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Register a new userHigher order Yang-Mills flow
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Authors
Casey Lynn Kelleher
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We define a family of functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for the critical dimensions. Consequently, we have an alternate proof of the convergence of Yang-Mills flow in dimensions 2 and 3 given by Rade and the bubbling criterion in dimension 4 of Struwe in the case where the initial flow data is smooth.
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Following work of Colding-Minicozzi, we define a notion of entropy for connections over $mathbb R^n$ which has shrinking Yang-Mills solitons as critical points. As in Colding-Minicozzi, this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stability in terms of the spectrum of a certain linear operator associated to the soliton. This leads furthermore to a gap theorem for solitons. These results point to a broader strategy of studying generic singularities of Yang-Mills flow, and we discuss the differences in this strategy in dimension $n=4$ versus $n geq 5$.
We study a new deformed Hermitian Yang-Mills Flow in the supercritical case. Under the same assumption on the subsolution as Collins-Jacob-Yau cite{cjy2020cjm}, we show the longtime existence and the solution converges to a solution of the deformed Hermitian Yang-Mills equation which was solved by Collins-Jacob-Yau cite{cjy2020cjm} by the continuity method.
We study singularity structure of Yang-Mills flow in dimensions $n geq 4$. First we obtain a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow, which leads to a stratification of the singular set. By a refined blowup analysis we obtain Yang-Mills connections or solitons as blowup limits at any point in the singular set.
We give a complete description of the global existence and convergence for the Ricci-Yang-Mills flow on $T^k$ bundles over Riemann surfaces. These results equivalently describe solutions to generalized Ricci flow and pluriclosed flow with symmetry.
We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.
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