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Zhou Zhang

Publications and source records attributed to Zhou Zhang.

At least 55 records · Page 3Linked to original sources
arXiv2020

On the precise asymptotics of Type-IIb solutions to mean curvature flow

In this paper, we study the precise asymptotics of noncompact Type-IIb solutions to the mean curvature flow. Precisely, for each real number $γ>0$, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics as $t\nearrow\infty$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the Type-IIb rate $(2t+1)^{(γ-1)/2}$. (2) In a neighbourhood of the tip, the Type-IIb blow-up of the solution converges to a translating soliton known as the bowl soliton. (3) Near spatial infinity, the hypersurface has a precise growth rate depending on $γ$.

math.DG↗
arXiv2019

Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II

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 infinity, the hypersurface approaches a collapsing cylinder at an exponential rate.

math.DG↗
arXiv2019

Mean curvature flow in Fuchsian manifolds

Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and the real line. In particular, we prove that there exists a large class of closed initial surfaces, as geodesic graphs over the totally geodesic surface $Σ$, such that the mean curvature flow exists for all time and converges to $Σ$. This is among the first examples of converging mean curvature flows of compact hypersurfaces in Riemannian manifolds. We also provide some useful calculations for the general warped product setting.

math.DG↗
arXiv2019

Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds

We investigate the mean curvature flows in a class of warped product manifolds with closed hypersurfaces fibering over $\mathbb{R}$. In particular, we prove that under natural conditions on the warping function and Ricci curvature bound for the ambient space, there exists a large class of closed initial hypersurfaces, as geodesic graphs over the totally geodesic hypersurface $Σ$, such that the mean curvature flow starting from $S_0$ exists for all time and converges to $Σ$.

math.DG↗
arXiv2016

Ricci flow on quasiprojective manifolds II

We study the Ricci flow on complete Kaehler metrics that live on the complement of a divisor in a compact complex manifold. In earlier work, we considered finite-volume metrics which, at spatial infinity, are transversely hyperbolic. In the present paper we consider three different types of spatial asymptotics: cylindrical, bulging and conical. We show that in each case, the asymptotics are preserved by the Kaehler-Ricci flow. We address long-time existence, parabolic blowdown limits and the role of the Kaehler-Ricci flow on the divisor.

math.DG↗
arXiv2015

Globally Existing Kähler-Ricci Flows

We consider the general Kähler-Ricci flows which exist for all time. The zeroth order control on the flow metric potential for various infinite time singularities is the focus. The possible semi-amplness for numerically effective classes serves as the main motivation.

math.DG↗
arXiv2015

The Representation of Line Dirac Delta Function Along a Space Curve

In this paper, we describe the line Dirac delta function of a curve in three-dimensional space in terms of the distance function to the curve. Its extension to level set formulation and plane curves are also developed. The main ideas can be applied for general dimension and codimension.

math.MG↗
arXiv2015

General Weak Limit for Kahler-Ricci Flow

Consider the Kahler-Ricci flow with finite time singularities over any closed Kahler manifold. We prove the existence of the flow limit in the sense of current towards the time of singularity. This answers affirmatively a problem raised by Tian on the uniqueness of the weak limit from sequential convergence construction. The notion of minimal singularity in the study of positive current, introduced by Demailly, comes up naturally. We also provide some discussion on the infinite time singularity case for comparison. The consideration can be applied to more flexible evolution equation of Kahler-Ricci flow type for any cohomology class. The study is related to general conjectures on the singularities of Kahler-Ricci flows.

math.DG↗
arXiv2015

Opportunistic Cooperative Channel Access in Distributed Wireless Networks with Decode-and-Forward Relays

This letter studies distributed opportunistic channel access in a wireless network with decode-and-forward relays. All the sources use channel contention to get transmission opportunity. If a source wins the contention, the channel state information in the first-hop channel (from the source to its relay) is estimated, and a decision is made for the winner source to either give up the transmission opportunity and let all sources start a new contention, or transmit to the relay. Once the relay gets the traffic, it may have a sequence of probings of the second-hop channel (from the relay to the destination). After each probing, if the second-hop channel is good enough, the relay transmits to the destination and completes the transmission process of the source; otherwise, the relay decides either to give up and let all sources start a new contention, or to continue to probe the second-hop channel. The optimal decision strategies for the two hops are derived in this letter. The first-hop strategy is a pure-threshold strategy, i.e., when the first-hop channel signal-to-noise ratio (SNR) is more than a threshold, the winner source should transmit to the relay, and subsequently the second-hop strategy should let the relay keep probing the second-hop channel until a good enough second-hop channel is observed. Simulation results show that our scheme is beneficial when the second-hop channels have larger average SNR.

cs.IT↗
arXiv2014

Convergence Results for Two Kahler-Ricci Flows

In this note, we provide some general discussion on the two main versions in the study of Kahler-Ricci flows over closed manifolds, aiming at smooth convergence to the corresponding Kahler-Einstein metrics with assumptions on the volume form and Ricci curvature form along the flow.

math.DG↗
arXiv2013

The Collapsing Rate of the Kähler-Ricci Flow with Regular Infinite Time Singularity

We study the collapsing behavior of the Kaehler-Ricci flow on a compact Kaehler manifold X admitting a holomorphic submersion X -> S coming from its canonical class, where S is a Kaehler manifold with dim S < dim X. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibers collapse at the optimal rate diameter ~ e^{-t/2}. Consequently, it leads to some analytic and geometric extensions to the regular case of Song-Tian's works on elliptic and Calabi-Yau fibrations. Its applicability to general Calabi-Yau fibrations with possibly singular fibers will also be discussed in local sense.

math.DG↗
arXiv2013

Asymptotics of complete Kahler metrics of finite volume on quasiprojective manifolds

Let X be a quasiprojective manifold given by the complement of a divisor $\bD$ with normal crossings in a smooth projective manifold $\bX$. Using a natural compactification of $X$ by a manifold with corners $\tX$, we describe the full asymptotic behavior at infinity of certain complete Kahler metrics of finite volume on X. When these metrics evolve according to the Ricci flow, we prove that such asymptotic behaviors persist at later time by showing the associated potential function is smooth up to the boundary on the compactification $\tX$. However, when the divisor $\bD$ is smooth with $K_{\bX}+[\bD]>0$ and the Ricci flow converges to a Kahler-Einstein metric, we show that this Kahler-Einstein metric has a rather different asymptotic behavior at infinity, since its associated potential function is polyhomogeneous with in general some logarithmic terms occurring in its expansion at the boundary.

math.DG↗
arXiv2012

Channel Exploration and Exploitation with Imperfect Spectrum Sensing in Cognitive Radio Networks

In this paper, the problem of opportunistic channel sensing and access in cognitive radio networks when the sensing is imperfect and a secondary user has limited traffic to send at a time is investigated. Primary users' statistical information is assumed to be unknown, and therefore, a secondary user needs to learn the information online during channel sensing and access process, which means learning loss, also referred to as regret, is inevitable. In this research, the case when all potential channels can be sensed simultaneously is investigated first. The channel access process is modeled as a multi-armed bandit problem with side observation. And channel access rules are derived and theoretically proved to have asymptotically finite regret. Then the case when the secondary user can sense only a limited number of channels at a time is investigated. The channel sensing and access process is modeled as a bi-level multi-armed bandit problem. It is shown that any adaptive rule has at least logarithmic regret. Then we derive channel sensing and access rules and theoretically prove they have logarithmic regret asymptotically and with finite time. The effectiveness of the derived rules is validated by computer simulation.

cs.IT↗
arXiv2011

Distributed Opportunistic Channel Access in Wireless Relay Networks

In this paper, the problem of distributed opportunistic channel access in wireless relaying is investigated. A relay network with multiple source-destination pairs and multiple relays is considered. All the source nodes contend through a random access procedure. A winner source node may give up its transmission opportunity if its link quality is poor. In this research, we apply the optimal stopping theory to analyze when a winner node should give up its transmission opportunity. By assuming the winner node has information of channel gains of links from itself to relays and from relays to its destination, the existence and uniqueness of an optimal stopping rule are rigorously proved. It is also found that the optimal stopping rule is a pure-threshold strategy. The case when the winner node does not have information of channel gains of links from relays to its destination is also studied. Two stopping problems exist, one in the main layer (for channel access of source nodes), and the other in the sub-layer (for channel access of relay nodes). An intuitive stopping rule, where the sub-layer and the main layer maximize their throughput respectively, is shown to be a semi-pure-threshold strategy. The intuitive stopping rule turns out to be non-optimal. An optimal stopping rule is then derived theoretically. Our research reveals that multi-user (including multi-source and multi-relay) diversity and time diversity can be fully utilized in a relay network by our proposed strategies.

cs.IT↗
arXiv2010

Ricci flow on quasiprojective manifolds

We consider the Kaehler-Ricci flow on complete finite-volume metrics that live on the complement of a divisor in a compact Kaehler manifold X. Assuming certain spatial asymptotics on the initial metric, we compute the singularity time in terms of cohomological data on X. We also give a sufficient condition for the singularity, if there is one, to be type-II.

math.DG↗
arXiv2010

Differential Harnack Estimates for Parabolic Equations

Let $(M,g(t))$ be a solution to the Ricci flow on a closed Riemannian manifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type $$\ppt f=Δf-f \ln f +Rf.$$ We also comment on an earlier result of the first author on positive solutions of the conjugate heat equation under the Ricci flow.

math.DG↗
arXiv2009

Kähler-Ricci Flow with Degenerate Initial Class

In an earlier work joint with X. X. Chen and G. Tian, we introduced the weak Kähler-Ricci flow for various geometric motivations. In the current work, we take further consideration on setting up the weak flow. Namely, the initial class is allowed to be no longer Kähler.

math.DG↗