【摘 要】
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In this paper,we consider the problem of the nonnegative scalar curvature(NNSC)-cobordism of Bartnik data(Σn-11,γ1,Hi)and(Σn-12,γ2,H2).We prove that given two metrics γ1 and γ2 on Sn-1(3≤n≤7)with H1 fixed,then(Sn-1,γ1,H1)and(Sn-1,γ2,H2)admit no NNSC-cobor
【机 构】
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Key Laboratory of Pure and Applied Mathematics,School of Mathematical Sciences,Peking University,Bei
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In this paper,we consider the problem of the nonnegative scalar curvature(NNSC)-cobordism of Bartnik data(Σn-11,γ1,Hi)and(Σn-12,γ2,H2).We prove that given two metrics γ1 and γ2 on Sn-1(3≤n≤7)with H1 fixed,then(Sn-1,γ1,H1)and(Sn-1,γ2,H2)admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough(see Theorem 1.3).Moreover,we show that for n=3,a much weaker condition that the total mean curvature ∫S2 H2dμγ2 is large enough rules out NNSC-cobordisms(see Theorem 1.2);if we require the Gaussian curvature of γ2 to be positive,we get a criterion for nonexistence of the trivial NNSC-cobordism by using the Hawking mass and the Brown-York mass(see Theorem 1.1).For the general topology case,we prove that(Σn-1,γ1,0)and(Σn-12,γ2,H2)admit no NNSC-cobordism provided the prescribed mean curvature H2 is large enough(see Theorem 1.5).
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