scholarly journals On the Convergence of the Elastic Flow in the Hyperbolic Plane

2020 ◽  
Vol 5 (1) ◽  
pp. 40-77
Author(s):  
Marius Müller ◽  
Adrian Spener
Keyword(s):  
Hyperbolic Space ◽  
Elastic Energy ◽  
Hyperbolic Plane ◽  
Blow Up ◽  
Type Inequality ◽  
Gradient Flow ◽  
Global Minimizer ◽  
Infinite Time ◽  
Closed Curves ◽  
Convergence Results

AbstractWe examine the L2-gradient flow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a constrained sharp Reilly-type inequality.

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.

Infinite time blow-up for half-harmonic map flow from R into S1

2021 ◽  
Vol 143 (4) ◽  
pp. 1261-1335
Author(s):  
Yannick Sire ◽  
Juncheng Wei ◽  
Youquan Zheng
Keyword(s):  
Blow Up ◽  
Harmonic Map ◽  
Infinite Time ◽  
Harmonic Map Flow

scholarly journals Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sun-Hye Park
Keyword(s):  
Time Delay ◽  
Global Existence ◽  
Wave Equations ◽  
Blow Up ◽  
Logarithmic Sobolev Inequality ◽  
Decay Rates ◽  
Infinite Time ◽  
Global Existence Of Solutions ◽  
Frictional Damping ◽  
T Tau

AbstractIn this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \alpha u_{t} (x,t) + \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$ u t t ( x , t ) − Δ u ( x , t ) + α u t ( x , t ) + β u t ( x , t − τ ) = u ( x , t ) ln | u ( x , t ) | γ . There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.

scholarly journals Blow up and globality of solutions for a nonautonomous semilinear heat equation with Dirichlet condition

2019 ◽  
Vol 53 (1) ◽  
pp. 57-72
Author(s):  
Marcos Josías Ceballos-Lira ◽  
Aroldo Pérez
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Mild Solution ◽  
Blow Up ◽  
Local Existence ◽  
Dirichlet Condition ◽  
Infinite Time ◽  
Diffusion Operator ◽  
Intrinsic Ultracontractivity ◽  
Semilinear Heat Equation

In this paper we prove the local existence of a nonnegative mild solution for a nonautonomous semilinear heat equation with Dirichlet condition, and give sucient conditions for the globality and for the blow up infinite time of the mild solution. Our approach for the global existence goes back to the Weissler's technique and for the nite time blow up we uses the intrinsic ultracontractivity property of the semigroup generated by the diffusion operator.

scholarly journals On the Kähler–Ricci flow near a Kähler–Einstein metric

2015 ◽  
Vol 2015 (699) ◽  
Author(s):  
Song Sun ◽  
Yuanqi Wang
Keyword(s):  
Critical Point ◽  
Ricci Flow ◽  
Complex Structure ◽  
Type Inequality ◽  
Gradient Flow ◽  
Einstein Metric ◽  
Fano Manifold ◽  
Kähler Metric ◽  
Canonical Class ◽  
Kahler Metric

AbstractOn a Fano manifold, we prove that the Kähler–Ricci flow starting from a Kähler metric in the anti-canonical class which is sufficiently close to a Kähler–Einstein metric must converge in a polynomial rate to a Kähler–Einstein metric. The convergence cannot happen in general if we study the flow on the level of Kähler potentials. Instead we exploit the interpretation of the Ricci flow as the gradient flow of Perelman's μ functional. This involves modifying the Ricci flow by a canonical family of gauges. In particular, the complex structure of the limit could be different in general. The main technical ingredient is a Lojasiewicz type inequality for Perelman's μ functional near a critical point.

scholarly journals Cross-ratio Dynamics on Ideal Polygons

2020 ◽  
Author(s):  
Maxim Arnold ◽  
Dmitry Fuchs ◽  
Ivan Izmestiev ◽  
Serge Tabachnikov
Keyword(s):  
Hyperbolic Space ◽  
Moduli Space ◽  
Hyperbolic Plane ◽  
Projective Line ◽  
Cross Ratio ◽  
Poisson Structures ◽  
Completely Integrable ◽  
The Cross

Abstract Two ideal polygons, $(p_1,\ldots ,p_n)$ and $(q_1,\ldots ,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $\alpha $-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = \alpha $ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $\alpha = -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is generically a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures and show that these relations, with different values of the constants $\alpha $, commute, in an appropriate sense. We investigate the case of small-gons and describe the exceptional ideal pentagons and hexagons that possess infinitely many $\alpha $-related polygons.

SHIMIZU’S LEMMA FOR COMPLEX HYPERBOLIC SPACE

1992 ◽  
Vol 03 (02) ◽  
pp. 291-308 ◽  
Author(s):  
JOHN R. PARKER
Keyword(s):  
Hyperbolic Space ◽  
Hyperbolic Plane ◽  
Discrete Group ◽  
Complex Hyperbolic Space ◽  
Necessary Condition ◽  
Higher Dimensional ◽  
Group Of Isometries ◽  
Vertical Translation

Shimizu’s lemma gives a necessary condition for a discrete group of isometries of the hyperbolic plane containing a parabolic map to be discrete. Viewing the hyperbolic plane as complex hyperbolic 1-space we generalise Shimizu’s lemma to higher dimensional complex hyperbolic space In particular we give a version of Shimizu’s lemma for subgroups of PU (n, 1) containing a vertical translation Partial generalisation to groups containing either an ellipto-parabolic map or non-vertical translations are also given together with examples that show full generalisation is not possible in these cases

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