scholarly journals General weak limit for Kähler–Ricci flow

2016 ◽  
Vol 18 (05) ◽  
pp. 1550079
Author(s):  
Zhou Zhang
Keyword(s):  
Evolution Equation ◽  
Ricci Flow ◽  
Weak Limit ◽  
Infinite Time ◽  
Time Singularity ◽  
Ricci Flows ◽  
Sequential Convergence ◽  
Minimal Singularity ◽  
Flow Limit ◽  
Time Singularities

Consider the Kähler–Ricci flow with finite time singularities over any closed Kähler manifold. We prove the existence of the flow limit in the sense of current toward the time of singularity. This answers affirmatively a problem raised by Tian in [New results and problems on Kähler–Ricci flow, Astérisque 322 (2008) 71–92] on the uniqueness of the weak limit from sequential convergence construction. The notion of minimal singularity introduced by Demailly in the study of positive current 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 Kähler–Ricci flow type for any cohomology class. The study is related to general conjectures on the singularities of Kähler–Ricci flows.

scholarly journals RICCI LOWER BOUND FOR KÄHLER–RICCI FLOW

2014 ◽  
Vol 16 (02) ◽  
pp. 1350053 ◽  
Author(s):  
ZHOU ZHANG
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Ricci Flow ◽  
Finite Time ◽  
Ricci Flows ◽  
Time Singularities

We provide general discussion on the lower bound of Ricci curvature along Kähler–Ricci flows over closed manifolds. The main result is the non-existence of Ricci lower bound for flows with finite time singularities and non-collapsed global volume. As an application, we give examples showing that positivity of Ricci curvature would not be preserved by Ricci flow in general.

scholarly journals The collapsing rate of the Kähler–Ricci flow with regular infinite time singularity

2015 ◽  
Vol 2015 (703) ◽  
Author(s):  
Frederick Tsz-Ho Fong ◽  
Zhou Zhang
Keyword(s):  
Ricci Flow ◽  
Kähler Manifold ◽  
Infinite Time ◽  
Time Singularity ◽  
Kahler Manifold ◽  
Compact Kähler Manifold

AbstractWe study the collapsing behavior of the Kähler–Ricci flow on a compact Kähler manifold

scholarly journals Infinite-time singularities of the Kähler–Ricci flow

2015 ◽  
Vol 19 (5) ◽  
pp. 2925-2948 ◽  
Author(s):  
Valentino Tosatti ◽  
Yuguang Zhang
Keyword(s):  
Ricci Flow ◽  
Infinite Time ◽  
Time Singularities

scholarly journals METRIC COMPATIBLE OR NON-COMPATIBLE FINSLER–RICCI FLOWS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250041 ◽  
Author(s):  
SERGIU I. VACARU
Keyword(s):  
Ricci Flow ◽  
Evolution Equations ◽  
Finsler Geometry ◽  
Flow Theory ◽  
Finsler Metric ◽  
Geometric Evolution ◽  
Ricci Flows ◽  
Canonical Metric ◽  
Self Consistent ◽  
Math Phys

There were elaborated different models of Finsler geometry using the Cartan (metric compatible), or Berwald and Chern (metric non-compatible) connections, the Ricci flag curvature, etc. In a series of works, we studied (non)-commutative metric compatible Finsler and non-holonomic generalizations of the Ricci flow theory [see S. Vacaru, J. Math. Phys. 49 (2008) 043504; 50 (2009) 073503 and references therein]. The aim of this work is to prove that there are some models of Finsler gravity and geometric evolution theories with generalized Perelman's functionals, and correspondingly derived non-holonomic Hamilton evolution equations, when metric non-compatible Finsler connections are involved. Following such an approach, we have to consider distortion tensors, uniquely defined by the Finsler metric, from the Cartan and/or the canonical metric compatible connections. We conclude that, in general, it is not possible to elaborate self-consistent models of geometric evolution with arbitrary Finsler metric non-compatible connections.

Interacting cosmological viscous fluid with Little Rip and Pseudo Rip solutions in f(R) gravity

2018 ◽  
Vol 15 (02) ◽  
pp. 1850028 ◽  
Author(s):  
R. D. Boko ◽  
M. J. S. Houndjo ◽  
J. Tossa
Keyword(s):  
Dark Matter ◽  
Dark Energy ◽  
Equation Of State ◽  
Viscous Fluid ◽  
Modified Gravity ◽  
Infinite Time ◽  
Two Fluids ◽  
Viscosity Models ◽  
Time Singularities ◽  
Analytic Expressions

In this paper, we investigate the evolution of the equation of state of the interacting viscous dark energy in [Formula: see text] gravity. We first consider the case when the dark energy does not interact with the dark matter and after, the case where there is a coupling between these dark components. The viscosity and the interaction between the two fluids are parameterized by constants [Formula: see text] and [Formula: see text] respectively for which a detailed investigation on the cosmological implications has been done. In the later part of the paper, we explore some bulk viscosity models with Little and Pseudo Rip infinite time singularities within [Formula: see text] modified gravity. We obtain analytic expressions for characteristic properties of these cosmological models.

A note on Ricci flow with Ricci curvature bounded below

2017 ◽  
Vol 2017 (726) ◽  
Author(s):  
Xiuxiong Chen ◽  
Fang Yuan
Keyword(s):  
Ricci Curvature ◽  
Ricci Flow ◽  
Ricci Flows ◽  
Bounded Below

AbstractIn this note, we will prove that given a sequence of Ricci flows

Topological drone navigation in uncertain environment

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Bhumeshwar Patle ◽  
Shyh-Leh Chen ◽  
Brijesh Patel ◽  
Sunil Kumar Kashyap ◽  
Sudarshan Sanap
Keyword(s):  
Evolution Equation ◽  
Scalar Curvature ◽  
Objective Function ◽  
Obstacle Avoidance ◽  
Ricci Flow ◽  
Dynamic Environments ◽  
Riemannian Metric ◽  
Content Type ◽  
Uncertain Environment ◽  
Planning Approach

Purpose With the increasing demand for surveillance and smart transportation, drone technology has become the center of attraction for robotics researchers. This study aims to introduce a new path planning approach to drone navigation based on topology in an uncertain environment. The main objective of this study is to use the Ricci flow evolution equation of metric and curvature tensor over angular Riemannian metric, and manifold for achieving navigational goals such as path length optimization at the minimum required time, collision-free obstacle avoidance in static and dynamic environments and reaching to the static and dynamic goals. The proposed navigational controller performs linearly and nonlinearly both with reduced error-based objective function by Riemannian metric and scalar curvature, respectively. Design/methodology/approach Topology and manifolds application-based methodology establishes the resultant drone. The trajectory planning and its optimization are controlled by the system of evolution equation over Ricci flow entropy. The navigation follows the Riemannian metric-based optimal path with an angular trajectory in the range from 0° to 360°. The obstacle avoidance in static and dynamic environments is controlled by the metric tensor and curvature tensor, respectively. The in-house drone is developed and coded using C++. For comparison of the real-time results and simulation results in static and dynamic environments, the simulation study has been conducted using MATLAB software. The proposed controller follows the topological programming constituted with manifold-based objective function and Riemannian metric, and scalar curvature-based constraints for linear and nonlinear navigation, respectively. Findings This proposed study demonstrates the possibility to develop the new topology-based efficient path planning approach for navigation of drone and provides a unique way to develop an innovative system having characteristics of static and dynamic obstacle avoidance and moving goal chasing in an uncertain environment. From the results obtained in the simulation and real-time environments, satisfactory agreements have been seen in terms of navigational parameters with the minimum error that justifies the significant working of the proposed controller. Additionally, the comparison of the proposed navigational controller with the other artificial intelligent controllers reveals performance improvement. Originality/value In this study, a new topological controller has been proposed for drone navigation. The topological drone navigation comprises the effective speed control and collision-free decisions corresponding to the Ricci flow equation and Ricci curvature over the Riemannian metric, respectively.

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