Wanxing Liu
In this short note, we offer an observation that the Miyaoka-Yau inequality holds for any compact Kähler manifold with nef canonical bundle, i.e. a smooth minimal model. It follows directly from the existence of cscK metrics in a neighborhood of the canonical class which was confirmed both by the work of Dyrefelt and Song using different approaches.
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On a compact Kähler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kähler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kähler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle, contains no rational curves and satisfies some Miyaoka-Yau type inequalities. In the course of the discussions, we attach a real value to any fixed Kähler class which, up to a constant factor depending only on the dimension of manifold,…
In this short note, we prove the Miyaoka-Yau inequality for minimal projective $n$-manifolds of general type by using Kähler-Ricci flow.
On Miyaoka-Yau Inequalities and Weil-Petersson Metrics
本文用Kähler-Ricci流重新证明Miyaoka-Yau不等式与斜率半稳定性，并揭示半丰沛典范线丛情形下Miyaoka-Yau量等于Weil-Petersson与Fubini-Study度量相交，等号条件为多重典范映射成为全纯纤维丛。
In this paper, we prove the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle. The key point of the proof is the estimate for the $L^2$-norm of the scalar curvature along the Kähler-Ricci flow.
After establishing suitable notions of stability and Chern classes for singular pairs, we use Kähler-Einstein metrics with conical and cuspidal singularities to prove the slope semistability of orbifold tangent sheaves of minimal log-canonical pairs of log general type. We then proceed to prove the Miyaoka-Yau inequality for all minimal pairs with standard coefficients. Our result in particular provides an alternative proof of the Abundance theorem for threefolds that is independent of positivity results for tangent sheaves.
On the Miyaoka-Yau inequality for manifolds with nef anti-canonical line bundle
本文在 K. Zhang 工作的基础上，结合 Fujita–Odaka 的 delta 不变量下界条件，研究具有 nef 反典范线丛的射影流形上的 Miyaoka–Yau 型不等式。
Equality in the Miyaoka-Yau inequality and uniformization of non-positively curved klt pairs
本文研究了紧Kähler klt对在orbifold Miyaoka-Yau不等式中等号成立的条件，证明了此时其orbifold泛覆盖为单位球或仿射空间。
The Miyaoka-Yau inequality for minimal Kähler klt spaces
本文通过在紧Kähler klt空间正则轨迹上得到自反Higgs层的广义Bogomolov不等式，建立所有极小Kähler klt空间的宫冈-丘不等式，并通过Gauduchon orbifolds上的分析结果推导奇异情形下无挠层的相关性质。
The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors
研究为具有大典范或反典范除子的奇异簇建立宫冈-丘不等式，定义非多极积，建立相关博戈莫洛夫-吉泽克型不等式，从而证明了特定射影klt簇的宫冈-丘不等式。
The Miyaoka-Yau inequality and the delta invariant for Fano varieties I -klt Fano case-
研究针对\(n\)维klt法诺簇建立宫冈 - 丘不等式，还受他人启发在孤子情形下证明其等变形式，用等变陈类和加权δ不变量取代相关量，得出允许凯勒 - 里奇孤子的光滑法诺流形的等变宫冈 - 丘不等式。
A Miyaoka-Yau inequality for hyperplane arrangements in $\mathbb{CP}^n$
研究超平面排列的稳定性条件，利用Bogomolov-Gieseker不等式证明二次形式非正，推导出交子空间乘数总和的下界，扩展Hirzebruch在复射影平面中线排列的条件。
In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature-dimension inequality $CDE'(n,K)$, which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, heat kernel bounds and Cheng's eigenvalue estimate. These are first kind of results on this direction for unbounded Laplacian on graphs.
We show that a problem by Yau can not be true in general. The counterexamples are constructed based on the recent work of Wu and Zheng.
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 partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound $R(X)$ to the existence of conical Kahler-Einstein metrics on a Fano manifold $X$. In particular, if $D\in |-K_X|$ is a smooth simple divisor and the Mabuchi $K$-energy is bounded below, then there exists a unique conical Kahler-Einstein metric satisfying $Ric(g) = βg + (1-β) [D]$ for any $β\in (0,1)$. We also construct unique smooth conical toric Kahler-Einstein metrics with $β=R(X)$ and a unique effective Q-divisor $D\in [-K_X]$ for all toric Fano manifolds. Finally we prove a Miyaoka-Yau type inequality f…
We introduce a modified non-linear heat equation $\partial_t u = Δu + Γu$ as a substitute of $\log P_t f$ where $P_t$ is the heat semigroup. We prove an exponential decay of $Γu$ under the Bakry Emery curvature condition $CD(K,\infty)$ and prove the Li-Yau inequality $-Δu_t \leq \frac{n}{2t}$ under the Bakry Emery curvature condition $CD(0,n)$. From this, we deduce the volume doubling property which solves a major open problem in discrete Ricci curvature. As an application, we show that there exist no expander graphs satisfying $CD(0,n)$.
In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel.
We give a proof to the Li-Yau-Hamilton type inequality claimed by Perelman on the fundamental solution to the conjugate heat equation. The rest of the paper is devoted to improving the known differential inequalities of Li-Yau-Hamilton type via monotonicity formulae.
Let $Ω$ be a compact and mean-convex domain with smooth boundary $Σ:=\partialΩ$, in an initial data set $(M^3,g,K)$, which has no apparent horizon in its interior. If $Σ$ is spacelike in a spacetime $(\E^4,g\_\E)$ with spacelike mean curvature vector $\mathcal{H}$ such that $Σ$ admits an isometric and isospin immersion into $\mathbb{R}^3$ with mean curvature $H\_0$, then: \begin{eqnarray*} \int\_Σ|\mathcal{H}|dΣ\leq\int\_Σ\frac{H\_0^2}{|\mathcal{H}|}dΣ. \end{eqnarray*} If equality occurs, we prove that there exists a local isometric immersion of $Ω$ in $\mathbb{R}^{3,1}$ (the Minkowski spaceti…
A Li-Yau inequality for the 1-dimensional Willmore energy
本研究将曲面的Li-Yau不等式推广至一维曲线情形，探讨涉及欧拉弹性能量等曲率泛函的类似结果，并应用于相关梯度流。
In this paper, we study Li-Yau gradient estimates for the solutions $u$ to the heat equation $\partial_tu=Δu$ on graphs under the curvature condition $CD(n,-K)$ introduced by Bauer et al. in \cite{BHLLMY}. As applications, we derive Harnack inequalities and heat kernel estimates on graphs. Also we present a type of Hamilton gradient estimates.
Chung-Grigor'yan-Yau's inequality describes upper bounds of eigenvalues of Laplacian in terms of subsets ("input") and their volumes. In this paper we will show that we can reduce "input" in Chung-Grigor'yan-Yau's inequality in the setting of Alexandrov spaces satisfying CD$(0,\infty)$. We will also discuss a related conjecture for some universal inequality among eigenvalues of Laplacian.
In a seminal paper published in 1980, P. C. Yang and S.-T. Yau proved an inequality bounding the first eigenvalue of the Laplacian on an orientable Riemannian surface in terms of its genus $γ$ and the area. The equality in Yang-Yau's estimate is attained for $γ=0$ by an old result of J. Hersch and it was recently shown by S. Nayatani and T. Shoda that it is also attained for $γ=2$. In the present article we combine techniques from algebraic geometry and minimal surface theory to show that Yang-Yau's inequality is strict for all genera $γ> 2$. Previously this was only known for $γ=1$. In the se…
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.
We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.
Li-Yau type inequality for curves in any codimension
本研究针对任意余维数欧氏空间中的浸入曲线建立Li-Yau型不等式，给出弯曲能下界，证明多数情形下的最优性及奇数重数平面闭曲线的非最优性，采用变分法结合相关分类与定理，还探讨了相关应用。
Let $(X,d,μ)$ be a $RCD^\ast(K, N)$ space with $K\in mathbb{R}$ and $N\in [1,\infty)$. Suppose that $(X,d)$ is connected, complete and separable, and $\supp μ=X$. We prove that the Li-Yau inequality for the heat flow holds true on $(X,d,μ)$ when $K\ge 0$. A Baudoin-Garofalo inequality and Harnack inequalities for the heat flows are established on $(X,d,μ)$ for general $K\in \mathbb{R}$. Large time behaviors of heat kernels are also studied.
A Dynamical Approach to the Berezin-Li-Yau Inequality
通过体积保持平均曲率流和Riesz均值$R_\Lambda(\Omega_t)$的新单调性原理，证明了凸域上的尖锐Berezin-Li-Yau不等式。
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。
We show that harmonic spinors obey a strengthened version of the well-known pointwise Kato inequality for sections of a vector bundle with a connection. We then prove a decay estimate for eigenspinors using this Kato-Yau estimate and resulting differential inequality. We briefly describe some applications to gauge theory---specifically to integral estimates which are used when gluing and ungluing PU(2) monopoles (math.DG/9907107).
We prove a linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow. We then use this sharp differential inequality to study the Liouville properties of the plurisubharmonic functions on complete Kaehler manifolds with nonnegative bisectional curvature.
In this note, we investigate the well-known Yau rigidity theorem for minimal submanifolds in spheres. Using the parameter method of Yau and the DDVV inequality verified by Lu, Ge and Tang, we prove that if $M$ is an $n$-dimensional oriented compact minimal submanifold in the unit sphere $S^{n+p}(1)$, and if $K_{M}\geq\frac{sgn(p-1)p}{2(p+1)},$ then $M$ is either a totally geodesic sphere, the standard immersion of the product of two spheres, or the Veronese surface in $S^4(1)$. Here $sgn(\cdot)$ is the standard sign function. We also extend the rigidity theorem above to the case where $M$ is a…
We prove a generalization of the Li-Yau estimate for a board class of second order linear parabolic equations. As a consequence, we obtain a new Cheeger-Yau inequality and a new Harnack inequality for these equations. We also prove a Hamilton-Li-Yau estimate, which is a matrix version of the Li-Yau estimate, for these equations. This results in a generalization of Huisken's monotonicity formula for a family of evolving hypersurfaces. Finally, we also show that all these generalizations are sharp in the sense that the inequalities become equalities for a family of fundamental solutions, which h…
We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to Ricci flow on a 2-dimensional closed manifold. This result can be regarded as a nonlinear version of the previous work of Y. Zheng and the author (Arch. Math. 94 (2010), 591-600).
Global existence for the Willmore flow with boundary via Simon's Li-Yau inequality
本文利用修改后的 Simon 的 Li-Yau 不等式和几何测度论工具，证明了初始能量低于特定阈值的带 Dirichlet 边界数据的圆柱旋转曲面 Willmore 流的全局存在性，并改进了平稳问题中 Willmore 极小化子的存在性结果。
A nonlinear Li-Yau inequality and its consequences
该研究针对非负Ricci曲率闭黎曼流形上的归一化抛物型p-拉普拉斯方程，证明了尖锐非线性Li-Yau不等式，推导了相关推论及尖锐全局Harnack不等式。
We study an algebraic inequality for nilpotent matrices and show some interesting geometric applications: (i) obtaining topological information for nilpotent polystable Higgs bundles over a compact Riemann surface; (ii) obtaining a sharp upper bound of the holomorphic sectional curvatures of the period domain and the Hodge metric on the Calabi-Yau moduli.
In this note we show how results in \cite{BaudoinBonnefont2016, BaudoinGarofalo2013, CoulhonJiangKoskelaSikora2017} yield the Cheng-Yau estimate on two classes of sub-Riemannian manifolds: Carnot groups and sub-Riemannian manifolds satisfying a generalized curvature-dimension inequality.
Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c_1=0 which are never Kahler; e.g., we produce such manifolds with b_1=0=b_3 and also with c_2.omega <0, answering questions posed by Smith-Thomas-Yau. Examples come from Riemannian geometry, via the Levi-Civita connection on Lambda^+. The…
Cheng-Yau logarithmic gradient estimates for a nonlinear elliptic equation on smooth metric measure spaces
本文在m-Bakry-Émery Ricci曲率有下界的完备光滑测度空间上，利用Nash-Moser迭代技术研究非线性椭圆方程，获得了正解的局部梯度估计，并推导出Liouville型定理和Harnack不等式，推广并改进了已有结果。
A short proof of Bernstein's theorem via the Omori--Yau maximum principle
本文利用Omori-Yau最大值原理，通过构造显式函数并应用逐点微分不等式，简洁地证明了三维空间中完整极小图必为平面的Bernstein定理。
In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a total mean curvature comparison theorem for isometric embeddings of a surface of positive Gaussian curvature. The Riemannian version corresponds to an earlier theorem of Shi and Tam \cite{st}. In this article, we generalize such an inequality to the case when the Gaussian curvature of the surface is allowed to be negative. This is done by an isometric embedding into the hyperboloid in the Minkowski s…
We derive a matrix version of Li \& Yau--type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R.~Hamilton did in~\cite{hamilton7} for the standard heat equation. We then apply these estimates to obtain some Harnack--type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.
The first eigenvalue of the Laplacian on orientable surfaces
本文推广Ros的论证方法，利用到任意n>0维复射影空间的全纯映射，对多数亏格大于3的可定向曲面，给出了Yang-Yau不等式的定量改进结果。
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…
Let $u$ be a harmonic function in the unit ball $B(0,1) \subset \mathbb{R}^n$, $n \geq 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that $H^{n-1}(\{u=0 \}\cap B) \geq c$. We prove Nadirashvili's conjecture as well as its counterpart on $C^\infty$-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact $C^\infty$-smooth Riemannian manifold $M$ (without boundary) of dimension $n$ there exists $c>0$ such that for any Laplace eigenfunction $φ_λ$ …
We give a short proof of the systolic inequality for the n-dimensional torus. The proof uses minimal hypersurfaces. It is based on the Schoen-Yau proof that an n-dimensional torus admits no metric of positive scalar curvature.
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