Zhang Zonglao
This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.
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In this short paper, we show there do not exist three-dimensional noncompact $κ$-solutions of Ricci flow that have positive curvature and satisfy a Type-I bound. This represents progress towards the proof of Perelman's conjecture that the only complete noncompact three-dimensional $κ$-solution with positive curvature is the Bryant soliton.
Nonexistence of solutions to $Δ_pu+Δ_qu+u^s|\nabla u|^t\leq 0$ on geodesically complete noncompact Riemannian manifolds
本文通过测试函数方法，在测地完备非紧黎曼流形上证明了含梯度项的不等式解的Liouville型定理，推广了已有结果，并讨论了高阶项在s<0时的影响。
Let $M$ be a compact Kähler manifold and $N$ be a subvariety with codimension greater than or equal to 2. We show that there are no complete Kähler--Einstein metrics on $M-N$. As an application, let $E$ be an exceptional divisor of $M$. Then $M-E$ cannot admit any complete Kähler--Einstein metric if blow-down of $E$ is a complex variety with only canonical or terminal singularities. A similar result is shown for pairs.
The goal of this article is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain pde estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian.
We prove that there is no faithful finite-dimensional representation by skew-hermitian matrices of a ``basic algebra of observables'' B on a noncompact symplectic manifold M. Consequently there exists no finite-dimensional quantization of any Lie subalgebra of the Poisson algebra C^\infty(M) containing B.
We prove that a complete hyperbolic 3-manifold of finite volume does not admit a properly embedded noncompact surface of finite topology with constant mean curvature greater than or equal to 1.
Sharp Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below
本文研究了非紧致黎曼流形上具有下界曲率的Sobolev和Moser-Trudinger不等式，得出了最佳常数。
Prescribing Chern scalar curvatures on noncompact manifolds
针对非紧非 Kähler 流形上的预定 Chern 标量曲率问题，论文借助新的解析条件建立 Kazdan-Warner 型方程的存在性结果，并据此给出 Ni 经典多重性定理的新证明。
In this article we study a class of prescribed curvature problems on complete noncompact Riemannian manifolds. To be precise, we derive local $C^0$-estimate under an asymptotic condition which is in effect optimal, and prove the existence of complete conformal metrics with prescribed curvature functions. A key ingredient of our strategy is Aviles-McOwen's result or its fully nonlinear version on the existence of complete conformal metrics with prescribed curvature functions on manifolds with boundary.
In this paper, we study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solitons of the mean curvature flow in Minkowski space. We also study the asymptotic behavior and the strict convexity of general solitons of such flows.
We continue the study, initiated by the first two authors in \cite{IW19}, of Type-II curvature blow-up in mean curvature flow of complete noncompact embedded hypersurfaces. In particular, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics near the "vanishing" time $T$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the rate $(T-t)^{-1}$. (2) In a neighbourhood of the tip, the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infi…
Let $(M^n,g),~n\ge 3$ be a noncompact complete Riemannian manifold with compact boundary and $f$ a smooth function on $\partial M$. In this paper we show that for a large class of such manifolds, there exists a metric within the conformal class of $g$ that is complete, has zero scalar curvature on $M$ and has mean curvature $f$ on the boundary. The problem is equivalent to finding a positive solution to an elliptic equation with a non-linear boundary condition with critical Sobolev exponent.
We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time $T$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter $γ>1/2$, the…
In this paper, we give the first detailed proof of the short-time existence of Deane Yang's local Ricci flow. Then using the local Ricci flow, we prove short-time existence of the Ricci flow on noncompact manifolds, whose Ricci curvature has global lower bound and sectional curvature has only local average integral bound. The short-time existence of the Ricci flow on noncompact manifolds with bounded curvature was studied by Wan-Xiong Shi in 1990s. As a corollary of our main theorem, we get the short-time existence part of Shi's theorem in this more general context.
We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures.
Prescribed Chern scalar curvature on complete noncompact Hermitian manifolds with nonpositive curvatures
本文研究了在非正曲率的完整非紧赫尔米特流形上规定陈标量曲率的问题，得出了存在常数负陈标量曲率度量的充分条件。
Some obstructions to positive scalar curvature on a noncompact manifold
本文研究非紧流形上存在完备正数量曲率度量的障碍，并分别对有限体积与无限体积情形给出判定条件。
We show that recent work of Ni and Wilking yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact Kähler--Ricci shrinkers by Feldman, Ilmanen, and Knopf exhibit that this result is sharp.
A nonexistence result for rotating mean curvature flows in $\mathbb{R}^{4}$
针对平均曲率流中可能作为奇点模型的旋转古老流，本文利用切流轴唯一性无法约束内部旋转这一观察，证明在 $\mathbb{R}^4$ 的古老非坍缩流中不存在此类旋转流。
In this paper, we introduce a monotonicity formula for the mean curvature flow which is related to self-expanders. Then we use the monotonicity to study the asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces.
Under suitable conditions near infinity and assuming boundedness of curvature tensor, we prove a no breathers theorem in the spirit of Ivey-Perelman for some noncompact Ricci flows. These include Ricci flows on asymptotically flat (AF) manifolds with positive scalar curvature. Since the method for the compact case faces a difficulty, the proof involves solving a new non-local elliptic equation which is the Euler-Lagrange equation of a scaling invariant log Sobolev inequality.
Gauss Curvature Flow on Surfaces of Revolution -- The Noncompact Case
本文研究了旋转曲面上高斯曲率流的非紧情形，处理了两个非紧情况下的边值问题，扩展了此前关于紧区间上正可微函数生成旋转曲面的研究。
Sharp Sobolev inequalities on noncompact Riemannian manifolds with bounded Ricci curvature
研究非紧致黎曼流形上带有有界瑞奇曲率的尖锐Sobolev不等式，通过等周轮廓的渐进行展和局部Sobolev不等式推导
In this note we show that a closed Riemannian manifold does not admit a fill-in with nonnegative scalar curvature if the mean curvature is point-wise large. Similar result also holds for fill-ins with a negative scalar curvature lower bound.
Remark about scalar curvature on certain noncompact manifolds
本文研究了非紧流形内部完整黎曼度量的标量曲率问题，给出了排除非负标量曲率的充分条件，并在高维情况下提供了正标量曲率度量的端点例子。
We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface $M$ in $\mathbb{R}^{4}$ with zero scalar curvature $S_2$, nonzero Gauss-Kronecker curvature and finite total curvature (i.e. $\int_M|A|^3<+\infty$).
Min-max construction of prescribed mean curvature hypersurfaces in noncompact manifolds
该研究建立了非紧流形中预定平均曲率超曲面的极小-极大理论，适用于紧集外不变号的预定函数，并借此证明了两类超曲面的新存在性结果。
We discuss an elementary consequence of the works of (1) Brett Kotschwar and Lu Wang and (2) Ovidiu Munteanu and Jiaping Wang.
In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative $N$-Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative $N$-Bakry Émery Ricci Curvature. In addition, we show that if $M^n$ is a complete, noncompact Riemannian manifold with nonnegative $N$-Bakry Émery Ricci curvature where $N>n$, then $H_{n-1}(M,\mathbb{Z})$ is $0$.
In this paper, we look for properties of gradient Yamabe solitons on top of warped product manifolds. Utilizing the maximum principle, we find lower bound estimates for both the potential function of the soliton and the scalar curvature of the warped product. By slightly modifying Li-Yau's technique so that we can handle drifting Laplacians, we were able to find three different gradient estimates for the warping function, one for each sign of the scalar curvature of the fiber manifold. As an application, we exhibit a nonexistence theorem for gradient Yamabe solitons possessing certain metric p…
We prove that a complete noncompact Kähler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}$^{n}$ and we show that the manifold is topologically {\bf R}$^{2n}$. In particular, when $M^{n}$ is a Kähler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to {\bf C}$^{2}$.
A differential Harnack inequality for noncompact evolving hypersurfaces
本文证明非紧致凸超曲面在对称曲率函数速度下流动的微分Harnack不等式，推广Andrews的紧致结果，并适用于奇点爆破产生的古代解。
We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc>= 0.
Let $(M,g)$ be a noncompact complete $n$-manifold with harmonic curvature and positive Sobolev constant. Assume that $L_2$ norms of Weyl curvature and traceless Ricci curvature are finite. We prove that $(M,g)$ is Einstein if $n \ge 5$ and $L_{n/2}$ norms of Weyl curvature and traceless Ricci curvature are small enough.
A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension $3$ which have positive sectional curvature and are $κ$-noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $\mathbb{R}^3$, and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in $\mathbb{R}^3$ which is strictly convex and noncollapsed.
We give a short proof of the following fact. Let $Σ$ be a connected, finitely connected, noncompact manifold without boundary. If $g$ is a complete Riemannian metric on $Σ$ whose Gaussian curvature $K$ is nonnegative at infinity, then $K$ must be integrable. In particular, we obtain a new short proof of the fact that if $Σ$ admits a complete metric whose Gaussian curvature is nonnegative and positive at one point, then $Σ$ is diffeomorphic to $\mathbb{R}^2$.
We obtain a Calabi-Yau type lower volume growth estimates for complete noncompact self-shrinkers of the mean curvature flow, more precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.
We consider the Ricci flow $\frac{\partial}{\partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rm\geq 0, |Rm(p)|\to 0, ~as ~d(o,p)\to 0.$ We prove that the Ricci flow on such a manifold is nonsingular in any finite time.
In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold $M$ of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have Euclidean volume growth and its scalar curvature decays to zero at infinity in the average sense, then $M$ is biholomorphic to $\C^2$. During the proof, we also discover an interesting gap phenomenon which says that a Kähler manifold as above automatically ha…
Nonexistence of the metric with positive intermediate curvatures on manifolds with boundary
本文证明维数不超过7的紧致带边流形在边界$m$-凸时不能有正$m$-中间曲率度量，并推广了相关结果到带边情形。
Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds
本文研究洛伦兹流形中从完备非紧类空超曲面出发的预定平均曲率流，给出短时间存在的充分条件，并讨论长时间存在性与收敛性，进而应用于闵可夫斯基时空和施瓦西时空未来零无穷远附近的相关问题。
Positive scalar curvature on products of noncompact manifolds
本文证明三个非紧流形的乘积可具有一致正标量曲率的完备度量，利用莫尔斯指标加性和无定向达斯手术构造，并给出反例。研究背景与问题：非紧流形乘积上的正标量曲率存在性。核心方法：莫尔斯指标加性与开莫尔斯-手术构造。主要贡献：证明正曲率度量存在并给出反例。
Nonexistence of stable discrete maps into some homogeneous spaces of nonnegative curvature
本文研究了从有限加权图到光滑黎曼流形的离散映射的加权长度或能量泛函的稳定性，并证明了到特定齐性空间（如正全纯截面曲率的Kähler C-空间及部分单连通紧致黎曼对称空间）不存在稳定离散极小浸入或非常数稳定离散调和映射。
This paper concerns the evolution of complete noncompact locally uniformly convex hypersurface in Euclidean space by curvature flow, for which the normal speed $Φ$ is given by a power $β\geq 1$ of a monotone symmetric and homogeneous of degree one function $F$ of the principal curvatures. Under the assumption that $F$ is inverse concave and its dual function approaches zero on the boundary of positive cone, we prove that the complete smooth strictly convex solution exists and remains a graph until the maximal time of existence. In particular, if $F=K^{s/n}G^{1-s}$ for any $s\in(0, 1]$, where $…
We generalize the famous result of Gromov and Lawson on the nonexistence of metric of positive scalar curvature on enlargeable manifolds to the case of foliations, without using index theorems on noncompact manifolds.
A nonexistence result for wing-like mean curvature flows in $\mathbb{R}^4$
本文排除了$\text{R}^4$中3维超曲面平均曲率流的非塌缩翼状奇点模型，证明古老非塌缩流的爆破极限非楔体，提出精细气泡片分析，为$\text{R}^4$古老非塌缩流分类奠定基础。
In this paper we prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability of Monge-Ampère type equations subject to certain boundary conditions depending on the value of $p$. The special case of $p=1$ was previously studied by Pogorelov and Chou-Wang. Here, we give some sufficient conditions for the solvability for general $p\neq1$.
We present a numerical study of the local stability of mean curvature flow of rotationally symmetric, complete noncompact hypersurfaces with Type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs "numerically global" (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviors under mean curvature flow. Firstly, there is a "near" class …
In this note we show that, under certain curvature positivity conditions (the weak $\operatorname{PIC}-2$ condition or the nonnegative bisectional curvature condition), a complete and noncompact expanding breather of the Ricci flow is also an expanding gradient Ricci soliton. This is the first no expanding breather theorem in the noncompact category.
Let $(M,g^{TM})$ be a noncompact (not necessarily complete) enlargeable Riemannian manifold in the sense of Gromov-Lawson and $F$ an integrable subbundle of $T M$ . Let $k^F$ be the leafwise scalar curvature associated to $g^F=g^{TM}|_F$. We show that if either $TM$ or $F$ is spin, then ${\rm inf}(k^F)\leq 0$. This generalizes the famous result of Gromov-Lawson on enlargeable manifolds to the case of foliations. It also extends an ansatz of Gromov on hyper-Euclidean spaces to general enlargeable Riemannian manifolds, as well as recent results on compact enlargeable foliated manifolds due to Be…
Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a closed manifold. The second result is on a complete noncompact manifold. To prove both results, we fully analyze the structure of the Lichnerowicz Laplacian on complex hyperbolic space. To prove the second result, we also define suitably weighted little Hölder spaces on a complete noncompact manifold and establish their interpolation properties.
Noncompact self-shrinkers for mean curvature flow with arbitrary genus
本文利用 min-max 方法严格证明了 Ilmanen 关于任意亏格非紧自收缩子存在的猜想，并在大亏格情形下证实其恰有一个渐近锥形端，同时给出奇数亏格双端情形的数值证据。
On the Prescribed Ricci Curvature of Noncompact Homogeneous Spaces with Two Isotropy Summands
本文对具有两个不可约求和项的单连通非紧齐性空间进行分类，并求解规定 Ricci 曲率问题。
In this paper we obtain three results concerning the geometry of complete noncompact positively curved Kähler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact Kähler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex dimension). The second one states that the curvature of a complete noncompact Kähler manifold with positive bisectional curvature decays at least linearly in the average sense. The third result is concerned with the relation between the volume growth and the curvature decay. We …
A Conformal Positive Mass Theorem with Noncompact Boundary
该研究通过积分不等式证明了在非紧边界条件下，两个符合相关度量的ADM质量的凸组合的正性，扩展了Batista和Lopes de Lima的结果。
On ends of finite-volume noncompact manifolds of nonpositive curvature
本文证实了一个猜想：在截面曲率介于-1和0之间的有限体积完备非紧流形上，若万有覆盖是能见流形，则每个端的根本群几乎幂零。
The goal of this article is to study the geometry of Bach-flat noncompact steady quasi-Einstein manifolds. We show that a Bach-flat noncompact steady quasi-Einstein manifold $(M^{n},\,g)$ with positive Ricci curvature such that its potential function has at least one critical point must be a warped product with Einstein fiber. In addition, the fiber has constant curvature if $n = 4.$
The Avoidance Principle for Noncompact Hypersurfaces Moving by Mean Curvature Flow
研究非紧超曲面在均曲率流中保持不相交的条件，证明在欧几里得空间或满足曲率有界条件的流形中，初始非零距离可保证曲面始终分离。
Nonexistence of isoperimetric sets in spaces of positive curvature
该研究构造了正曲率黎曼流形及凸集中小体积下等周集不存在的例子，证明维度下界3为最优，并首次给出无等周集的非塌缩非负曲率空间。
Let $(M,g)$ be a noncompact complete Bach-flat manifold with positive Yamabe constant. We prove that $(M,g)$ is flat if $(M, g)$ has zero scalar curvature and sufficiently small $L_{2}$ bound of curvature tensor. When $(M, g)$ has nonconstant scalar curvature, we prove that $(M, g)$ is conformal to the flat space if $(M, g)$ has sufficiently small $L_2$ bound of curvature tensor and $L_{4/3}$ bound of scalar curvature.
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