Gregor Giesen, Peter M. Topping
We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.
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In this paper we construct a Ricci de Turck flow on any incomplete Riemannian manifold with bounded curvature. The central property of the flow is that it stays uniformly equivalent to the initial incomplete Riemannian metric, and in that sense preserves any given initial singularity structure. Together with the corresponding result by Shi for complete manifolds, this gives that any (complete or incomplete) manifold of bounded curvature can be evolved by the Ricci de Turck flow for a short time.
We show uniqueness of Ricci flows starting at a surface of uniformly negative curvature, with the assumption that the flows become complete instantaneously. Together with the more general existence result proved in [10], this settles the issue of well-posedness in this class.
We give a simple proof of an extension of the existence results of Ricci flow of G.Giesen and P.M.Topping [GiT1],[GiT2], on incomplete surfaces with bounded above Gauss curvature without using the difficult Shi's existence theorem of Ricci flow on complete non-compact surfaces and the pseudolocality theorem of G.Perelman [P1] on Ricci flow. We will also give a simple proof of a special case of the existence theorem of P.M.Topping [T] without using the existence theorem of W.X.Shi [S1].
On a smooth closed Riemannian manifold, we show short time existence of smooth solutions to the $(α,β)$-Ricci-Yamabe flow, which is a natural generalization of the Ricci flow and the Yamabe flow. We also establish some long time existence theorems for the closed $(α,β)$-Ricci-Yamabe flow by estimating its curvatures.
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.
In this paper we prove the existence of Type II singularities for the Ricci flow on $S^{n+1}$ for all $n\geq 2$.
In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of Simon \cite{Simon2015}.
Let $(M^n, g)$ be a compact $n$-dim ($n\geq 2$) manifold with nonnegative Ricci curvature, and if $n\geq 3$ we assume that $(M^n, g)\times \mathbb{R}$ has nonnegative isotropic curvature. The lower bound of the Ricci flow's existence time on $(M^n, g)$ is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows' maximal existence times, which was firstly proved by E. Cabezas-Rivas and B. Wilking. We also get an interior curvature estimates for $n= 3$ under $Rc\geq 0$ assumption among others. Combining these results, we proved the short time exist…
We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with large coupling constant.
We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions $n+1\ge 3$, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow "pushes away" from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. In the …
Convergence of Ricci flow and long-time existence of Harmonic map heat flow
本文建立了渐近紧致可积shrinker的ancient Ricci flow及相应奇点情形下harmonic map heat flow的长时间存在性，提供全局parabolic gauge并证明切流唯一性，同时分类了ancient flows并得出奇点最优收敛速率。
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.
This note illustrates the Ricci flow method based on the Cao.H.D's paper[1] and Yau.S.T's paper[4], and tries to explain the method in detail, especially in some calculations. Jian Song and Weinkove's note[9] used some other estimates to obtain the result, this paper will explain some of their estimates as well. This note was a seminar lecture note in 2022 summer when the author was giving lectures on the geometry analysis seminar reasearching the Ricci flow method. The part of the imporatnt zero order estimate is going to be added in a few days. The level of the author is limited, if there ar…
We study the generalized Kähler-Ricci flow on complex surfaces with nondegenerate Poisson structure, proving long time existence and convergence of the flow to a weak hyperKähler structure.
The first part of the paper discusses a second-order quasilinear parabolic equation in a vector bundle over a compact manifold $M$ with boundary $\partial M$. We establish a short-time existence theorem for this equation. The second part of the paper is devoted to the investigation of the Ricci flow on $M$. We propose a new boundary condition for the flow and prove two short-time existence results.
Existence of 5 minimal tori in 3-spheres of positive Ricci curvature
本研究针对B. White关于黎曼3维球面至少含5个嵌入极小环面的猜想，结合极小极大理论与平均曲率流思路，证明了正Ricci曲率3维球面中该猜想成立。
We give a complete description of the global existence and convergence for the Ricci-Yang-Mills flow on $T^k$ bundles over Riemann surfaces. These results equivalently describe solutions to generalized Ricci flow and pluriclosed flow with symmetry.
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min-max theory for the area functional to prove this conjecture in the positive Ricci curvature setting. More precisely, we show that every compact Riemannian manifold with positive Ricci curvature and dimension at most seven contains infinitely many smooth, closed, embedded minimal hypersurfaces. In the last section we mention some open problems related with the geometry of these minimal hypersurfaces.
In this paper we establish the existence of extremals for the Log Sobolev functional on complete non-compact manifolds with Ricci curvature bounded from below and strictly positive injectivity radius, under a condition near infinity. When Ricci curvature is also bounded from above we get exponential decay at infinity of the extremals. As a consequence of these analytical results we establish, under the same assumptions, that non-trivial shrinking Ricci solitons support a gradient Ricci soliton structure. On the way, we prove two results of independent interest: the existence of a distance-like…
We prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature. We do not require upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger-Gromoll convex exhaustion and solving the singular initial value problem for the Ricci flow on these closed manifolds, we obtain a sequence of closed solutions of the Ricci flow with nonnegative complex sectional curvature which subconverge to a solution of the Ricci flow on the open manifold. Furthermore, we find an optimal volume growth condition which guarantees long time exi…
The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified proof. In the later of 80's, Shi \cite{Sh1} generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ri…
We consider a closed cohomogeneity one Riemannian manifold $(M^n,g) $ of dimension $n\geq 3$. If the Ricci curvature of $M$ is positive, we prove the existence of infinite nodal solutions for equations of the form $-Δ_g u + λu = λu^q$ with $λ>0$, $q>1$. In particular for a positive Einstein manifold which is of cohomogeneity one or fibers over a cohomogeniety one Einstein manifold we prove the existence of infinite nodal solutions for the Yamabe equation, with a prescribed number of connected components of its nodal domain.
In this note, we prove that on an $n$-dimensional compact toric manifold with positive first Chern class, the Kähler-Ricci flow with any initial $(S^1)^n$-invariant Kähler metric converges to a Kähler-Ricci soliton. In particular, we give another proof for the existence of Kähler-Ricci solitons on a compact toric manifold with positive first Chern class by using the Kähler-Ricci flow.
In this paper, by limiting twisted conical Kähler-Ricci flows, we prove the long-time existence and uniqueness of cusp Kähler-Ricci flow on compact Kähler manifold $M$ which carries a smooth hypersurface $D$ such that the twisted canonical bundle $K_M+D$ is ample. Furthermore, we prove that this flow converge to a unique cusp Kähler-Einstein metric.
In this paper, we prove the long-time existence and uniqueness of the conical Kähler-Ricci flow with weak initial data which admits $L^{p}$ density for some $p>1$ on Fano manifold. Furthermore, we study the convergence behavior of this kind of flow.
In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of the paper. We remark that in this paper we will mostly be dealing with non-singular Ricci flows. The existence of surgeries will not play an important role.
We apply ideas from viscosity theory to establish the existence of a unique global weak solution to the generalized Kahler-Ricci flow in the setting of commuting complex structures. Our results are restricted to the case of a smooth manifold with smooth background data. We discuss the possibility of extending these results to more singular settings, pointing out a key error in the existing literature on viscosity solutions to complex Monge-Ampere equations/Kahler-Ricci flow.
We consider the initial value problem $u_t = Δ\log u$, $u(x,0) = u_0(x)\ge 0$ in $\R^2$, corresponding to the Ricci flow, namely conformal evolution of the metric $u (dx_1^2 + dx_2^2)$ by Ricci curvature. It is well known that the maximal (complete) solution $u$ vanishes identically after time $T= \frac 1{4π} \int_{\R^2} u_0 $. Assuming that $u_0$ is compactly supported we describe precisely the Type II vanishing behavior of $u$ at time $T$: we show the existence of an inner region with exponentially fast vanishing profile, which is, up to proper scaling, a {\em soliton cigar solution}, and th…
Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces
本研究在先前证明以黎曼曲面搭配非原子Radon测度为共形因子的Ricci流存在性的基础上，进一步证明其唯一性，二者结合实现了这类粗糙曲面的典范光滑化并正则化其无穷远几何。
In this paper, we establish a framework for the analysis of linear parabolic equations on conical surfaces and use them to study the conical Ricci flow. In particular, we prove the long time existence of the conical Ricci flow for general cone angle and show that this solution has the optimal regularity, namely, the time derivatives of the conformal factor are bounded and for each fixed time, the conformal factor has an explicit asymptotic expansion near the cone points.
In this paper, we establish the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in $\mathbb{R}^3$ as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex surfaces whose metrics converge uniformly to the metric of the initial surface in intrinsic sense.
We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with some special restriction conditions. The restriction conditions are motivated by convexity results for Perelman's $\mathcal{W}$-functional over convex subsets inside adequate subspaces of Riemannian metrics. We show indeed that the Soliton-Ricci flow is generated by the gradient flow of the restriction of Perelman's $\mathcal{W}$-functional over such subspaces…
In this paper we discuss the existence and non--existence of weak solutions to parametric equations involving the Laplace-Beltrami operator $Δ_g$ in a complete non-compact $d$--dimensional ($d\geq 3$) Riemannian manifold $(\mathcal{M},g)$ with asymptotically non--negative Ricci curvature and intrinsic metric $d_g$. Namely, our simple model is the following problem $$ \left\{ \begin{array}{ll} -Δ_gw+V(σ)w=λα(σ)f(w) & \mbox{ in } \mathcal{M}\\ w\geq 0 & \mbox{ in } \mathcal{M} \end{array}\right. $$ where $V$ is a positive coercive potential, $α$ is a positive bounded function, $λ$ is a real para…
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