We develop a parabolic formulation of the Uhlenbeck gauge fixing theorem and use it to prove the maximal existence of the Yang-Mills flow in vector bundles over compact $n$-dimensional Riemannian manifolds for $n \geq 4$. We apply Moser's iteration technique \cite{Mo1} to give a new proof of the bound of the curvature. Using an idea of Hamilton on the Ricci flow \cite{Ha}, we also provide a new proof on uniform estimates of higher order derivatives. Furthermore, we investigate the blow-up of the Yang-Mills flow at the maximal existence time and improve an asymptotical result of Hong-Tian \cite{HT1}.
We study a family of coupled geometric evolution equations describing the deformation of a Riemannian metric and a non-abelian gauge field on closed manifolds, which generalizes the Ricci--Yang--Mills flow. We derive interior curvature estimates, find a preserved integral curvature condition, and discover a scaling inv...
We consider the cone Yang-Mills functional recently introduced by Tseng and Zhou. This functional generalizes the Yang-Mills functional by coupling it to a closed 2-form which can represent a geometric structure of interest, e.g. a symplectic form or Ricci form. While the perspective of dimensional reduction is not alw...
We improve a parabolic version of Uhlenbeck's gauge fixing theorem for the Yang-Mills flow. This allows us to prove new existence of weak solutions of the Yang-Mills flow having with better properties than Struwe's weak solution on four manifolds in \cite {St2}. Finally, we apply the new parabolic version of Uhlenbeck'...
This paper considers a four-dimensional Riemannian manifold with an additional circulant tensor structure $Q$. We prove that such a manifold is almost-Einstein if the curvature tensor preserves the action of the structure on the last two arguments. Additionally, we examine an associated Riemannian product manifold $(M,...
Iva Dokuzova, G. Dzhelepov, D. Razpopov· 0 citations
We establish a Bochner formula for double forms in terms of the curvature operator of the second kind. As an application, we prove that a complete Riemannian manifold of dimension $n \ge 4$ with harmonic Weyl tensor and $\frac{3(n-1)}{4}$-nonnegative curvature operator of the second kind is either globally conformally...
In this paper, we introduce a concept of modified curvature measures for h-convex domains in hyperbolic space and prove a shifted version of the generalized Alexandrov theorem, which characterizes geodesic balls in hyperbolic space via modified curvature measures. This result generalizes the classical work of Kohlmann...
Xin-Xiu Zhang· 1 citation
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