Yuli Rudyak
Let $S$ be a set of critical points of a smooth real-valued function on a closed manifold $M$. Generalizing a well-known result of Lusternik--Schnirelmann, Reeken~[R] proved that $\cat S \geq \cat M$. Here we prove a generalization of Reeken"s inequality for gradient-like flows on compact spaces.
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In this paper, we study the gradient Ricci soliton equation on a complete Riemannian manifold. We show that under a natural decay condition on the Ricci curvature, the Ricci soliton is Ricci-flat and ALE.
We prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension $n=2$. This then implies that the mean curvature flow exists for all time and converges to a translating solution.
In this short note, using an argument by Munteanu and Wang, we provide an alternative proof of the fact first obtained by Lei Ni that shrinking gradient Kähler-Ricci solitons with positive bisectional curvature must be compact.
Remarks on almost Riemann solitons with gradient or torse-forming vector field
本文研究黎曼流形中近Riemann孤立子与近Ricci孤立子的关联，利用Bochner公式给出梯度型近Riemann孤立子的λ函数显式表达式，并推导螺管式、扭形势向量场情形下的曲率相关性质。
Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons
本文证明在标量曲率满足R≥(n-1)λ的收缩梯度Ricci孤子中，任何紧致双侧光滑稳定极小超曲面的第二基本形式和法向Ricci曲率必须为零，且面积极小超曲面的存在会导致流形分裂。
In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and $L^p$-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under $L^p$ conditions on the relevant quantities.
For a Hamiltonian action of a compact group $U$ of isometries on a compact Kähler manifold $Z$ and a compatible subgroup $G$ of $U^{\mathbb{C}}$, we prove that for any closed $G$--invariant subset $Y\subset Z$ the image of the gradient map $μ_{\mathfrak{p}}(Y)$ is independent of the choice of the invariant Kähler form $ω$ in its cohomology class $[ω]$.
We study the set of critical points of a solution to $Δu = λ\cdot u$ and in particular components of the critical set that have codimension 1. We show, for example, that if a second Neumann eigenfunction of a simply connected polygon $P$ has infinitely many critical points, then $P$ is a rectangle.
Remarks on Einstein solitons with certain types of potential vector field
该研究针对黎曼流形中势向量场为梯度、螺向或共圆型的近Einstein孤立子展开，明确了λ函数的表达式并辅以示例，还推导了Ricci曲率张量满足特定对称条件时的几何性质，丰富了相关微分几何研究。
We show that a rescale limit at any degenerate singularity of Ricci flow in dimension 3 is a steady gradient soliton. In particular, we give a geometric description of type I and type II singularities.
In this paper, we study gradient Ricci expanding solitons $(X,g)$ satisfying $$ Rc=cg+D^2f, $$ where $Rc$ is the Ricci curvature, $c<0$ is a constant, and $D^2f$ is the Hessian of the potential function $f$ on $X$. We show that for a gradient expanding soliton $(X,g)$ with non-negative Ricci curvature, the scalar curvature $R$ has at least one maximum point on $X$, which is the only minimum point of the potential function $f$. Furthermore, $R>0$ on $X$ unless $(X,g)$ is Ricci flat. We also show that there is exponentially decay for scalar curvature for $ε$-pinched complete non-compact expandin…
Remarks on Semistable Points and Nonabelian Convexity of Gradient Maps
本文研究了实约化群在Kähler流形上的作用，利用梯度映射证明了半稳定点集合的开放性与连通性，并给出了紧Lagrange子流形及二轨道簇的非阿贝尔凸性定理。
Recently, Sogge-Zelditch and Colding-Minicozzi gave new power law lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for eigenfunctions and a growth vs. volume relation we proved in a previous work.
Remarks on almost $η$-Ricci solitons in $(\varepsilon)$-para Sasakian manifolds
本文研究满足特定曲率条件的$(\u03b5)$-para Sasakian流形中几乎$\u03b7$-Ricci孤立子，给出梯度情形下Ricci曲率范数估计与标量曲率表达式，并证明Codazzi型Ricci算子下类空/类时流形对应膨胀/收缩孤立子。
We mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. In section 2, we estimate the area growth of level sets and the volume growth of sublevel sets of a Ricci potential. In section 3, we show that the scalar curvature of such solitons approaches zero at infinity. In section 4, we investigate the geometry of such solitons at infinity, e.g., the tangent cone, the asymptotic behavior, etc.
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.
The main purpose of this short note is to point out that the negative gradient flow for the prescribed $\mathbf Q$-curvature problem on $S^n$ can be extended to handle the case that the $\mathbf Q$-curvature candidate $f$ may change signs.
For stationary harmonic maps between Riemannian manifolds, we provide a necessary and sufficient condition for the uniform interior and boundary gradient estimates in terms of the total energy of maps. We also show that if analytic target manifolds do not carry any harmonic S^2, then the singular sets of stationary maps are m \leq n - 4 rectifiable. Both of these results follow from a general analysis on the defect measures and energy concentration sets associated with a weakly converging sequence of stationary harmonic maps.
We revisit the question of existence and regularity of minimizers to weighted least gradient problems on a fixed bounded domain, subject to a Dirichlet boundary condition, in the case where the boundary data is continuous and the weight function is C^2 and bounded away from zero. Under suitable geometric conditions on the domain in R^n we construct continuous solutions of the above variational problem in any dimension n>=2, by extending the Sternberg-Williams-Ziemer technique to this setting of inhomogeneous variations. We show that the level sets of the constructed minimizer are minimal surfa…
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