arXiv subjects

Kai Zheng

Publications and source records attributed to Kai Zheng.

190 records · Page 11Linked to original sources

arXiv2015

Regularity scales and convergence of the Calabi flow

We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson's conjectural picture for the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra assumption that the scalar curvature is uniformly bounded.

math.DG↗

arXiv2014

$I$-properness of Mabuchi's $K$-energy

Over the space of Kähler metrics associated to a fixed Kähler class, we first prove the lower bound of the energy functional $\tilde E^β$, then we provide the criterions of the geodesics rays to detect the lower bound of $\tilde {\mathfrak J}^β$-functional. They are used to obtain the properness of Mabuchi's $K$-energy. The criterions are examined by showing the convergence of the negative gradient flow of $\tilde {\mathfrak J}^β$-functional.

math.AP↗

arXiv2014

Kähler non-collapsing, eigenvalues and the Calabi flow

We first proved a compactness theorem of the Kähler metrics, which confirms a prediction of Chen. Then we prove several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed by Chen-Li-Wang to prove the small energy theorems of the Calabi flow.

math.DG↗

arXiv2014

Energy-Efficient Flow Scheduling and Routing with Hard Deadlines in Data Center Networks

The power consumption of enormous network devices in data centers has emerged as a big concern to data center operators. Despite many traffic-engineering-based solutions, very little attention has been paid on performance-guaranteed energy saving schemes. In this paper, we propose a novel energy-saving model for data center networks by scheduling and routing "deadline-constrained flows" where the transmission of every flow has to be accomplished before a rigorous deadline, being the most critical requirement in production data center networks. Based on speed scaling and power-down energy saving strategies for network devices, we aim to explore the most energy efficient way of scheduling and routing flows on the network, as well as determining the transmission speed for every flow. We consider two general versions of the problem. For the version of only flow scheduling where routes of flows are pre-given, we show that it can be solved polynomially and we develop an optimal combinatorial algorithm for it. For the version of joint flow scheduling and routing, we prove that it is strongly NP-hard and cannot have a Fully Polynomial-Time Approximation Scheme (FPTAS) unless P=NP. Based on a relaxation and randomized rounding technique, we provide an efficient approximation algorithm which can guarantee a provable performance ratio with respect to a polynomial of the total number of flows.

cs.NI↗

arXiv2013

GreenDCN: a General Framework for Achieving Energy Efficiency in Data Center Networks

The popularization of cloud computing has raised concerns over the energy consumption that takes place in data centers. In addition to the energy consumed by servers, the energy consumed by large numbers of network devices emerges as a significant problem. Existing work on energy-efficient data center networking primarily focuses on traffic engineering, which is usually adapted from traditional networks. We propose a new framework to embrace the new opportunities brought by combining some special features of data centers with traffic engineering. Based on this framework, we characterize the problem of achieving energy efficiency with a time-aware model, and we prove its NP-hardness with a solution that has two steps. First, we solve the problem of assigning virtual machines (VM) to servers to reduce the amount of traffic and to generate favorable conditions for traffic engineering. The solution reached for this problem is based on three essential principles that we propose. Second, we reduce the number of active switches and balance traffic flows, depending on the relation between power consumption and routing, to achieve energy conservation. Experimental results confirm that, by using this framework, we can achieve up to 50 percent energy savings. We also provide a comprehensive discussion on the scalability and practicability of the framework.

cs.NI↗

arXiv2013

The Dirichlet and the weighted metrics for the space of Kahler metrics

In this work we study the intrinsic geometry of the space of Kahler metrics under various Riemannian metrics. The first part is on the Dirichlet metric. We motivate its study, we compute its curvature, and we make links with the Calabi metric, the K-energy, the degenerate complex Hessian equation. The second part is on the weighted metrics, for which we investigate as well their geometric properties.

math.DG↗

arXiv2013

Geodesics in the space of Kähler cone metrics

In this paper, we study the Dirichlet problem of the geodesic equation in the space of Kähler cone metrics $\mathcal H_\b$; that is equivalent to a homogeneous complex Monge-Ampère equation whose boundary values consist of Kähler metrics with cone singularities. Our approach concerns the generalization of the space defined in Donaldson \cite{MR2975584} to the case of Kähler manifolds with boundary; moreover we introduce a subspace $\mathcal H_C$ of $\mathcal H_\b$ which we define by prescribing appropriate geometric conditions. Our main result is the existence, uniqueness and regularity of $C^{1,1}_\b$ geodesics whose boundary values lie in $\mathcal H_C$. Moreover, we prove that such geodesic is the limit of a sequence of $C^{2,\a}_\b$ approximate geodesics under the $C^{1,1}_\b$-norm. As a geometric application, we prove the metric space structure of $\mathcal H_C$.

math.AP↗

arXiv2010

Stability of Calabi flow near an extremal metric

We prove that on a Kähler manifold admitting an extremal metric $ω$ and for any Kähler potential $φ_0$ close to $ω$, the Calabi flow starting at $φ_0$ exists for all time and the modified Calabi flow starting at $φ_0$ will always be close to $ω$. Furthermore, when the initial data is invariant under the maximal compact subgroup of the identity component of the reduced automorphism group, the modified Calabi flow converges to an extremal metric near $ω$ exponentially fast.

math.DG↗

arXiv2010

Pseudo-Calabi Flow

We first define Pseudo-Calabi flow, as {equation*} {{aligned}{{\partial φ}\over {\partial t}}&= -f(φ), \triangle_varphi f(φ) &= S(φ) - \ul S.{aligned}. \end{equation*} Then we prove the well-posedness of this flow including the short time existence, the regularity of the solution and the continuous dependence on the initial data. Next, we point out that the $L^\infty$ bound on Ricci curvature is an obstruction to the extension of the pseudo-Calabi flow. Finally, we show that if there is a cscK metric in its Kähler class, then for any initial potential in a small $C^{2,α}$ neighborhood of it, the pseudo-Calabi flow must converge exponentially to a nearby cscK metric.

math.DG↗

arXiv2010

Stability of Kähler-Ricci flow in the space of Kähler metrics

In this paper, we prove that on a Fano manifold $M$ which admits a Kähler-Ricci soliton $(\om,X)$, if the initial Kähler metric $\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak Kähler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if $\vphi_0$ is also $K_X$-invariant, then the weak modified Kähler-Ricci flow converges exponentially to a unique Kähler-Ricci soliton nearby. Especially, if the Futaki invariant vanishes, we may delete the $K_X$-invariant assumption. The methods based on the metric geometry of the space of the Kähler metrics are potentially applicable to other stability problem of geometric flow near a critical metric.

math.DG↗