The short time existence and the evolution equation of curvatures

2002 ◽  
pp. 15-23
Keyword(s):  
Evolution Equation ◽  
Short Time ◽  
Time Existence

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 A parabolic flow of balanced metrics

2017 ◽  
Vol 2017 (723) ◽  
Author(s):  
Lucio Bedulli ◽  
Luigi Vezzoni
Keyword(s):  
Vector Bundle ◽  
Evolution Equation ◽  
Existence And Uniqueness ◽  
General Criterion ◽  
Balanced Metrics ◽  
Time Solution ◽  
Parabolic Flow ◽  
Short Time

AbstractWe prove a general criterion to establish existence and uniqueness of a short-time solution to an evolution equation involving “closed” sections of a vector bundle, generalizing a method used by Bryant and Xu [

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.

scholarly journals Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

2006 ◽  
Vol 181 (3) ◽  
pp. 449-504 ◽  
Author(s):  
Olivier Alvarez ◽  
Philippe Hoch ◽  
Yann Le Bouar ◽  
Régis Monneau
Keyword(s):  
Existence And Uniqueness ◽  
Dislocation Dynamics ◽  
Uniqueness Of The Solution ◽  
Short Time ◽  
Time Existence

The elastic flow of curves on the sphere

2018 ◽  
Vol 3 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Anna Dall’Acqua ◽  
Tim Laux ◽  
Lin ◽  
Paola Pozzi ◽  
Adrian Spener
Keyword(s):  
Critical Points ◽  
Elastic Energy ◽  
Gradient Flow ◽  
Closed Curves ◽  
Long Time Existence ◽  
Long Time ◽  
Short Time ◽  
Time Existence

Abstract We consider closed curves on the sphere moving by the L2-gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.

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