Singularities of first kind in the harmonic map and Yang-Mills heat flows

2002 ◽  
Vol 242 (1) ◽  
pp. 47-62 ◽  
Author(s):  
Andreas Gastel
Keyword(s):  
Harmonic Map ◽  
Yang Mills ◽  
Heat Flows

scholarly journals Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

2021 ◽  
Author(s):  
Ahmad Afuni
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Map ◽  
Local Regularity ◽  
Yang Mills ◽  
Heat Flows ◽  
Singular Sets ◽  
Regularity Results ◽  
Local Monotonicity ◽  
Partial Differ ◽  
Singular Time

AbstractWe establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

Heat ball formulæ for k-forms on evolving manifolds

2019 ◽  
Vol 12 (2) ◽  
pp. 135-156 ◽  
Author(s):  
Ahmad Afuni
Keyword(s):  
Vector Bundle ◽  
Heat Kernel ◽  
Harmonic Map ◽  
Yang Mills ◽  
Heat Flows ◽  
Local Monotonicity ◽  
Superlevel Sets

AbstractWe establish a local monotonicity identity for vector bundle-valued differential k-forms on superlevel sets of appropriate heat kernel-like functions. As a consequence, we obtain new local monotonicity formulæ for the harmonic map and Yang–Mills heat flows on evolving manifolds. We also show how these methods yield local monotonicity formulæ for the Yang–Mills–Higgs flow.

ENERGY CONCENTRATION FOR 2-DIMENSIONAL RADIALLY SYMMETRIC EQUIVARIANT HARMONIC MAP HEAT FLOWS

2011 ◽  
Vol 13 (04) ◽  
pp. 675-695 ◽  
Author(s):  
MICHIEL BERTSCH ◽  
REIN VAN DER HOUT ◽  
JOSEPHUS HULSHOF
Keyword(s):  
Energy Method ◽  
Harmonic Map ◽  
Single Bubble ◽  
Energy Concentration ◽  
Singularity Formation ◽  
Heat Flows ◽  
Comparison Arguments

We give a description of singularity formation in terms of energy quanta for 2-dimensional radially symmetric equivariant harmonic map heat flows. Adapting Struwe's energy method we first establish a finite bubble tree result with a discrete multiple of energy quanta disappearing in the singularity. We then use intersection-comparison arguments to show that the bubble tree consists of a single bubble only and that there is a well defined scale RBHK(t) ↓ 0 in which the solution converges to the standard harmonic map.

Some conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang-Mills theory

1982 ◽  
Vol 91 (3) ◽  
pp. 441-452 ◽  
Author(s):  
H. C. J. Sealey
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
Differential Forms ◽  
Finite Energy ◽  
Energy Functional ◽  
Conformal Transformations ◽  
Second Variation ◽  
Yang Mills

In (5) it is shown that if m ≽ 3 then there is no non-constant harmonic map φ: ℝm → Sn with finite energy. The method of proof makes use of the fact that the energy functional is not invariant under conformal transformations. This fact has also allowed Xin(9), to show that any non-constant-harmonic map φ:Sm → (N, h), m ≽ 3, is not stable in the sense of having non-negative second variation.

Local regularity and compactness for the p-harmonic map heat flows

2018 ◽  
Vol 11 (3) ◽  
pp. 223-255 ◽  
Author(s):  
Masashi Misawa
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
Geometric Analysis ◽  
Regularity Criterion ◽  
Local Regularity ◽  
Stationary Case ◽  
Heat Flows ◽  
Regularity Estimate ◽  
Energy Bound ◽  
Scaled Energy

AbstractWe study a geometric analysis and local regularity for the evolution of {{p}}-harmonic maps, called {{p}}-harmonic map heat flows. Our main result is to establish a criterion for a uniform local regularity estimate for regular {{p}}-harmonic map heat flows, devising some new monotonicity-type formulas of a local scaled energy. The regularity criterion obtained is almost optimal, comparing with that of the corresponding stationary case. As application we show a compactness of regular {{p}}-harmonic map heat flows with energy bound.

scholarly journals Łojasiewicz–Simon gradient inequalities for analytic and Morse–Bott functions on Banach spaces

2020 ◽  
Vol 2020 (765) ◽  
pp. 35-67 ◽  
Author(s):  
Paul M. N. Feehan ◽  
Manousos Maridakis
Keyword(s):  
Heat Flow ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Harmonic Map ◽  
Nonlinear Evolution Equations ◽  
Energy Functions ◽  
Yang Mills ◽  
Link Type ◽  
Geometric Problems ◽  
Harmonic Map Heat Flow

AbstractWe prove several abstract versions of the Łojasiewicz–Simon gradient inequality for an analytic function on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, that of the well-known infinite-dimensional version of the gradient inequality due to Łojasiewicz [S. Łojasiewicz, Ensembles semi-analytiques, (1965), Publ. Inst. Hautes Etudes Sci., Bures-sur-Yvette. LaTeX version by M. Coste, August 29, 2006 based on mimeographed course notes by S. Łojasiewicz, https://perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf] and proved by Simon [L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 1983, 3, 525–571]. We prove that the optimal exponent of the Łojasiewicz–Simon gradient inequality is obtained when the function is Morse–Bott, improving on similar results due to Chill [R. Chill, On the Łojasiewicz–Simon gradient inequality, J. Funct. Anal. 201 2003, 2, 572–601], [R. Chill, The Łojasiewicz–Simon gradient inequality in Hilbert spaces, Proceedings of the 5th European-Maghrebian workshop on semigroup theory, evolution equations, and applications 2006, 25–36], Haraux and Jendoubi [A. Haraux and M. A. Jendoubi, On the convergence of global and bounded solutions of some evolution equations, J. Evol. Equ. 7 2007, 3, 449–470], and Simon [L. Simon, Theorems on regularity and singularity of energy minimizing maps, Lect. Math. ETH Zürich, Birkhäuser, Basel 1996]. In [P. M. N. Feehan and M. Maridakis, Łojasiewicz–Simon gradient inequalities for harmonic maps, preprint 2019, https://arxiv.org/abs/1903.01953], we apply our abstract gradient inequalities to prove Łojasiewicz–Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Those inequalities generalize those of Kwon [H. Kwon, Asymptotic convergence of harmonic map heat flow, ProQuest LLC, Ann Arbor 2002; Ph.D. thesis, Stanford University, 2002], Liu and Yang [Q. Liu and Y. Yang, Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds, Ark. Mat. 48 2010, 1, 121–130], Simon [L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 1983, 3, 525–571], [L. Simon, Isolated singularities of extrema of geometric variational problems, Harmonic mappings and minimal immersions (Montecatini 1984), Lecture Notes in Math. 1161, Springer, Berlin 1985, 206–277], and Topping [P. M. Topping, Rigidity in the harmonic map heat flow, J. Differential Geom. 45 1997, 3, 593–610]. In [P. M. N. Feehan and M. Maridakis, Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions, preprint 2019, https://arxiv.org/abs/1510.03815v6; to appear in Mem. Amer. Math. Soc.], we prove Łojasiewicz–Simon gradient inequalities for coupled Yang–Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. Those inequalities generalize that of the pure Yang–Mills energy function due to the first author [P. M. N. Feehan, Global existence and convergence of solutions to gradient systems and applications to Yang–Mills gradient flow, preprint 2016, https://arxiv.org/abs/1409.1525v4] for base manifolds of arbitrary dimension and due to Råde [J. Råde, On the Yang–Mills heat equation in two and three dimensions, J. reine angew. Math. 431 1992, 123–163] for dimensions two and three.

scholarly journals The harmonic map and Killing fields for self-dual SU(3) Yang-Mills equations

1984 ◽  
Vol 17 (5) ◽  
pp. L227-L230 ◽  
Author(s):  
P Kernsten ◽  
R Martini
Keyword(s):  
Harmonic Map ◽  
Yang Mills ◽  
Killing Fields

scholarly journals Adiabatic Limit in Yang–Mills Equations in R⁴

2019 ◽  
pp. 449-454
Author(s):  
Armen G. Sergeev
Keyword(s):  
Loop Space ◽  
Harmonic Map ◽  
Riemann Sphere ◽  
Dimensional Space ◽  
Adiabatic Limit ◽  
Yang Mills

Our goal is to present an approach to the proof of the harmonic spheres conjecture based on the adiabatic limit construction. This construction allows to associate with an arbitrary Yang–Mills G-field on the Euclidean 4-dimensional space a harmonic map of the Riemann sphere to the loop space of the group G

Sign in / Sign up

Export Citation Format

Share Document