Matches in SemOpenAlex for { <https://semopenalex.org/work/W2948657514> ?p ?o ?g. }
- W2948657514 abstract "A result (Corollary 4.3) in an article by Uhlenbeck (1985) asserts that the $W^{1,p}$-distance between the gauge-equivalence class of a connection $A$ and the moduli subspace of flat connections $M(P)$ on a principal $G$-bundle $P$ over a closed Riemannian manifold $X$ of dimension $dgeq 2$ is bounded by a constant times the $L^p$ norm of the curvature, $|F_A|_{L^p(X)}$, when $G$ is a compact Lie group, $F_A$ is $L^p$-small, and $p>d/2$. While we prove that this estimate holds when the Yang-Mills energy function on the space of Sobolev connections is Morse-Bott along the moduli subspace $M(P)$ of flat connections, it does not hold when the Yang-Mills energy function fails to be Morse-Bott, such as at the product connection in the moduli space of flat $mathrm{SU}(2)$ connections over a real two-dimensional torus. However, we prove that a useful modification of Uhlenbeck's estimate always holds provided one replaces $|F_A|_{L^p(X)}$ by a suitable power $|F_A|_{L^p(X)}^lambda$, where the positive exponent $lambda$ reflects the structure of non-regular points in $M(P)$. The proof of our refinement involves gradient flow and Morse theory for the Yang-Mills energy function on the quotient space of Sobolev connections and a Lojasiewicz distance inequality for the Yang-Mills energy function. A special case of our estimate, when $X$ has dimension four and the connection $A$ is anti-self-dual, was proved by Fukaya (1998) by entirely different methods. Lastly, we prove that if $A$ is a smooth Yang-Mills connection with small enough energy, then $A$ is necessarily flat." @default.
- W2948657514 created "2019-06-14" @default.
- W2948657514 creator A5064826043 @default.
- W2948657514 date "2019-06-10" @default.
- W2948657514 modified "2023-09-27" @default.
- W2948657514 title "Morse theory for the Yang-Mills energy function near flat connections" @default.
- W2948657514 cites W116259974 @default.
- W2948657514 cites W1479864490 @default.
- W2948657514 cites W1485995001 @default.
- W2948657514 cites W1498173400 @default.
- W2948657514 cites W1502181184 @default.
- W2948657514 cites W1509448450 @default.
- W2948657514 cites W1511103314 @default.
- W2948657514 cites W1515679419 @default.
- W2948657514 cites W1519284264 @default.
- W2948657514 cites W1522020530 @default.
- W2948657514 cites W1524106747 @default.
- W2948657514 cites W1535843651 @default.
- W2948657514 cites W1539838732 @default.
- W2948657514 cites W1541939020 @default.
- W2948657514 cites W1544585913 @default.
- W2948657514 cites W1545761024 @default.
- W2948657514 cites W1557324374 @default.
- W2948657514 cites W1559922687 @default.
- W2948657514 cites W1567771969 @default.
- W2948657514 cites W1569557141 @default.
- W2948657514 cites W1570973162 @default.
- W2948657514 cites W1573809402 @default.
- W2948657514 cites W1587685671 @default.
- W2948657514 cites W1597469717 @default.
- W2948657514 cites W1597660739 @default.
- W2948657514 cites W1602756358 @default.
- W2948657514 cites W1603757492 @default.
- W2948657514 cites W1608275713 @default.
- W2948657514 cites W1616916263 @default.
- W2948657514 cites W1626457863 @default.
- W2948657514 cites W1645828065 @default.
- W2948657514 cites W1661689961 @default.
- W2948657514 cites W1866311589 @default.
- W2948657514 cites W1968936725 @default.
- W2948657514 cites W1974849843 @default.
- W2948657514 cites W1976600010 @default.
- W2948657514 cites W1984208744 @default.
- W2948657514 cites W1987318918 @default.
- W2948657514 cites W1997148845 @default.
- W2948657514 cites W2003062603 @default.
- W2948657514 cites W2006045182 @default.
- W2948657514 cites W2010605600 @default.
- W2948657514 cites W2015082070 @default.
- W2948657514 cites W2018216665 @default.
- W2948657514 cites W2024768109 @default.
- W2948657514 cites W2030721298 @default.
- W2948657514 cites W2032317001 @default.
- W2948657514 cites W2035275755 @default.
- W2948657514 cites W2036856385 @default.
- W2948657514 cites W2041355119 @default.
- W2948657514 cites W2042684878 @default.
- W2948657514 cites W204424931 @default.
- W2948657514 cites W2047701972 @default.
- W2948657514 cites W2053441708 @default.
- W2948657514 cites W2053909528 @default.
- W2948657514 cites W2054440554 @default.
- W2948657514 cites W2054863674 @default.
- W2948657514 cites W2057924490 @default.
- W2948657514 cites W2058429718 @default.
- W2948657514 cites W2059300827 @default.
- W2948657514 cites W2066074909 @default.
- W2948657514 cites W2066260750 @default.
- W2948657514 cites W2068707449 @default.
- W2948657514 cites W2069722898 @default.
- W2948657514 cites W2075915700 @default.
- W2948657514 cites W2080803951 @default.
- W2948657514 cites W2083411489 @default.
- W2948657514 cites W2089465567 @default.
- W2948657514 cites W2092272609 @default.
- W2948657514 cites W2116163539 @default.
- W2948657514 cites W2124367079 @default.
- W2948657514 cites W2134723900 @default.
- W2948657514 cites W2135557790 @default.
- W2948657514 cites W2136226659 @default.
- W2948657514 cites W2156105801 @default.
- W2948657514 cites W2167675215 @default.
- W2948657514 cites W2203786939 @default.
- W2948657514 cites W2292127841 @default.
- W2948657514 cites W2314653975 @default.
- W2948657514 cites W2329975073 @default.
- W2948657514 cites W2332294454 @default.
- W2948657514 cites W238131626 @default.
- W2948657514 cites W2392223763 @default.
- W2948657514 cites W2497367977 @default.
- W2948657514 cites W2500739438 @default.
- W2948657514 cites W2521318420 @default.
- W2948657514 cites W2530549011 @default.
- W2948657514 cites W2576873271 @default.
- W2948657514 cites W2605449853 @default.
- W2948657514 cites W2726184532 @default.
- W2948657514 cites W2735221503 @default.
- W2948657514 cites W2899917663 @default.
- W2948657514 cites W2909984845 @default.
- W2948657514 cites W2922435852 @default.