Geometry And Topology

Algebraic Geometry Bucharest 1982. Proc. conf by L. Badescu, D. Popescu

By L. Badescu, D. Popescu

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D. §5. 1 in [CS]). 9) above without the term Ra. Proposition A . l For any e > 0 and any 0 < R < RM, there exists a constant 8 > 0 (6 < | ) depending on e,R and the geometry of M, such that ifO

Therefore, (N, a) > 0 or (JV, a) = 0. 2 holds. In particular, when n = 3, the author and Wan [8] proved that the assertion due to Nomizu and Smyth [26] is true without the assumption on the sectional curvature. 3 (Cheng and Wan [8]). Let Mn be an oriented complete hypersurface in E 4 with constant scalar curvature. If the mean curvature is constant, then Mn is isometric to a standard round sphere, the Euclidean three-space E 3 or the Riemannian product Sk(c) x E 3 _ f c for k = 1,2. In 1982, Yau [34] proposed to study oriented compact hypersurfaces with constant scalar curvature in E™+1 and conjectured the following Yau's conjecture .

Ls Ya{BRl(xn)} -°° ^ sv '^°° so we obtain « ~ " ( S W < c ^ t f - " • (^y)"-'2+"> • ( j ^ L ) - . f /2 • {V-logRi)n+i~0/2 = 0. for any /? > 0. D. Now we shall prove the following main theorem. Main Theorem Let P(Mn, G) be a principal bundle over a n-dimen- sional (n > 4) compact Riemannian manifold (Mn,g) with compact, simple Lie structure group G. Let At be a regular Yang-Mills flow with an initial connection AQ for any time t > 0.

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