A Compactness Theorem for Yang-Mills Connections

2004 ◽  
Vol 47 (4) ◽  
pp. 624-634 ◽  
Author(s):  
Xi Zhang
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Compact Riemannian Manifold ◽  
Compactness Theorem ◽  
Yang Mills ◽  
Uniformly Bounded

AbstractIn this paper, we consider Yang-Mills connections on a vector bundle E over a compact Riemannian manifold M of dimension m > 4, and we show that any set of Yang-Mills connections with the uniformly bounded -norm of curvature is compact in C∞ topology.

On a gauge-invariant functional

2010 ◽  
Vol 53 (1) ◽  
pp. 143-151
Author(s):  
Cătălin Gherghe
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Compact Riemannian Manifold ◽  
Second Variation ◽  
Gauge Invariant ◽  
Variation Formula ◽  
First Variation ◽  
Yang Mills ◽  
Invariant Functional ◽  
The Second Variation

AbstractWe define a new functional which is gauge invariant on the space of all smooth connections of a vector bundle over a compact Riemannian manifold. This functional is a generalization of the classical Yang-Mills functional. We derive its first variation formula and prove the existence of critical points. We also obtain the second variation formula.

The Palais–Smale conditions for the Yang–Mills functional

1988 ◽  
Vol 108 (3-4) ◽  
pp. 189-200
Author(s):  
D. R. Wilkins
Keyword(s):  
Riemannian Manifold ◽  
Principal Bundle ◽  
Compact Riemannian Manifold ◽  
Hilbert Manifold ◽  
Yang Mills ◽  
Palais Smale Condition ◽  
Dimension 2

SynopsisWe consider the Yang–Mills functional denned on connections on a principal bundle over a compact Riemannian manifold of dimension 2 or 3. It is shown that if we consider the Yang–Mills functional as being defined on an appropriate Hilbert manifold of orbits of connections under the action of the group of principal bundle automorphisms, then the functional satisfies the Palais–Smale condition.

Eigenvalue pinching for Riemannian vector bundles

1999 ◽  
Vol 1999 (511) ◽  
pp. 73-86 ◽  
Author(s):  
Peter Petersen ◽  
Chadwick Sprouse
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Vector Bundles ◽  
Compact Riemannian Manifold ◽  
Small Eigenvalue

Abstract We investigate some very general pinching results for eigensections with small eigenvalue of a Riemannian vector bundle. In particular, this gives pinching results for the eigenvalues of 1-forms on a compact Riemannian manifold, along with other applications.

Curvature Tensor under the Yamabe Flow

2012 ◽  
Vol 459 ◽  
pp. 514-517
Author(s):  
Jing Fang Shen ◽  
Jia Jun Yang ◽  
Neng Xi
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Curvature Tensor ◽  
Compact Riemannian Manifold ◽  
Injectivity Radius ◽  
Time Slice ◽  
Yamabe Flow ◽  
Flow Curvature ◽  
Bounded Below ◽  
Uniformly Bounded

Yamabe flow, curvature, tensor, compact. Abstract. This paper focuses a compact Riemannian manifold Mn evolving under the Yamabe flow and proves that if the Ricci curvature is uniformly bounded under the flow for all times that t from 0 to T and the injectivity radius is bounded below at each time slice, then the curvature tensor is uniformly bounded.

scholarly journals Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

1997 ◽  
Vol 20 (2) ◽  
pp. 397-402 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Function ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Smooth Functions ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The spectral functionΘ(t)=∑i=1∞exp(−tλj), where{λj}j=1∞are the eigenvalues of the negative Laplace-Beltrami operator−Δ, is studied for a compact Riemannian manifoldΩof dimension “k” with a smooth boundary∂Ω, where a finite number of piecewise impedance boundary conditions(∂∂ni+γi)u=0on the parts∂Ωi(i=1,…,m)of the boundary∂Ωcan be considered, such that∂Ω=∪i=1m∂Ωi, andγi(i=1,…,m)are assumed to be smooth functions which are not strictly positive.

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

EIGENVALUES OF GEOMETRIC OPERATORS RELATED TO THE WITTEN LAPLACIAN UNDER THE RICCI FLOW

2017 ◽  
Vol 59 (3) ◽  
pp. 743-751
Author(s):  
SHOUWEN FANG ◽  
FEI YANG ◽  
PENG ZHU
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Ricci Flow ◽  
Compact Riemannian Manifold ◽  
Laplacian Operator ◽  
Witten Laplacian ◽  
Normalized Ricci Flow ◽  
Geometric Operator

AbstractLet (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.

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