality, involving the volume entropy, in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
The curvature dimension condition CD(K,N), pioneered by Sturm and Lott--Villani, is a synthetic notion of having curvature bounded below and dimension bounded above, in the non-smooth setting. This condition implies a suitable generalization of the Brunn--Minkowski inequality, denoted by BM(K,N). In this paper, we address the converse implication in the setting of weighted Riemannian manifolds, proving that BM(K,N) is in fact equivalent to CD(K,N). Our result allows to characterize the curvature dimension condition without using neither the optimal transport nor the differential structure of t…
We present Chen-Ricci inequality and improved Chen-Ricci inequality for curvature like tensors. Applying our improved Chen-Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms and C-totally real submanifolds of Sasakian space forms.
A logarithmic Sobolev inequality for closed submanifolds with constant length of mean curvature vector
该研究在微分几何领域证明了非负截面曲率流形中具常长度平均曲率向量的闭子流形满足对数Sobolev不等式，丰富了子流形几何的分析工具。
We will show the Cheeger-Colding segment inequality for manifolds with integral Ricci curvature bound. By using this segment inequality, the almost rigidity structure results for integral Ricci curvature will be derived by a similar method as in \cite{CC1}. And the sharp Hölder continuity result of \cite{CoN} holds in the limit space of manifolds with integral Ricci curvature bound.
We introduce a new version of a curvature-dimension inequality for non-negative curvature. We use this inequality to prove a logarithmic Li-Yau inequality on finite graphs. To formulate this inequality, we introduce a non-linear variant of the calculus of Bakry and Émery. In the case of manifolds, the new calculus and the new curvature-dimension inequality coincide with the common ones. In the case of graphs, they coincide in a limit. In this sense, the new curvature-dimension inequality gives a more general concept of curvature on graphs and on manifolds. We show that Ricci-flat graphs have a…
We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t μ- P_t ν\|_1 \leq \frac{W_1(μ,ν)}{\sqrt{t}} \] for all probability measures $μ,ν$ where $P_t$ is the heat semigroup and $W_1$ is the $\ell_1$-Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincaré inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the the eigenvalue estimate $λ_1 \geq \log(2)/\operatorname{diam}^2$.
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 have a specific Euclid-like property. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature. This extends a recent result of S. Brendle with Euclidean setting.
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 present in this note a lower bound for the Calabi functional in a given Kähler class. This yields an integral inequality for constant scalar curvature metrics, which can be viewed as a refined version of Yau's Chern number inequality.
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…
For an asymptotically Poincare-Einstein manifold with a lower Ricci curvature bound, we establish a sharp inequality relating the type II Yamabe invariant of the interior and the Yamabe invariant of its conformal infinity
We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary.
We give a simple proof of a recent result due to Agostiniani, Fogagnolo and Mazzieri.
In this paper, we prove several Poincaré inequalities of fractional type on conformally flat manifolds with finite total Q-curvature. This shows a new aspect of the $Q$-curvature on noncompact complete manifolds.
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.
Sobolev Inequality on Manifolds With Asymptotically Nonnegative Bakry-Émery Ricci Curvature
本文受已有工作启发，在具有密度且Bakry-Émery Ricci曲率渐近非负的流形上证明了Sobolev不等式。
In this paper, we study the power of Gaussian curvature flow of a compact convex hypersurface and establish its Harnack inequality when the power is negative. In the Harnack inequality, we require that the absolute value of the power is strictly positive and strictly less than the inverse of the dimension of the hypersurface.
We study a pointwise inequality for submanifolds in real space forms involving the scalar curvature, the normal scalar curvature and the mean curvature. We translate it into an algebraic problem, allowing us to prove a slightly weaker version of it. We also prove the conjecture for certain types of submanifolds of $\mathbb C^n$.
We state and prove a Chern-Osserman Inequality in terms of the volume growth for minimal surfaces properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity.
We show a connection between the $CDE'$ inequality and the $CDψ$ inequality. In particular, we introduce a $CD_ψ^φ$ inequality as a slight generalization of $CDψ$ which turns out to be equivalent to $CDE'$ with appropriate choices of $φ$ and $ψ$. We use this to prove that the $CDE'$ inequality implies the classical $CD$ inequality on graphs, and that the $CDE'$ inequality with curvature bound zero holds on Ricci-flat graphs.
Spacelike surfaces in Generalized Robertson-Walker spacetimes whose mean curvature function satisfies a natural nonlinear inequality are analyzed. Several uniqueness and nonexistence results for such compact spacelike surfaces are proved. In the nonparametric case, new Calabi-Bernstein type problems are solved as a consequence.
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.
A sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature
本文利用黎曼几何标准比较方法，在渐近非负曲率的完备非紧流形中为闭超曲面建立了锐利几何不等式，并简化了非负里奇曲率情形下Willmore型不等式的证明。
Sobolev Inequality In Manifolds With Lower Quadratic Curvature Decay
本文利用ABP方法，在具有下二次曲率衰减的完备黎曼流形中，为紧致区域和子流形建立了Sobolev不等式。
The Wintgen inequality (1979) is a sharp geometric inequality for surfaces in the 4-dimensional Euclidean space involving the Gauss curvature (intrinsic invariant) and the normal curvature and squared mean curvature (extrinsic invariants), respectively. In the present paper we obtain a Wintgen inequality for statistical surfaces.
A 2-systolic inequality on non-rational compact Kähler surfaces with positive scalar curvature
本文研究了具有非常数全纯映射到正亏曲线的紧正标量曲率凯勒表面的2-系统不等式，证明了此类表面必为纤维化 over 复曲线的有理面。
In this paper, we obtain a basic Chen's inequality for a C-totally real submanifold in a generalized $(κ,μ)$-contact space forms involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvature on the right hand side. Inequalities between the squared mean curvature and Ricci curvature and between the squared mean curvature and $k$-Ricci curvature are also obtained.
Generalized existence of extremizers for the sharp $p$-Sobolev inequality on Riemannian manifolds with nonnegative curvature
研究非负曲率黎曼流形上sharp p-塞伯利格不等式极值函数的广义存在性，通过非光滑方法和Cheeger能量收敛性分析，揭示极值函数的全局行为。
Let $(M,d)$ be a separable and complete geodesic space with curvature lower bounded, by $κ\in \mathbb R$, in the sense of Alexandrov. Let $μ$ be a Borel probability measure on $M$, such that $μ\in\mathcal P_2(M)$, and that has at least one barycenter $x^{*}\in M$. We show that for any geodesically $α$-convex function $f:M\to \mathbb R$, for $α\in \mathbb R$, the inequality \[f(x^*)\le \int_M (f -\fracα{2}d^2(x^*,.))\,{\rm d}μ,\] holds provided $f$ is locally Lipschitz at $x^*$ and either positive or in $L^1(μ)$. Our proof relies on the properties of tangent cones at barycenters and on the exis…
Fenchel-Willmore-Chen inequality under lower bounds on weighted intermediate Ricci curvature
针对具1-Bakry-Émery k-里奇曲率下界的光滑度量测度空间，建立Fenchel-Willmore-Chen不等式并给出统一证明，同时证明非负1-加权里奇曲率的Sobolev与等周不等式。
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.
We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--Émery (via energy and Γ_2$-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $L^2$-Wasserstein distance.
In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the $C^\infty$ topology. As an application, we prove a Minkowski type inequality for star-shaped, $F$-mean convex hypersurfaces.
We analyze an upper bound on the curvature of a Riemannian manifold, using "root-Ricci" curvature, which is in between a sectional curvature bound and a Ricci curvature bound. (A special case of root-Ricci curvature was previously discovered by Osserman and Sarnak for a different but related purpose.) We prove that our root-Ricci bound implies Günther's inequality on the candle function of a manifold, thus bringing that inequality closer in form to the complementary inequality due to Bishop.
A Willmore-type inequality for hypersurfaces with asymptotic or integral Ricci curvature bounds
本文研究完备非紧 Riemann 流形有界区域的几何不等式，在渐近或积分 Ricci 曲率界条件下建立 Willmore 型不等式，并推广恢复 Jin-Yin 的近期结果。
Quantitative stability in distribution for the Sobolev inequality under curvature dimension condition
研究${\sf CD}(N - 1,N)$度量测度空间中临界Sobolev不等式的定量稳定性。在最优常数与圆球相近时，证明几乎极值函数的累积分布在Wasserstein距离上接近圆球上Aubin - Talenti气泡的分布，对数Sobolev不等式和谱隙也有类似结果。
Harnack Inequality for $f$-Mean Curvature Flow
本文研究了$f$-均曲率流的Harnack不等式，证明了其作为加权面积泛函梯度流的估计。
On fill-ins with scalar curvature bounded from below and an inequality of Hijazi-Montiel-Roldán
本文研究了标量曲率下界为$-n(n-1)$的旋量流形填充问题，通过结合Hijazi-Montiel-Roldán不等式与Bär定理，证明了Gromov猜想，并提供了该不等式的新证明方法。
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.
The Log-Sobolev inequality for a submanifold in manifolds with asymptotic non-negative intermediate Ricci curvature
本文在具有渐近非负中间里奇曲率和欧氏体积增长的完备非紧黎曼流形上，证明了子流形的尖锐对数-Sobolev不等式，推广了Dong-Lin-Lu等人的已有结果。
A Poincaré-Einstein metric $g$ is called non-degenerate if there are no non-zero infinitesimal Einstein deformations of $g$, in Bianchi gauge, that lie in $L^2$. We prove that a 4-dimensional Poincaré-Einstein metric is non-degenerate if it satisfies a certain chiral curvature inequality. Write $R_+$ for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if $R_+$ is negative definite then $g$ is non-degenerate. This is a chiral generalisation of a result due to Biquard and Lee, that a Poincaré-Einstein metric of negative sectional curvature is non-degenerate
In this article, a proof of the interpolation inequality along geodesics in $p$-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel-Brascamp-Lieb inequality for general Riemannian and Finsler manifolds and led Lott-Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli's recently developed calculus on metric measure spaces, even a $q$-Laplacian comparison theorem holds on $q$-infinitesimal conve…
Willmore-type inequality for closed hypersurfaces in complete manifolds with Ricci curvature bounded below
本文在Ricci曲率有下界的完备流形中建立闭超曲面的Willmore型不等式，推广经典结果至负曲率情形，并应用于双曲空间得到测地球面的刻画。
We introduce and study the conical curvature-dimension condition, $CCD(K,N)$, for graphs. We show that $CCD(K,N)$ provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincaré inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs. Another application of the conical curvature-dimension analysis is finding a sharp estimate on the curvature of complete graphs.
We settle the case of equality for the relative isoperimetric inequality outside any arbitrary convex set with not empty interior.
We obtain an explicit formula for comparing total curvature of level sets of functions on Riemannian manifolds, and develop some applications of this result to the isoperimetric problem in spaces of nonpositive curvature.
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。
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