Pinched流形上Yang-Mills场的不稳定性和孤立性

INSTABILITY AND ISOLATION OF YANGMILLS FIELDS OVER PINCHED MANIFOLDS

一个黎曼流形Ｍ称为δｐｉｎｃｈｅｄ，如果Ｍ的所有截面曲率ＫＭ满足δＫ＜ＫＭ≤Ｋ，０＜δ≤１，Ｋ＝Ｃｏｎｓｔ＞０。本文证明了：紧致单连通的０．９１Ｐｉｎｃｈｅｄ黎曼流形Ｍｎ（ｎ＞４）上任何平行的ＹａｎｇＭｉｌｓ场是不稳定的。还证明了：假设Ｍｎ是δＰｉｎｃｈｅｄ的黎曼流形，ｎ≥３，δ＜１／（ｎ－１），则点态满足‖Ｒ‖２＜ｎ（ｎ－１）（ｎ－２）２δｎ１＋δ－１２的ＹａｎｇＭｉｌｓ场Ｒ是平凡的。这些结论推广了Ｂｏｕｒｇｕｉｇｎｏｎ和Ｌａｗｓｏｎ的重要结果．

A Riemannian manifold is called δpinched if its sectional curvature is between the interval(δk,k] with constant k>0 and 0<δ≤1. It is shown that any paralell YangMills field over a compact simply connected 0.91pinched Riemannian manifold M n,n>4 is unstable. It is also shown that any YangMills field R on a compact simply connected δpinched Riemannian manifold M n n>2,that satisfies the pointwise contition ‖R ‖ 2<n(n-1)(n-2) 2δn1+δ-1 2 δ<1n-1,is trival. These generalize the results due to Bourguignon and Lawson.