scholarly journals Deforming metrics of foliations

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Vladimir Rovenski ◽  
Robert Wolak
Keyword(s):  
Heat Flow ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Existence Results ◽  
Long Time Existence ◽  
Long Time ◽  
Main Observation ◽  
The Mean ◽  
Conformal Flow ◽  
Time Existence

AbstractLet M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D ⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D ⊥-closed. Assuming that D ⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H.

scholarly journals Generalized Lagrangian mean curvature flows: The cotangent bundle case

2019 ◽  
Vol 2019 (750) ◽  
pp. 97-121 ◽  
Author(s):  
Knut Smoczyk ◽  
Mao-Pei Tsui ◽  
Mu-Tao Wang
Keyword(s):  
Mean Curvature ◽  
Cotangent Bundle ◽  
Curvature Vector ◽  
Lagrangian Submanifold ◽  
Mean Curvature Vector ◽  
Long Time Existence ◽  
Almost Kähler Manifold ◽  
Generalized Mean Curvature ◽  
Generalized Mean ◽  
Time Existence

Abstract In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel {(n,0)} -form, just like the Calabi–Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.

scholarly journals Orlicz–Minkowski flows

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
Paul Bryan ◽  
Mohammad N. Ivaki ◽  
Julian Scheuer
Keyword(s):  
Final Section ◽  
Gauss Curvature ◽  
Stationary Solutions ◽  
Existence Results ◽  
Minkowski Problem ◽  
Parabolic Approximation ◽  
Long Time Existence ◽  
Long Time ◽  
And Behavior ◽  
Time Existence

AbstractWe study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if they exist, solve the regular Orlicz–Minkowski problems. As an application, we obtain old and new existence results for the regular even Orlicz–Minkowski problems; the corresponding $$L_p$$ L p version is the even $$L_p$$ L p -Minkowski problem for $$p>-n-1$$ p > - n - 1 . Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz–Minkowski problems; the $$L_p$$ L p versions are the even $$L_p$$ L p -Minkowski problem for $$p>0$$ p > 0 and the $$L_p$$ L p -Minkowski problem for $$p>1$$ p > 1 . In the final section, we use a curvature flow with no global term to solve a class of $$L_p$$ L p -Christoffel–Minkowski type problems.

scholarly journals Energy convexity of intrinsic bi-harmonic maps and applications I: Spherical target

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Paul Laurain ◽  
Longzhi Lin
Keyword(s):  
Heat Flow ◽  
Uniform Convergence ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Spherical Target ◽  
Long Time Existence ◽  
Long Time ◽  
Harmonic Map Heat Flow ◽  
Time Existence

AbstractIn this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball {B_{1}\subset\mathbb{R}^{4}} into the sphere {\mathbb{S}^{n}}. In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. We also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into {\mathbb{S}^{n}}, which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that was until now only known assuming the non-positivity of the target manifolds by Lamm [26]. Further, we establish the previously unknown result that the energy convexity along the flow yields uniform convergence of the flow.

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
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