scholarly journals Convergence of the Weak Kähler–Ricci Flow on Manifolds of General Type

2019 ◽  
Author(s):  
Tat Dat Tô
Keyword(s):  
Ricci Flow ◽  
General Type ◽  
Einstein Metric ◽  
Canonical Class ◽  
Open Set ◽  
Long Time Existence ◽  
Long Time ◽  
Viscosity Sense ◽  
Monge Ampere ◽  
Time Existence

Abstract We study the Kähler–Ricci flow on compact Kähler manifolds whose canonical bundle is big. We show that the normalized Kähler–Ricci flow has long-time existence in the viscosity sense, is continuous in a Zariski open set, and converges to the unique singular Kähler–Einstein metric in the canonical class. The key ingredient is a viscosity theory for degenerate complex Monge–Ampère flows in big classes that we develop, extending and refining the approach of Eyssidieux–Guedj–Zeriahi.

THE SASAKI–RICCI FLOW

2010 ◽  
Vol 21 (07) ◽  
pp. 951-969 ◽  
Author(s):  
KNUT SMOCZYK ◽  
GUOFANG WANG ◽  
YONGBING ZHANG
Keyword(s):  
Ricci Flow ◽  
Einstein Metrics ◽  
Positive Case ◽  
Einstein Metric ◽  
Well Posedness ◽  
Long Time Existence ◽  
Null Case ◽  
Long Time ◽  
Time Existence

In this paper, we introduce the Sasaki–Ricci flow to study the existence of η-Einstein metrics. In the positive case any η-Einstein metric can be homothetically transformed to a Sasaki–Einstein metric. Hence it is an odd-dimensional counterpart of the Kähler–Ricci flow. We prove its well-posedness and long-time existence. In the negative or null case the flow converges to the unique η-Einstein metric. In the positive case the convergence remains in general open. The paper can be viewed as an odd-dimensional counterpart of Cao's results on the Kähler–Ricci flow.

scholarly journals A Parabolic Monge–Ampère Type Equation of Gauduchon Metrics

2017 ◽  
Vol 2019 (17) ◽  
pp. 5497-5538 ◽  
Author(s):  
Tao Zheng
Keyword(s):  
Smooth Function ◽  
Existence And Uniqueness ◽  
Type Equation ◽  
Uniqueness Of Solution ◽  
Long Time Existence ◽  
Hermitian Manifolds ◽  
Long Time ◽  
Monge Ampere ◽  
Time Existence

Abstract We prove the long time existence and uniqueness of solution to a parabolic Monge–Ampère type equation on compact Hermitian manifolds. We also show that the normalization of the solution converges to a smooth function in the smooth topology as $t$ approaches infinity which, up to scaling, is the solution to a Monge–Ampère type equation. This gives a parabolic proof of the Gauduchon conjecture based on the solution of Székelyhidi, Tosatti, and Weinkove to this conjecture.

scholarly journals Combinatorial Ricci Flows with Applications to the Hyperbolization of Cusped 3-Manifolds

2021 ◽  
Author(s):  
Ke Feng ◽  
Huabin Ge ◽  
Bobo Hua ◽  
Xu Xu
Keyword(s):  
Ricci Flow ◽  
Hyperbolic Metric ◽  
Effective Algorithm ◽  
Hyperbolic Structure ◽  
Ricci Flows ◽  
Long Time Existence ◽  
The Hyperbolic Metric ◽  
Long Time ◽  
Combinatorial Ricci Flow ◽  
Time Existence

Abstract In this paper, we adopt combinatorial Ricci flow to study the existence of hyperbolic structure on cusped 3-manifolds. The long-time existence and the uniqueness for the extended combinatorial Ricci flow are proven for general pseudo 3-manifolds. We prove that the extended combinatorial Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature, and the flow converges exponentially fast in this case. For an ideally triangulated cusped 3-manifold admitting a complete hyperbolic metric, the flow provides an effective algorithm for finding the hyperbolic metric.

scholarly journals The parabolic Monge–Ampère equation on compact almost Hermitian manifolds

2020 ◽  
Vol 2020 (761) ◽  
pp. 1-24 ◽  
Author(s):  
Jianchun Chu
Keyword(s):  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Limit Function ◽  
Uniqueness Of Solutions ◽  
Long Time Existence ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Long Time ◽  
Monge Ampere ◽  
Time Existence

AbstractWe prove the long time existence and uniqueness of solutions to the parabolic Monge–Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in {C^{\infty}} topology as {t\rightarrow\infty}. Up to scaling, the limit function is a solution of the Monge–Ampère equation. This gives a parabolic proof of existence of solutions to the Monge–Ampère equation on almost Hermitian manifolds.

scholarly journals Long-Time Existence for Semi-linear Beam Equations on Irrational Tori

2021 ◽  
Author(s):  
Joackim Bernier ◽  
Roberto Feola ◽  
Benoît Grébert ◽  
Felice Iandoli
Keyword(s):  
Long Time Existence ◽  
Beam Equations ◽  
Long Time ◽  
Time Existence

Evolving inextensible and elastic curves with clamped ends under the second-order evolution equation in ℝ2

2018 ◽  
Vol 3 (1) ◽  
pp. 14-18 ◽  
Author(s):  
Chun-Chi Lin ◽  
Yang-Kai Lue
Keyword(s):  
Evolution Equation ◽  
Convergent Subsequence ◽  
Second Order ◽  
Plane Curves ◽  
Smooth Solutions ◽  
Fixed Position ◽  
Elastic Curves ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

Abstract For any given C2-smooth initial open curves with fixed position and fixed tangent at the boundary points, we obtain the long-time existence of smooth solutions under the second-order evolution of plane curves. Moreover, the asymptotic limit of a convergent subsequence is an inextensible elastica.

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