scholarly journals Short-time persistence of bounded curvature under the Ricci flow

2017 ◽  
Vol 24 (2) ◽  
pp. 427-447
Author(s):  
Brett Kotschwar
Keyword(s):  
Ricci Flow ◽  
Bounded Curvature ◽  
Short Time

scholarly journals Ricci flow on manifolds with boundary with arbitrary initial metric

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tsz-Kiu Aaron Chow
Keyword(s):  
Ricci Flow ◽  
Positive Curvature ◽  
Curvature Operator ◽  
Manifolds With Boundary ◽  
Convex Boundary ◽  
Uniqueness Of The Solution ◽  
Positive Time ◽  
Natural Boundary Conditions ◽  
Short Time ◽  
Time Existence

Abstract In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

scholarly journals The Long-Time Behavior of the Ricci Tensor under the Ricci Flow

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Christian Hilaire
Keyword(s):  
Ricci Flow ◽  
Ricci Tensor ◽  
Closed Manifold ◽  
Time Behavior ◽  
Dimensional Manifold ◽  
Long Time Behavior ◽  
Bounded Curvature ◽  
3 Dimensional ◽  
Long Time ◽  
Uniformly Bounded

We show that, given an immortal solution to the Ricci flow on a closed manifold with uniformly bounded curvature and diameter, the Ricci tensor goes to zero as . We also show that if there exists an immortal solution on a closed 3-dimensional manifold such that the product of the curvature and the square of the diameter is uniformly bounded, then this solution must be of type III.

scholarly journals POSITIVITY OF RICCI CURVATURE UNDER THE KÄHLER–RICCI FLOW

2006 ◽  
Vol 08 (01) ◽  
pp. 123-133 ◽  
Author(s):  
DAN KNOPF
Keyword(s):  
Ricci Curvature ◽  
Ricci Flow ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Bounded Curvature ◽  
Complex Dimension

In each complex dimension n ≥ 2, we construct complete Kähler manifolds of bounded curvature and non-negative Ricci curvature whose Kähler–Ricci evolutions immediately acquire Ricci curvature of mixed sign.

scholarly journals Ricci flow with bounded curvature integrals

2021 ◽  
Vol 314 (2) ◽  
pp. 283-309
Author(s):  
Shota Hamanaka
Keyword(s):  
Ricci Flow ◽  
Bounded Curvature

Some local maximum principles along Ricci flows

2020 ◽  
pp. 1-20
Author(s):  
Man-Chun Lee ◽  
Luen-Fai Tam
Keyword(s):  
Maximum Principle ◽  
Ricci Flow ◽  
Direct Proof ◽  
Local Maximum ◽  
Maximum Principles ◽  
Kernel Estimates ◽  
Bounded Curvature ◽  
Noncompact Manifolds ◽  
Ricci Flows ◽  
Dirichlet Heat Kernel

Abstract In this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$ for $t>0$ for some constant ${\alpha }>0$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t>0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature. By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t>0$ .

scholarly journals Bergman kernel along the Kähler–Ricci flow and Tian’s conjecture

2016 ◽  
Vol 2016 (717) ◽  
Author(s):  
Wenshuai Jiang
Keyword(s):  
Recent Work ◽  
Ricci Curvature ◽  
Ricci Flow ◽  
Bergman Kernel ◽  
Fano Manifolds ◽  
Bergman Kernels ◽  
Complex Dimension ◽  
Short Time

AbstractIn this paper, we study the behavior of Bergman kernels along the Kähler–Ricci flow on Fano manifolds. We show that the Bergman kernels are equivalent along the Kähler–Ricci flow for short time under certain condition on the Ricci curvature of the initial metric. Then, using a recent work of Tian and Zhang, we can solve a conjecture of Tian for Fano manifolds of complex dimension at most 3.

scholarly journals Uniqueness and short time regularity of the weak Kähler–Ricci flow

2017 ◽  
Vol 305 ◽  
pp. 953-993 ◽  
Author(s):  
Eleonora Di Nezza ◽  
Chinh H. Lu
Keyword(s):  
Ricci Flow ◽  
Short Time
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